1. science
2. mathematics
3. geometry

Arithmetic of curves
Daniel Miller ∗
fall 2013
Contents
Disclaimer
3
Notation and conventions
3
Motivation: plane curves
4
1
2
3
Jacobians and abelian varieties
1.1 Jacobians over C . . . . . . . . . . . . .
1.2 Abelian varieties over arbitrary fields
1.3 Albanese varieties . . . . . . . . . . . .
1.4 Picard schemes . . . . . . . . . . . . .
1.5 Recovering a curve from its jacobian .
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6
7
9
9
10
12
The Mordell-Weil theorem
2.1 Statement and generalizations . . . . . . .
2.2 Plan of the proof . . . . . . . . . . . . . . .
2.3 Group cohomology . . . . . . . . . . . . .
2.4 Selmer groups and weak Mordell-Weil . .
2.5 Crash course in algebraic number theory
2.6 Reduction of abelian varieties . . . . . . .
2.7 Restricted ramification . . . . . . . . . . .
2.8 Torsion and weak Mordell-Weil . . . . . .
2.9 Tate-Shafarevich groups . . . . . . . . . .
2.10 Weil heights . . . . . . . . . . . . . . . . .
2.11 N´eron-Tate heights . . . . . . . . . . . . .
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14
14
15
17
20
21
24
26
27
29
31
33
Curves and abelian varieties over finite fields
3.1 Tate modules . . . . . . . . . . . . . . . . .
3.2 Endomorphisms of abelian varieties . . .
3.3 Abelian varieties over finite fields . . . . .
3.4 Characteristic polynomial of Frobenius .
3.5 Zeta functions . . . . . . . . . . . . . . . .
3.6 Honda-Tate theory . . . . . . . . . . . . .
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37
37
39
43
43
46
47
∗ based
on a course taught by David Zywina at Cornell
1
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3.7
3.8
3.9
3.10
4
5
Curves and their jacobians . . . .
The Weil conjectures . . . . . . .
Generalizing the Weil conjectures
Computing zeta functions . . . .
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49
51
52
53
Birch and Swinnerton-Dyer conjecture
4.1 L-functions of elliptic curves . . . . . . . . . . . . .
4.2 Conductors . . . . . . . . . . . . . . . . . . . . . . .
4.3 Modularity . . . . . . . . . . . . . . . . . . . . . . .
4.4 Small analytic rank . . . . . . . . . . . . . . . . . .
4.5 The strong Birch and Swinnerton-Dyer conjecture
4.6 Predicting the order of X . . . . . . . . . . . . . .
4.7 Average orders of Selmer groups . . . . . . . . . .
4.8 The congruent number problem . . . . . . . . . . .
4.9 The Sato-Tate conjecture . . . . . . . . . . . . . . .
4.10 Some computations . . . . . . . . . . . . . . . . . .
4.11 The Sato-Tate conjecture and Haar measures . . .
4.12 Motives and the refined Sato-Tate conjecture . . .
4.13 The Bloch-Kato conjecture . . . . . . . . . . . . . .
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54
55
57
59
61
62
63
64
65
67
69
70
71
73
Some theorems of Faltings
5.1 Background and Tate’s conjecture . . .
5.2 Image of Frobenius for number fields
5.3 L-function of an abelian variety . . . .
5.4 Tate conjectures and isogenies . . . . .
5.5 Finiteness theorems . . . . . . . . . . .
5.6 Proof of the Mordell conjecture . . . .
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73
73
75
77
78
79
81
A Brief introduction to e´ tale cohomology
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82
2
Disclaimer
These notes originated in a course on the arithmetic of curves taught by David Zywina at
Cornell. However, a significant amount of material (including, no doubt, many errors) has
been added since then, so they are far from an exact reflection of what he covered in class.
Moreover, the notation has been changed in many places (sometimes significantly) as has
the order in which material is covered. Most significantly, the tone of these notes differs
drastically from the perspective Zywina took in class, with the notes being much more cohomological and scheme-theoretic than Zywina’s generally elementary (and pedagogically
correct) approach. Any errors in these notes are entirely my fault.
The original computations for the class were done using the commercial computer
algebra system Magma. For these notes, everything has been reworked into Sage, an opensource alternative designed for for number theorists. Nearly all of the Sage code used is
contained (using the LATEX package sagetex) in the source code for this document, which
may be found at https://github.com/dkmiller/7390.
Todo Clarify definition of local factors in the L-function of an abelian variety. This should
involve a clarification of the relationship between the “geometric Frobenius” on a variety
over a finite field, and the “arithmetic Frobenius” as an element of an absolute Galois group.
Notation and conventions
We write N, Z, Q, . . . for the natural numbers, integers, . . . .
A nice variety over a field k is a smooth, projective, geometrically integral variety over k.
¯ denote its separable (resp. algebraic)closure. We will write
If k is a field, ks (resp. k)
s
Gk = Gal(k /k) for the absolute Galois group of k.
Let X be a scheme over a finite field Fq . The Frobenius of X is the morphism Φ X : X → X
that is the identity on the underlying topological space, and x 7→ x q on the structure sheaf.
The Frobenius in GFq = Gal(Fq /Fq ) is the arithmetic Frobenius, induced by x 7→ x q on Fq .
We will write Frobq for the arithmetic Frobenius.
If v is a finite place of a global field k, then k v denotes the completion of k with respect to
v. Most of our notation here is standard, with the exception of κv = ov /pv for the residue
field of v.
If k is a global field and v is a finite place, Frobv denotes the arithmetic Frobenius
associated to v. We will sometimes take Frobv to be a single (non-canonical) element of Gk ,
or the elements’ entire conjugacy class. This should not cause any confusion.
The symbol H• is used to denote cohomology. Unless decorated by a descriptive
subscript, it should be taken to mean e´ tale cohomology.
If E is a module over a ring A and x ∈ E, then we will write E/x instead of E/( A · x ). In
particular, for a ∈ A, A/a is the quotient of A by the ideal generated by A.
All abelian groups are tacitly taken to modules over Z, so the previous convention
applies. Even if an abelian group G is written multiplicatively, G/n will be used to denote
G/( G n ).
3
Motivation: plane curves
Fir a non-constant polynomial f ( x, y) ∈ Q[ x, y]. Assume f is geometrically irreducible,
that is, irreducible in the ring Q[ x, y]. We can define the curve C over Q determined by the
equation f ( x, y) = 0. For now, we will think of C in terms of its functor of points. To any
Q-algebra A, we set C ( A) = {( a, b) ∈ A2 : f ( a, b) = 0}. As a scheme, C = Spec (Q[ x, y]/ f ).
Some big questions are:
1. Is C (Q) = ∅?
2. Is C (Q) finite?
3. Can we compute C (Q)?
None of these questions can be answered in full generality.
0.0.1 Example. Let f = x2 + y2 − 1, i.e. C is the circle. It is well-known that
C (Q) =
1 − t2
2t
,
2
1 + t 1 + t2
: t ∈ Q ∪ {(−1, 0)}
This can be proved in the usual manner by choosing the point (−1, 0) in C, and then drawing
lines with rational slopes through (−1, 0). As a Riemann surface, C (C) is a sphere with two
points removed.
0.0.2 Example. Let f = x2 + y2 + 1. Then C (R) = ∅, hence C (Q) = ∅. But as a Riemann
surface, C (C) is the same sphere with two points removed. Thus the geometry of C does
not necessarily determine C (Q).
0.0.3 Example. Let f = x4 + y4 − 1. Then it is a theorem of Fermat that C (Q) = {(±1, 0), (0, ±1)}.
It is a (much harder) theorem of Wiles that if Cn is the curve given by f = x n + yn − 1 for
n > 3, then Cn (Q) = {(±1, 0), (0, ±1)}. In fact, this is the celebrated “Fermat’s last theorem,”
proved in [Wil95].
0.0.4 Example (Stoll). Consider
C : y2 = 82342800x6 − 470135160x5 + 52485681x4 + 2396040466x3
+ 567207969x2 − 985905640x + 247747600.
This is a curve of genus 2, i.e. C (C) is a punctured two-holed torus. It turns out that
#C (Q) > 642 [Sto09, §6]. However, by a theorem of Faltings, #C (Q) is finite. This example
exhibits the largest known number of rational points of a genus two curve over Q. If we
take C : y2 = f ( x ) with f ∈ Q[ x ] a “random” sextic polynomial, then the expectation is that
C (Q) = ∅.
It is natural to ask whether for each g > 2 there exists a number Bg such that whenever a
curve C over Q has genus g, we have #C (Q) 6 Bg . This is not even known for g = 2.
0.0.5 Example. Let C : y2 = x3 + 875x. Note that C (Q) has the obvious point (0, 0), and
one can do a bit of work to show that this is the only point.
4
0.0.6 Example. Let C : y2 = x3 + 877x. Then C (Q) once again contains (0, 0). A computer
search showed that (0, 0) is the only point on C of height 6 1000000. For now, the height of
a solution is just the largest absolute value of the numerator / denominator of a solution
written in reduced fractions. But other methods lead us to expect many more solutions
(infinitely many, in fact). Let E be the projective curve over Q obtained by adjoining a point
O to C. As a Riemann surface, E(C) is just a torus.
We can give E the structure of an abelian variety over Q. That is, we can give E the
structure of a commutative algebraic group (the multiplication operation m : E × E → E
is given by regular functions), with O being the identity. Later, we will think of E as
being identified with its jacobian Jac( E) via the choice of a single point O ∈ E(Q). Since
E is a commutative algebraic group, E(Q) is an abelian group (with identity O). It is a
theorem of Mordell that E(Q) is finitely generated. We know the structure of such groups:
E(Q) ' A × Zr , where A is finite, and r = rk E is the (algebraic) rank of E.
In general, the group A is (conjecturally) computable. In our case, A = Z/2. It is much
harder to compute the algebraic rank of E. The Birch and Swinnerton-Dyer conjecture says
that r agrees with the order of vanishing r 0 of a certain holomorphic function L( E, s) at s = 1.
Sometimes, r 0 can be computed. In our example, a computation shows that r 0 = 1, so we
should expect E(Q) ' Z/2 × Z. In particular, E(Q) should be infinite. One can show (using
other methods) that E(Q) = h(0, 0), ( x0 , y0 )i, where
x0 =
375494528127162193105504069942092792346201
6215987776871505425463220780697238044100
For details, see [BC84]. One method to construct such large solutions for a rank one elliptic
curve uses Heegner points.
In general, we will take a curve C over Q, consider its jacobian J, and study J (Q). This
will be a group, and its structure heavily influences C (Q). For example, we could study
how the rank of J (Q) affects C (Q). In the case that C = E is an elliptic curve, Jac( E) = E,
so studying the curve and studying its jacobian is the same thing. The “average rank of an
elliptic curve” is not known, nor is there a general consensus on what is should be. Some
expect the rank of a random curve to be 0 or 1, both with probability 12 . Others suppose that
elliptic curves over Q have rank 2 with nonzero probability as well. It was proven recently
(see [BS10a, §1]) that
7
1
lim sup 2
∑ rk(Ea,b ) 6 6 ,
4B
B→∞
| a|,|b|6B
4a3 +27b2 6=0
where Ea,b is the elliptic curve over Q defined by y2 = x3 + ax + b.
It is natural to ask whether there is a global bound for rk E(Q) as E ranges over all
elliptic curves over Q. It is known that there are curves with rank at least 28, but their exact
ranks are not known [Duj]. The largest known rank is 19.
Our assumption that f ( x, y) is irreducible is not a serious one. For example, if f =
y2 − x2 , then we can factor f as ( x + y)( x − y), and then treat the solutions to x + y = 0
and x − y = 0 separately. Another example is f = x2 + y2 , which only factors over Q(i ) as
(y + ix )(y − ix ), and we the rational points lie in the intersection of the two components
over Q(i ). In general, a curve over Q will be the union of finitely many geometrically
irreducible components, each of which is defined over some finite extension of Q. Also, the
assumption that f ∈ Q[ x, y] is not serious, as every curve is birational to a plane curve.
5
Let C be a curve over Q. Then C (C) r {singular points} is a compact Riemann surface
with points removed, i.e. it is a torus with g handles with finitely many points removed.
Call this g the genus of C.
0.0.7 Theorem (Faltings, conjectured by Mordell). If C is a curve over Q with g > 2, then C (Q)
is finite.
For curves of genus g < 2, #C (Q) can be infinite. In Faltings’ theorem, Q can be replaced
by any field finitely generated over Q.
Now let C be a smooth projective curve of genus g over F p . We are interested in #C (F pn ),
which is obviously computable for each n. Define the zeta function of C to be the formal
power series
!
tn
Z (C, t) = exp ∑ #C (F pn )
.
n
n>1
0.0.8 Theorem (Weil). If C is a smooth projective curve of genus g over F p , then
Z (C, t) =
P(C, t)
,
(1 − t)(1 − pt)
2g
where P(C, t) ∈ Z[t] has degree 2g. Moreover, if we write P(C, t) = ∏i=1 (1 − αi t), then for each i,
we have |αi | = p1/2 .
The second statement in the theorem is called the Riemann hypothesis for C. It can be used
to obtain explicit bounds on the size of C (F pn ) as n → ∞. For example, we can compute
tn
∑ #C(F pn ) n
= log Z (C, t)
n >0
= − log(1 − t) − log(1 − pt) + ∑ log(1 − αt t)
i
=
∑
n >0
2g
pn + 1 − ∑ αin
i =1
!
tn
n
.
2g
Therefore, #C (F pn ) = pn + 1 − ∑i=1 αin . It follows easily that |#C (F pn ) − ( pn + 1)| 6 2gpn/2 .
This is equivalent to saying
pn − 2gpn/2 + 1 6 #C (F pn ) 6 pn + 2gpn/2 + 1.
In particular, setting n = g = 1, we obtain #C (F p ) > ( p1/2 − 1)2 > 0, hence C (F p ) 6= ∅.
1
Jacobians and abelian varieties
At first, we will work over C and treat curves as Riemann surfaces. By [Har77, I.6.12] and
[Jos06, 5.8.7], the category of smooth projective curves over C is equivalent to the category
of compact connected Riemann surfaces, there is no loss of information here. Let’s start
with some general definitions.
6
1.0.9 Definition. A variety over a field k is a separated scheme of finite type over Spec(k ). We call
a variety X over k nice if it is smooth, projective, and geometrically integral.
Recall that X/k is geometrically integral if Xk¯ = X ×k Spec(k¯ ) is integral. A curve is a
variety of dimension one. If we are interested in C (k) for general curves, it is sufficient to
consider nice curves. If X is a variety, we can consider its functor of points h X : Algk → Set
which assigns to a k-algebra A the set X ( A) of “A-valued points.” This extends to a functor
h X : Sch◦k → Set which is defined by h X (Y ) = homk (Y, X ).
Earlier, for f ∈ Q[ x, y] and C defined by the equation f = 0, we defined C ( A) =
{( a, b) ∈ A2 : f ( a, b) = 0} for any Q-algebra A. Note that
C ( A) = {( a, b) ∈ A2 : f ( a, b) = 0}
= homAlgQ (Q[ x, y]/ f , A)
= homSchk (Spec A, Spec(Q[ x, y]/ f )) ,
so this agrees with our general definition. By the Yoneda Lemma, the functor h X determines
X up to isomorphism.
1.1
Jacobians over C
For the rest of this section, let C be a nice curve over C. Set X = C (C); this is a compact
connected Riemann surface. So topologically, X is a many-handled torus. Let Λ = H1 ( X, Z),
the first singular homology group of X, which can be identified with π1 ( X )ab . We will
regard elements of Λ are equivalence classes [γ] for γ ∈ π1 ( X ).
It is a theorem of algebraic topology that Λ ' Z2g for some g > 0; we will call g the
genus of X (and also of C). Let K be the field of meromorphic functions on X. For f ∈ K,
at any point x ∈ X, we can write f in local coordinates as zn ( a0 + a1 z + · · · ) where a0 6= 0
and n ∈ Z. We call n = ordx ( f ) the order of vanishing of f at x. The degree of f is the sum
deg( f ) = ∑ x∈X ordx ( f ). If f is holomorphic instead of just meromorphic, then deg( f ) = 0
because f is constant, but the converse fails.
We can identify K with the function field of C, i.e. the set of rational maps C → A1 .
Certainly rational maps yield meromorphic functions, and it is a basic theorem of Riemann
surface theory that meromorphic functions are in fact algebraic. Moreover, if C is nice, then
C can be recovered from K. To do this, pick some x ∈ K r C. If we let A be the integral
closure of C[ x ] in K, then Spec( A) will be a smooth affine curve. Pick some embedding of
Spec( A) into projective space; the closure of its image will be a projective curve C 0 (possibly
with singularities) with function field K. We can resolve the singularities of C 0 to obtain
a smooth projective curve C 00 with function field K. By [Har77, I.6.12], this induces an
anti-equivalence between the category of extensions K/C of transcendence degree one and
the category of nice curves over C with surjective morphisms.
One might ask whether the singular homology H1 ( X, Z) can be defined “algebraically.”
Essentially, the answer is no – that is, there is no known algebraic definition for H1 ( X, Z)
that gives the “right” answers. On the other hand, H1 ( X, Q) is naturally isomorphic to
the dual of the algebraic de Rham cohomology H1dR ( X/Q), and H1 ( X, Z` ) is naturally
isomorphic to the dual of the `-adic cohomology H1et ( X, Z` ). Both of these isomorphisms
are hard theorems – the first due to Grothendieck [Gro66], the second originally due to
Artin, and proved in [Del77, I 4.6.3].
7
With C as before, let V = Ω1 ( X ) = H0 ( X, Ω1 ) = H1dR ( X ) be the first analytic de
Rham cohomology of C. This is a complex vector space of dimension g, so we get a nontopological definition of g. We can consider Ω1 as the sheaf of (algebraic) differentials, and
g = dimC H0 ( X, Ω1 ), giving us a purely algebraic definition of g. There is a natural pairing
H1 ( X, Z) ⊗ H1dR ( X ) → C, defined by
[γ] ⊗ ω 7→
Z
ω
γ
This pairing is nondegenerate and C-linear in the second component, It induces an injection
Φ : Λ ,→ V ∨ .
1.1.1 Definition (analytic). The Jacobian of X is Jac( X ) = V ∨ /Φ(Λ).
Is is known that Φ(Λ) is a lattice in V ∨ , i.e. it is discrete and V ∨ /Φ(Λ) is compact. There
is theRAbel-Jacobi map j : X → Jac( X ) defined as follows. Fix x0 ∈ X; we send x ∈ X to
ω 7→ [ x ,x] ω, where [ x0 , x ] denotes some path from x0 to x. A different choice of [ x0 , x ] will
0
differ by a closed loop, i.e. an element of H1 ( X, Z). So j : X → Jac X is well-defined. Note
that Jac X is a compact complex Lie group.
After choosing a basis for V ∨ , we have Jac X ' C g /L, where L ' Z2g . As a real Lie
group, Jac X is isomorphic to (S1 )2g . We care about Jac X because, despite its analytic
definition, it is in fact a projective variety.
1.1.2 Theorem. For X a compact Riemann surface, Jac X is algebraic, i.e. there exists a variety
J defined over C such that J (C) ' Jac X as complex manifolds. Moreover, the group operation
on Jac X is algebraic, i.e. there is a morphism m : J ×C J → J such that J (C) × J (C) → J (C)
corresponds to the addition law on Jac X.
Proof. See [Mila, I.18]. Essentially, X is the analytification of a curve C, and one proves that
Jac X (defined analytically) is isomorphic as a complex manifold to the analytification of
Jac C (defined algebraically).
Let Div X be the free abelian group generated by the points of X. There is a map
deg : Div X → Z, defined by ∑ n x · x 7→ ∑ n x . We define Div◦ X by the short exact sequence
0
/ Div◦ X
/ Div X
/Z
/ 0.
There is also a map div : K × → Div X, where div( f ) = ∑ x ordx ( f ) · x. It is a basic fact that
deg(div( f )) = 0, so we can define the Picard group of X to be Pic X = Div X/ div(K × ) and
Pic◦ X = Div◦ ( X )/ div(K × ).
Let M be the sheaf of meromorphic functions on X. It is a basic fact that Div( X ) =
H0 ( X, M × /O × ) and Pic( X ) = H1 ( X, O × ). Indeed, the first equality is often taken to
ˇ
be a definition as in [Har77, III.6], and the second is a straightforward exercise in Cech
◦ 1
cohomology. An example where the Picard group is easily determined is Pic (P ) = 0.
The Abel-Jacobi map j : X → Jac X extends naturally to a homomorphism j : Div◦ X →
Jac X.
1.1.3 Theorem (Jacobi). The map j : Div◦ X → Jac X is surjective.
1.1.4 Theorem (Abel). The kernel of j : Div◦ X → Jac X is div(K × ).
8
∼
It follows that j induces an isomorphism j : Pic◦ X −
→ Jac X. Note that Pic◦ X parameterizes invertible sheaves (line bundles) on X of degree zero.
Note that in general, C g /L for some lattice L need not be algebraic if g > 1. In the future,
we’ll try to define Jac C for a curve C over any field. The variety Jac C will be a nice variety,
i.e. smooth, projective and geometrically integral. We will use this to give an “algebraic
definition of H1 ( X, Z/n).”
1.2
Abelian varieties over arbitrary fields
Recall that a variety X/k is nice if it is smooth, projective, and geometrically connected.
1.2.1 Definition. Let k be a field. An abelian variety over k is a nice group variety over k.
In other words, there are morphisms m : A × A → A, i : A → A, e : Spec(k) → A such
that the induced maps m∗ : h A × h A → h A etc. turn h A into a group-valued functor. In
particular, A( X ) is an “honest group” for each k-scheme X.
1.2.2 Example. The general linear group GL(n) is a group variety, but not nice (at least, not
in the technical sense) because it is not projective. More generally, no linear algebraic group
is an abelian variety, for the same reason.
Note that abelian varieties are not required to be commutative, but this is in fact the
case. This is easy to see over C. If A/C is an abelian variety, then G = A(C) is a compact
connected complex Lie group. Let g be its Lie algebra. Consider the composite map
ad
f : G −→ GL(g) ,→ End(g), where ad : G → GL(g) is the adjoint representation. After
picking a basis for End(g), the components of f are entire holomorphic functions on a
compact complex manifold, hence locally constant. Since G is connected, f is constant, i.e.
the adjoint representation of G is trivial. But ker(ad) = Z ( G ), so G is commutative. The
case A/k for arbitrary k of characteristic zero follows from the Lefschetz principle, or one
can just prove commutativity directly using a “rigidity principle” for maps on projective
varieties [Mila, I.1.4].
1.3
Albanese varieties
We’d like to describe the jacobian J of a nice curve C/k with C (k) 6= ∅. It will be an abelian
variety over k of dimension g, the genus of C. So far we’ve only defined the genus of a curve
C/k with k ⊂ C. For an arbitrary field k and a curve C/k, set g(C ) = h0 (Ck¯ , Ω1 ) = h1 (OC ).
In general, if F is some sheaf on a scheme X over k, we write hi ( X, F ) or hi (F ) for
dimk Hi ( X, F ).
1.3.1 Definition (Albanese). Let C/k be a curve with fixed x0 ∈ C (k ). The jacobian of C is an
abelian variety J = Jac C with a morphism j : C → J taking x0 to 0, such that for any morphism
f : C → A to an abelian variety A with f ( x0 ) = 0, there is a unique f˜ : J → A making the
following diagram commute:
j
C
/J
A
f
9
f˜
Since ( J, j) is the solution of a universal problem, it is unique up to unique isomorphism.
Our definition can be made much more concise. Let AbVk be the category of abelian varieties
over k, and let Vark,∗ be the category of “nice pointed varieties” over k, i.e. nice varieties
X/k with chosen x ∈ X (k). Forgetting the group structure gives an inclusion functor
ι : AbVk → Vark . Our definition of Jac C can be rephrased as saying that j : C → Jac C
induces a natural isomorphism
homVark,∗ (C, ιA) ' homAbVark (Jac C, A).
It turns out that for any nice pointed variety X, there is an abelian variety A = Alb X,
the Albanese variety of X, with a morphism j : X → Alb X that induces a similar natural
isomorphism (with Alb X in place of Jac C). So taking jacobian may be seen as the left-adjoint
to the forgetful functor from abelian varieties to pointed varieties. For a proof that Alb X
exists, see [Moc12, A.11].
In general, the map X → Alb X need not be an embedding. For example, if C is a curve
of genus 0, then Alb X = 0 by [Mila, I.3.9]. On the other hand, if the genus g > 1, then C → J
is an embedding. The map C (k¯ ) → Pic◦ (Ck¯ ) sends a point x to the divisor [ x ] − [ x0 ]. If
[ x ] − [ x0 ] = [y] − [ x0 ] in Pic◦ (Ck¯ ), then [ x ] − [y] = div( f ) for some rational map f : C → P1 .
If x 6= y, then f would have a unique simple zero and poll, which would imply that f is
birational. But this is impossible, so x 6= y implies j( x ) 6= j(y).
If C (k) = ∅, we can still define J = Jac C. It will be a k-variety with a morphism
j : C × C → J such that j(∆) = 0. We require that for any abelian variety A over k with
f : C × C → A such that f (∆) = 0, there is a unique lift f˜ : J → A of f . The map j should
be thought of “( x, y) 7→ j( x ) − j(y),” even though an embedding C → J may not be defined
over k.
1.4
Picard schemes
Another approach to defining J = Jac C involves the Picard group. Recall that over C, we
proved that J (C) ' Pic◦ (C ). One might hope that J satisfies J ( L) = Pic◦ (CL ) for all field
extensions L/k. This works if C (k ) 6= ∅, but not otherwise. For, if J (ks ) = Pic◦ (Cks ), then
we would have Pic◦ (C ) = J (k) = J (ks )Gk = Pic◦ (Cks )Gk . But this does not always hold.
For a curve C over k, we will define an abelian variety Pic◦ (C ) in terms of its functor of
points. To do this, we need to define Picard groups for arbitrary schemes. For any scheme
ˇ
X, the Picard group of X is Pic( X ) = H1 ( X, OX× ). It is straightforward to show (using Cech
cohomology) that Pic( X ) is isomorphic to the group of isomorphism classes of invertible
sheaves on X, with group operation induced by tensor product, i.e. [L ] + [L 0 ] = [L ⊗ L 0 ].
Let C be a nice curve. If D = ∑ Dx · x is a divisor on C, the degree of D is deg D = ∑ Dx ∈
Z. Since C is smooth, Cartier divisors are Weil divisors, so deg induces a well-defined map
Pic(C ) → Z. For T an arbitrary scheme, define Pic◦ (C × T ) to be the subset of Pic(C × T )
consisting of invertible sheaves L with deg(Lt ) = 0 for all t ∈ T. That is, the following
sequence is exact
0
/ Pic◦ (C × T )
/ Pic(C × T )
/
∏ Z,
t∈ T
◦
where the last map is c 7→ (deg(ct ))t∈T . Now we define the functor PicC
: Sch◦k → Ab that
◦
sends T to Pic (C ×k T )/ Pic( T ).
10
◦
is represented by J = Jac C.
1.4.1 Theorem. If C (k) 6= ∅, then PicC
Proof. See [Mila, III.1.2].
0
In general, we might have C (k) = ∅. We will have J ( L) = Pic◦ (CL0 )Gal( L /L) where
is any separable extension with C ( L0 ) 6= ∅. Let J = Jac C and j : C → J be the
standard embedding. Let C g = C × · · · × C (g-fold product), and consider the map f :
C g → J, f ( x1 , . . . , x g ) = j( x1 ) + · · · + j( x g ). The symmetric group Sg acts on C g , and f is
Sg -equivariant. The quotient Symg C = C g /Sg exists, and has a map Symg C → J. This is
birational. Weil defined a “rational group law” on Symg C using the Riemann-Roch theorem,
and then showed that this induces an “honest group law” on a nice variety birational to
Symg C. For more details on Weil’s construction (and proofs), see [Mila, III.7].
Now suppose X is an arbitrary scheme. Recall that Pic( X ) = H1 ( X, OX× ); this classifies
invertible sheaves on X, where the group operation on sheaves is ⊗. If X is integral, Pic( X )
is isomorphic to the class group Cl( X ) of Cartier divisors (Weil divisors if X is a nice curve).
This is easy to prove. Recall that if M is the sheaf of rational functions on X, then the group
of Cartier divisors is Div( X ) = H0 ( X, M × /O × ). The short exact sequence
L0 /L
1 → O × → M × → M × /O × → 1
induces a long exact sequence in sheaf cohomology:
0 → H0 (O × ) → H0 (M × ) → Div( X ) → Pic( X ) → H1 (M × ) → · · ·
If X is integral, the sheaf M × is flasque, so H1 (M × ) = 0. It follows that Pic( X ) =
Div( X )/ H0 (M × ) = Cl( X ).
For X/k an arbitrary scheme, consider the functor PicX : Sch◦k → Ab given by PicX ( T ) =
Pic( X ×k T )/ Pic( T ). This is not in general representable. However, if X is a nice k-variety,
then the fppf-sheafification of PicX is representable [Kle05, 4.1.38]. Even better, if X (k) 6= ∅,
then PicX is representable [Kle05, 2.5]. We will also denote the representing scheme by
PicX , and we call PicX the Picard scheme of X. It is not a variety, but it is a disjoint union of
ind-varieties [Kle05, 4.8]. More precisely, choose a very ample line bundle L on X. If F is
any coherent sheaf on X, write F (n) = F ⊗ L ⊗n . Recall that the Euler characteristic of F is
χ(F ) = ∑(−1)i hi (F ). By [Gro61, 2.5.3], there is a (unique) polynomial φ ∈ Q[t] such that
χ (F (n)) = φ(n) for all n ∈ Z; set hL (F ) = φ. We call φ the Hilbert polynomial of F . The
Hilbert polynomial hL (F ) does depend on L .
If X (k) 6= ∅, we can define for x ∈ X (k) the modified Picard functor
∼
PicX,x ( T ) = {(L , i ) : L ∈ Pic( XT ) , i : x T∗ L −
→ OT }/ ∼ .
There is an obvious map Pic( XT ) → PicX/x ( T ) given by L 7→ L ⊗ ( x ◦ f )∗T L −1 , where
f : X → Spec(k) denotes the structure morphism. It is a good exercise to prove that this
∼
induces an isomorphism PicX −
→ PicX,x .
φ
Denote by PicX the functor which assigns to a scheme T the subset of PicX ( T ) consisting
φ
of invertible sheaves F on X × T with hL (Ft ) = φ for all t ∈ T. By [Kle05, 6.20], PicX
φ
represents a clopen subscheme of PicX , and PicX is covered by the PicX . Moreover, the
φ
PicX are varieties. We can do even better. If we let PicdX send T to the subset of PicX ( T )
consisting of invertible sheaves F with deg hL (Ft ) = d for all t ∈ T, then the PicdX form
11
a cover of PicX by clopen subvarieties. Just as the genus of a curve is the dimension of
its jacobian, there is a natural isomorphism H1 (OX ) ' Lie(PicX ), from which we deduce
dim(PicX ) = h1 (OX ) when X is nice [Kle05, 5.11].
Unlike the case when X is a curve, it is not always true that PicX (k¯ )/ Pic◦X (k¯ ) = Z. In
general, we set NS( X ) = PicX (k¯ )/ Pic◦X (k¯ ), and call NS( X ) the N´eron-Severi group of X.
Suppose X = A is already an abelian variety over k. Then we have
Pic◦A (k¯ ) = c ∈ Pic( Ak¯ ) : t∗a c = c for all a ∈ A(k¯ )
where t a : Ak¯ → Ak¯ is translation by a. See [Mila, I.8.4] for a partial proof.
For an abelian variety A over k, the dual of A is defined to be A∨ = Pic◦A . Each c ∈ Pic( A)
gives a map ϕc : A → A∨ . At the level of points, it is defined as a 7→ t∗a c − c, where if
c = [L ], the class t∗a c − c ∈ A∨ (k¯ ) = Pic◦ ( A) is represented by [t∗a L ⊗ L −1 ]. It turns out
that A∨∨ ' A, so calling A∨ the dual of A is rather natural. The map ϕc : A → A∨ is a
homomorphism of abelian varieties. If c is ample (i.e. the map from A to some projective
space induced by n · c for n 0 is an embedding) then ϕc is an isogeny, where
1.4.2 Definition. A homomorphism ϕ : A → B is an isogeny if it is surjective with finite kernel.
It is not at all obvious, but “A is isogenous to B” is an equivalence relation on abelian
varieties. The relation is clearly reflexive and transitive. To see that it is symmetric, suppose
we have an isogeny ϕ : A → B. For any ample c ∈ Pic( A∨ ), d ∈ Pic( B), the composite
B
ϕd
/ B∨
ϕ∨
/ A∨
ϕc
/ A∨∨
∼
/A
is an isogeny.
1.4.3 Definition. Let A be an abelian variety. A polarization of A is an isogeny of the form
ϕc : A → A∨ for some ample c ∈ Pic( A).
The duality theory of abelian varieties is very rich. A good place to start is Chapter VII
of [vdGM].
1.5
Recovering a curve from its jacobian
Let k be a field, C/k a nice curve, and J = Jac C its jacobian. What does (the arithmetic of)
J tell us about (the arithmetic of) C? In particular, can we recover C from J? In general,
J does not determine C. For example, if g = g(C ) = 0, then J = 0. However, there are
(non-algebraically closed) fields k for which there are nice curves C over k with g(C ) = 0
(hence Jac C = 0 = Jac P1 ), but C 6' P1k . There are more difficult examples with genus g > 0.
Suppose we add some data. Assume g > 2 and C (k) 6= ∅. This gives us a map j : C → J
determined by x 7→ 0 for some distinguished x ∈ C (k). Consider θ = j(C ) + · · · + j(C ),
where there are g − 1 terms in the sum. It turns out that θ is an irreducible ample divisor of
J. Thus θ induces a polarization ϕθ : J → J ∨ .
1.5.1 Theorem (Torelli). If C, C 0 are nice curves over a field k with ( J, θ ) ' ( J 0 , θ 0 ), then C ' C 0
over k.
Proof. See [Mila, III.13] for a rather unenlightening proof.
12
It is known that ϕθ : J → J ∨ is actually an isomorphism. We call a polarization A → A∨
principal if it is an isomorphism. The famous Schottky problem asks what pairs ( A, c), with c
inducing a principal polarization, come from jacobians.
This question has an easy partial answer if we are willing to use heavy machinery.
Consider the functor M g which sends a scheme S to the set of isomorphism classes of
curves of genus g over S. The functor M g is unfortunately not representable (there is no fine
moduli spaces for curves), but it is nearly so (there is a coarse moduli space). That is, there is
a scheme (a variety, actually) Mg together with a natural transformation M g → h Mg such
that M g ( L) → h Mg ( L) is a bijection whenever L is an algebraically closed field, and such
that M g → h Mg is initial among all morphisms from M g to representable functors. For a
proof, see [DFK94, 5]. Along the same lines, we can let A g be the functor which assigns
to a scheme S the set of isomorphism classes of principally polarized abelian schemes of
dimension g over S. The functor A g has a coarse moduli space A g .
The operation “take jacobian with its canonical polarization” induces a natural transformation j : M g → A g , and Torelli’s theorem can be rephrased as saying that j is injective.
Shottky’s question asks what the image of j is. To see that it cannot be all of A g , simply
g ( g +1)
note that dim( Mg ) = 3g − 3, while dim( A g ) = 2 . For g > 3, A g has greater dimension
¯ ) → A g (Q
¯ ) cannot possibly be surjective. On the other hand, for
than Mg , so j : Mg (Q
g 6 3, all principally polarized abelian varieties are jacobians (possibly after a change of the
polarization).
One might hope that all abelian varieties are at least isogenous to jacobians. While this
is true for dimension d 6 3, it is not true in general. In fact, for all d > 3, there exists an
abelian variety of dimension d over Q which is not isogenous to a jacobian. This was proven
recently in [Tsi12]
1.5.2 Example (group law on elliptic curves). Let k be field of characteristic not 2 or 3. Let E
be an elliptic curve of the form y2 = x3 + ax + b with 4a3 + 27b2 6= 0. That is, E is the subset
of P2k given by
x12 x2 − x03 − ax0 x22 − bx23 .
The choice of O = (0 : 1 : 0) ∈ E(k ) induces an embedding j : E → Jac( E) which is an
isomorphism by the Riemann-Roch theorem. We would like to relate the induced group
operation on E with the classical definition using chords and tangents.
Let P, Q ∈ E(k¯ ). If we assume P, Q 6= O, then we can write P = ( P0 : P1 : 1), Q = ( Q0 :
Q1 : 1). Assume P0 6= Q0 . Then there is an obvious rational function (canonical up to scale),
whose zero-set is a line containing both P and Q. Indeed, we put
` P,Q ( x0 : x1 : x2 ) =
Q1 − P1 x0
x
P Q − P0 Q1
·
− 1+ 1 0
.
Q0 − P0 x2
x2
Q0 − P0
The function ` P,Q has P, Q and a third point R as simple zeros, and one can verify directly
that div(` P,Q ) = P + Q + R − 3O = ( P − O) + ( Q − O) + ( R − O). Recall that (Jac E)(k¯ ) =
Pic◦ ( Ek¯ ), and the map j : E → Jac( E) corresponds to P 7→ P − O. Thus j( P) + j( Q) + j( R) =
div(` P,Q ) = 0 in Pic◦ ( Ek¯ ), i.e. j( P) + j( Q) = − j( R).
It is well-known that R can be written as a rational function of P and Q, so the chordtangent law defines a rational map m : E × E → E with j(m( P)) + j(m( Q)) = m( j( P), j( Q)).
It follows that m is defined everywhere.
13
2
2.1
The Mordell-Weil theorem
Statement and generalizations
The goal of this section is to give an essentially complete proof of the Mordell-Weil theorem.
Throughout the section, k will denote a field, often √
a number field (finite field extension of Q).
Important examples are the quadratic fields k = Q( d) and cyclotomic fields Q(ζ n ).
2.1.1 Theorem (Mordell-Weil). Let A be an abelian variety over a number field k. Then the abelian
group A(k) is finitely generated.
This is clearly false if k = C and A 6= 0, for then A(C) is a complex Lie group, hence uncountable. In fact, whenever k is a local field, A(k ) is a Lie group over k, hence uncountable.
The Mordell-Weil theorem does holds whenever k is finitely generated over its prime field.
In this case, basic algebra shows that A(k) = A(k )tors ⊕ Zx1 ⊕ · · · ⊕ Zxr , where the xi are
linearly independent over Z. We call r = rk A the rank of A. Mordell proved the theorem
for A an elliptic curve over Q, demonstrating an assertion of Poincar´e.
2.1.2 Example. Let E ⊂ P2Q be the projective closure of the affine curve defined by y =
x3 + 2x + 3, with O = (0 : 1 : 0) the point at infinity. The curve has a group law such
that a + b + c = 0 if and only if a, b, c are colinear. Alternatively, let J = Jac E, The point
O ∈ E(Q) induces an embedding E ,→ J sending O to 0. This is an isomorphism, and we
can use it to transfer the group structure of J to E. The curve E has an
obvious
rational point
(−1, 0) of order two. Another rational points is (3, 6). Their sum is 14 , − 15
16 . One can show
that E(Q) = h(−1, 0)i ⊕ h(3, 6)i, where (−1, 0) has order two and (3, 6) has infinite order.
So E(Q) = Z/2 ⊕ Z.
2.1.3 Example. Let E/Q be the projective closure of y2 + y = x3 + x2 − 2x. We claim that
E(Q) = h(0, 0), (1, 0)i ' Z⊕2 . As an exercise, try to find ten more points in E(Q).
A result that motivated Weil is the following conjecture of Mordell (now a theorem of
Faltings).
2.1.4 Theorem (Faltings). If C is a nice curve over a number field k with genus g > 2, then C (k)
is finite.
Mordell’s conjecture fails if g 6 1. For g = 0, P1 has lots of rational points, and we have
seen examples of elliptic curves with infinitely many rational points. Here is a heuristic.
Assume C (k) 6= ∅ and consider the canonical embedding C ,→ J. We have C (k ) = C ∩ J (k ).
The set C has positive codimension in J, and J (k) is a finitely generated abelian group. So
C (k ) is the intersection of two “sparse” subsets of J. One would expect this forces C (k)
to be small. This heuristic is validated by the following theorem, originally known as the
Mordell-Lang conjecture.
2.1.5 Theorem (Faltings). Let A be an abelian variety over an algebraically closed field k of
characteristic zero, and let Γ be a finitely generated subgroup of A(k). If X ⊂ A is a subvariety, then
there is a finite set S ⊂ Γ and a finite set { Bs : s ∈ S} of abelian subvarieties of A such that
X (k) ∩ Γ =
[
(s + Bs (k) ∩ Γ) .
s∈S
14
Proof. See [McQ95] for a proof in the case k = C. The general case follows by the Lefschetz
principle. McQuillan actually proves the theorem for a broader class of group varieties than
abelian varieties.
2.1.6 Corollary. Let A be an abelian variety over C, and let C be a nice curve in A of genus g > 2.
Let Γ be a finitely generated subgroup of A(C). Then C (C) ∩ Γ is finite.
Proof. Since C has genus g > 2, it cannot contain a nontrivial abelian variety. Thus each
Bs = 0, so the theorem yields C (C) ∩ Γ = S for some finite set S ⊂ Γ.
There is a relative version of the Mordell-Lang conjecture known as the Lang-N´eron
theorem. Let K/k be a regular field extension, that is, k¯ ∩ K = k and K/k is separable. If A is
an abelian variety defined over K, then there is an abelian variety trK/k ( A) defined over k
together which a morphism τ : trK/k ( A)K → A that is initial among abelian varieties B/k
with morphisms BK → A. One calls trK/k ( A) the K/k-trace of A. Intuitively, trK/k ( A) is the
smallest abelian subvariety of A defined over k. A proof of the following theorem can be
found in [Con06].
2.1.7 Theorem (Lang-N´eron). Let K/k be a finitely generated regular extension, and let A be an
abelian variety over K. Then the group A(K )/ trK/k ( A)(k) is finitely generated.
This implies the Mordell-Weil theorem for finitely generated fields. If K is a finitely
generated field, let k be the algebraic closure of the prime field of K within K. Then K/k is
regular, and by the usual Mordell-Weil theorem, trK/k ( A)(k ) is finitely generated, so A(K )
has a finitely generated subgroup with finitely generated quotient. It follows that A(K ) is
finitely generated.
2.2
Plan of the proof
Let A be an abelian variety over a number field k. Our proof has three parts:
1. construct a “height function” | · | : A(k) → R>0 with good properties
2. prove the weak Mordell-Weil theorem: A(k )/nA(k ) is finite for all n > 2
3. show that 1 and 2 formally imply the full Mordell-Weil theorem
We will prove 3 here, spend the next several sections on 2, and finally construct a good
height function in section 2.11. For the sake of space, write A(k)/n instead of A(k)/nA(k).
2.2.1 Lemma. Let A be an abelian group with a function | · | : A → R>0 such that for some n > 2,
• for all c > 0, the set Bc = { x ∈ A : | x | 6 c} is finite
• | x − y| 6 | x | + |y| and |nx | = n| x | for all x, y ∈ A, n ∈ N
• the group A/n is finite
Then A is finitely generated.
15
Proof. Let { ai } be a (finite) set of coset representatives for A/n. Let c = 2 sup{| ai |}. We
claim that A is generated by the finite set Bc . This will be shown “by descent.” Fix n > 2
and let x1 ∈ A. By our assumptions, we can write x1 = yi1 + nx2 with { ai }, and one see that
| x 1 | + | y i1 |
1
c
1
| x − y i1 | 6
6 | x1 | +
.
n 1
n
n
2n
= yir + nxr with yir ∈ { ai }, we can continue this process, obtaining the
| x2 | =
Setting xr+1
inequality
1
1
c
1
| x r +1 | 6 | x r | +
6 r | x1 | +
n
2n
n
2
1
1
+···+ r
n
n
c6
| x1 | c
+ .
nr
2
For r 0, the quantity on the right is less than c. Thus x1 is in the subgroup of A generated
by Bc and { ai }. Since | ai | 6 c for each i, we have A = h Bc i.
To prove the weak Mordell-Weil theorem, we will use group cohomology extensively.
For some motivation, let A be an abelian variety over a field k of characteristic zero. Let
A[n] be the kernel of ·n : A(k¯ ) → A(k¯ ). We can take Gk -invariants of A[n], resulting in an
exact sequence:
/ A[n]Gk
/ A ( k ) n / A ( k ).
0
We are interested in continuing this exact sequence to the right, i.e. in constructing a long
exact sequence
0
A[n]Gk
A(k)
H1 ( Gk , A[n])
H1 Gk , A(k¯ )
A(k)
H1 Gk , A(k¯ )
···
This fits into a general framework of derived functors, but we will mostly just use the
concrete definition of H1 ( Gk , −) using cocycles and coboundaries.
Here is a very brief explanation of the construction of | · |. Here we require k to be a
number field. There is a height function on Pn (k¯ ) defined by
h ( x0 : · · · : x n ) =
1
log sup{| xi |v },
[ L : Q] ∑
v
where L/k is some extension containing the xi and | · |v is the normalized absolute value
associated with v. For a very ample divisor c on A, we get an embedding ϕc : A → Pn ,
and thus a height function hc : A(k¯ ) → R>0 . As it stands, this is not uniquely defined,
but there is a unique way of adjusting hc by a bounded function on A(k¯ ) to get a function
b
hc : A(k¯ ) → R such that
1 b
( x, y) 7→ h x, yic =
hc ( x + y) − b
hc ( x ) − b
hc (y)
2
b
is bilinear. One calls hc the N´eron-Tate height associated with c. The function | · | in the proof
can be taken to be | · |c = b
h1/2
for any very ample even divisor c.
c
Our proof of the Mordell-Weil theorem is very nearly effective. Given a set of generators
for A(k)/n, it gives an algorithm for finding a set of generators for A(k ). Moreover, one can
choose any integer n. Most people use n = 2 when doing computations.
16
2.3
Group cohomology
¯ ) the absolute Galois
Let k be a number field, k¯ an algebraic closure of k, and Gk = Gal(k/k
group of k. This is a profinite group (compact, totally connected Hausdorff group) with a
¯ ) where L ranges over the
basis of neighborhoods of 1 being the groups GL = Gal(k/L
finite extensions of k. Let A be an abelian variety over k of dimension d > 1. There
is no harm in thinking of k = Q and A as an elliptic curve. The abelian group A(k¯ )
naturally has a continuous Gk -action. One way to see this is to embed A into some huge
projective space and let Gk act on each coordinate. A “fancier” way to see this is to note that
A(k¯ ) = hom(Spec(k¯ ), A), so if σ ∈ Gk , ∈ A(k¯ ), the point σ ( x ) is the composite
Spec(k¯ )
σ∗
/ Spec(k¯ )
x
% A
σ ( x )∗
n
The homomorphism A −
→ A that sends x to n · x is an isogeny. Let A[n] be the n-torsion
subgroup of A(k¯ ), i.e.
A[n] = { x ∈ A(k¯ ) : n · x = 0}
If k were a field of positive characteristic p and p | n, it would be better to think of A[n] as
the scheme A × A 0 via n : A → A. In any case, there is an exact sequence
/ A[n]
0
/ A(k¯ )
n
/ A(k¯ )
/ 0.
Recall that if G is an arbitrary group acting on some abelian group M, we define the module
of G-invariants of M by
M G = {m ∈ M : σm = m for all σ ∈ G }.
Taking Gk -invariants of the above short exact sequence, we get an exact sequence
0
/ A[n]Gk
/ A(k)
n
/ A ( k ).
We don’t usually have exactness on the right because the functor (−)Gk is not right-exact.
For example, if k is a number field, the n-th power map (−)n : k¯ × → k¯ × is surjective, but
(−)n : (k¯ × )Gk = k× → k× is not.
We can define H• ( Gk , A[n]) using the formalism of derived functors. Consider the
category Gk -Mod of (discrete) abelian groups with continuous Gk -action. This is an abelian
category with enough injectives. The functor Γ = (−)Gk : Gk -Mod → Ab is left-exact, we
we define the group cohomology as the derived functors of Γ, i.e. H• ( Gk , −) = R• Γ. In
particular, H0 ( Gk , M ) = M Gk for any discrete Gk -module M.
Choose x ∈ A(k). Then x = n · y for some y ∈ A(k¯ ). Take σ ∈ Gk . Then x = σx, so
σ(n · y) = n · σ (y). Then n · (σy − y) = n · σy − n · y = 0, so σy − y ∈ A[n]. Thus we have
a map (not a usually a homomorphism) ϕ : Gk → A[n], σ 7→ σy − y. Take σ, τ ∈ Gk . Then
one computes
ϕ(στ ) = στ (y) − y
= σ(τy − y) + σy − y
= σϕ(τ ) + ϕ(σ).
17
So ϕ is a homomorphism precisely when the action of Gk on A[n] is trivial. Moreover, there
is a number field L/k such that GL ⊂ Gk fixes y. In particular, ϕ( GL ) = 0, i.e. the map
ϕ : Gk → A[n] is continuous. Suppose we choose some y0 distinct from y with x = n · y0 . We
could define ϕ0 : Gk → A[n] by σ 7→ σy0 − y0 . Since n(y − y0 ) = 0, we have y0 − y ∈ A[n],
hence y0 = y + α for some α ∈ A[n]. We now have
ϕ0 (σ ) = σy0 − y0
= σ(y + α) − (y + α)
= σy − y + σα − α
= ϕ(σ) + σα − α.
Maps Gk → A[n] of the form σ 7→ σα − α will be called coboundaries.
Suppose we have another point x 0 ∈ A(k). Choose a y0 with x 0 = ny0 . Then x + x 0 =
n(y + y0 ), and the point x + x 0 gives rise to a map Gk → A[n], σ 7→ σ (y + y0 ) − (y + y0 ); this
map is just ϕ + ϕ0 , where ϕ0 : σ 7→ σy0 − y.
2.3.1 Definition. Let G be a profinite group, M a discrete G-module. The group Z1 ( G, M ) of
1-cocycles consists of continuous maps ϕ : G → M such that ϕ(στ ) = σϕ(τ ) + ϕ(σ ). The group
B1 ( G, M ) of 1-coboundaries consists of continuous maps ϕ : G → M of the form σ 7→ σα − α
for some α ∈ M. The first cohomology of G with coefficients in M is
H1 ( G, M) = Z1 ( G, M )/B1 ( G, M ).
One can prove using a canonical projective resolution of M that this agrees with the
f
g
derived functor definition. It is not hard to check directly that if 0 → M0 −
→M−
→ M00 → 0
is an exact sequence of Gk -modules, then there is a natural exact sequence:
G
f
( G, M0 )
f∗
M0
0
MG
g
M00
G
δ
H
1
1
H ( G, M )
g∗
H1 ( G, M00 ).
where δ( x ) is defined as follows. Choose a lift x˜ in M of x, and let δ( x )(σ ) = f −1 (σ x˜ − x˜ ).
One can check that this is a cocycle, and that choosing a different x˜ changes δ( x )(σ ) by a
coboundary.
If G is a discrete group, then the cohomology H• ( G, M) can be interpreted as an Extgroup. It is easy to see that the category of G-module is equivalent to the category of
Z[ G ]-modules, and that M G ' homZ[G] (Z, G ). It follows that H• ( G, M ) ' Ext•Z[G] (Z, M ).
If G is a profinite group, then Z[ G ] is the wrong ring to use. Instead, one considers the
completed group ring
ZJGK = lim Z[ G/N ]
←−
N /G
N open
where we give each Z[ G/N ] the discrete topology and ZJGK the inverse limit topology.
The category of discrete G-modules with continuous action is equivalent to the category of
discrete ZJGK-modules with continuous action, M G ' homZJGK (Z, M ), whence H• ( G, M ) '
Ext•ZJGK (Z, M ).
18
In the case that G = Gk , one can interpret the groups H• ( G, M) as a special case of e´ tale
cohomology. Recall that a morphism f : X → S of schemes is e´tale if it is flat and unramified.
The e´ tale site of S, denoted Se´ t , is the full subcategory of SchS consisting of X → S that are
e´ tale. A collection { f i : Ui → X } in Se´ t is a cover if the images of the f i cover X. With this
notion of a cover, Se´ t is a (subcanonical) site, so we can talk about sheaves and cohomology
on Se´ t . The main example is:
2.3.2 Example. Let k be a field, and write ke´ t for Spec(k)e´ t . Then the objects of ke´ t are all of the
form Spec( L1 ) t · · · t Spec( Ln ) → Spec(k) for a finite family of separable field extensions
L1 , . . . , Ln of k. One can check that the category Sh(ke´ t ) of abelian sheaves on ke´ t is equivalent
to the category of discrete Gk -modules, via the functor F 7→ Fk¯ = lim L/k F (Spec L),
−→
where L ranves over all finite Galois extensions of k. If M is a Gk -module, then there is a
e determined by M
e (Spec L) = M GL . Since M G ' H0 (ke´ t , M
e ),
corresponding e´ tale sheaf M,
•
•
e
we obtain that H ( Gk , M) ' H (ke´ t , M). For more details, see [Del77, I 2.4].
n
In our case, from the long exact sequence associated to 0 → A[n] → A(k¯ ) −
→ A(k¯ ) → 0
yields a short exact sequence
0
/ A(k)/n
δ
/ H1 ( Gk , A[n])
/ H1 Gk , A(k¯ ) [n]
/ 0.
We will try to prove that A(k)/n is finite by embedding it into a group we know is finite.
Unfortunately H1 ( Gk , A[n]) is infinite, so we cannot just use the above exact sequence to
show that A(k)/n is finite.
In general, suppose G is a (commutative) group scheme over k. For example, G could
be the multiplicative group Gm , or an abelian variety A. If G is any commutative group
scheme over k, we will use G [n] to denote the fiber product G ×G 0, where the map G → G
is “multiply by n.” This conflicts with our earlier convention that A[n] = A(k¯ )[n], but
¯
no confusion should arise from this. The
group G (k) is a Gk -module, so we can consider
•
¯
the cohomology groups H Gk , G (k) . We know that these are isomorphic to the e´ tale
]
]
cohomology groups H• ke´ t , G
(k¯ ) , and one can check quite easily that G
(k¯ ) is just G,
regarded as a sheaf on ke´ t via its functor of points. It follows that H• Gk , G (k¯ ) = H• (ke´ t , G ),
and we will identify the two without comment in the future.
To simplify the notation, we will often write k for ke´ t , i.e. H• (k, G ) = H• (ke´ t , G ). In this
context, Hilbert’s theorem 90 says that for k a field, H1 (k, Gm ) = 0. If we write µn = Gm [n],
n
then Kummer theory starts with the short exact sequence 1 → µn → Gm −
→ Gm → 1 and
uses Hilbert’s theorem 90 together with the long exact sequence in sheaf cohomology to
derive H1 (k, µn ) = k× /n.
There is an alternate description of H1 (k, A). If G is an arbitrary commutative algebraic
group over k, a principal homogeneous space (also called a torsor) for G over k is a variety X/k
¯
together with a morphism G × X → X which, on k-valued
points, is a simply transitive
group action. That is, if we write g + x for the image of ( g, x ) in X, we require that
• g + (h + x ) = ( g + h) + x
• 0+x = x
• for all x, y ∈ X (k¯ ), there is a unique g ∈ G (k¯ ) such that g + x = y
19
More generally, if S is a scheme and G is an abelian sheaf on Se´ t , a torsor for G is a sheaf
of sets T with a group action G × T → T that, e´ tale-locally on S, is isomorphic to G as a
sheaf with left G -action. Two torsors T , T 0 are isomorphic if there is a sheaf isomorphism
T → T 0 that commutes with the action of G . One can show (see e.g. [Del77, IV 1.1]) that
H1 (Se´ t , G ) is naturally isomorphic as a pointed set to the set of isomorphism classes of
G -torsors.
Thus if G is a commutative algebraic group over k, there is a natural bijection between
the set of isomorphism classes of G-torsors and H1 (k, G ). There is a non-abelian version
of this. If G is an arbitrary sheaf of groups over Se´ t , then one can still define the notion of
a G -torsor. An identical theorem holds, except that one must define the “cohomology set”
H1 (Se´ t , G ). For details, see [Sko01]. There is a non-abelian version of Hilbert’s theorem 90:
it says that H1 (k, SLn ) = H1 (k, GLn ) = 0 for all n [Ser79, X.1].
2.4
Selmer groups and weak Mordell-Weil
Recall that if G is a profinite group (e.g Gk for some field k) and M is a discrete abelian group
with continuous G-action, we directly defined the first cohomology group H1 ( G, M) =
Z1 ( G, M )/B1 ( G, M ), where
Z1 ( G, M) = { ϕ : G → M continuous : ϕ(στ ) = σϕ(τ ) + ϕ(σ)}
and B1 ( G, M ) consisted of all ϕ : G → M of the form σ 7→ σx − x. Note that if G acts trivially
on M, then H1 ( G, M) = homcts ( G, M ). If f : M → M0 is G-equivariant, then there is an
obvious map f ∗ : H1 ( G, M) → H1 ( G, M0 ) given by f ∗ [ ϕ] = [ f ◦ ϕ]. Moreover, if f : G 0 → G
is a continuous group homomorphism, then we have a map f ∗ : H1 ( G, M ) → H1 ( G 0 , M)
given by f ∗ [ ϕ] = ϕ ◦ f , where we regard G 0 as acting on M via f .
Let k be a number field, A be an abelian variety over k, and n > 2. The short exact
n
sequence 0 → A[n] → A −
→ A → 0 of group schemes induces an exact sequence
0
/ A(k)/n
/ H1 (k, A[n])
/ H1 (k, A)[n]
/ 0.
We are trying to prove that A(k)/n is finite. Since H1 (k, A[n]) can be infinite, this is not
immediate.
2.4.1 Example. Let K = Q, and let E be the elliptic curve defined by y2 = ( x − a)( x −
¯ ) = {0, ( a, 0), (b, 0), (c, 0)}. Thus
b)( x − c) for distinct a, b, c ∈ Q. Then E[2](Q
H1 (Q, E[2]) = hom( GQ , E[2])
= hom( GQ , Z/2 × Z/2)
= hom( GQ , Z/2) × hom( GQ , Z/2)
This is easily seen to be infinite (either using global class field theory, or by noting that Q
has lots of Galois extensions of degree 2). In fact, using Kummer theory, one can show that
H1 (Q, E[2]) = (Q× /2)⊕2 , which is F2 -vector space of dimension ℵ0 .
Our goal is to find a finite subgroup of H1 (k, A[n]) containing A(k)/n. We haven’t really
used the fact that k is a number field yet – everything so far works for any perfect field. That
changes when we start looking at completions of k.
20
As before, let k be a number field, and let k v denote the completion of k at a place v. One
can show that k v will either be R, C, or a finite extension of some Q p . Choose k v ⊃ k; this
gives a homomorphism Gkv → Gk , where σ 7→ σ |k¯ . It turns out that the map is injective (this
is an easy corollary of Krasner’s lemma). As an example, if k = Q, k v = R, then GR = {1, c}
where c : C → C is complex conjugation. The image of c in GQ is some element of order
two.
The functoriality of H• (−, −) applied to the injection Gkv → Gk gives us a map H1 (k, A[n]) →
H1 (k v , A[n]), where we regard A[n] as a group scheme over k v by base extension. We now
have a commutative diagram:
0
/ A(k)/n
/ H1 (k, A[n])
∏ A(kv )/n
/ ∏ H1 (k v , A[n])
/ H1 (k, A)[n]
/0
∏ H (kv , A)[n]
/0
β
0
/
α
v
/
v
1
v
The n-Selmer group of A over k is
!
Seln ( A) = β
−1
(im α) = ker H (k, A[n]) → ∏ H (k v , A)[n] .
1
1
v
Similarly, we define the Tate-Shafarevich group of A by the exact sequence
0
/ X( A )
/ H1 (k, A)
/
∏ H1 ( k v , A ) .
v
Putting these two definitions together, we obtain a short exact sequence
0
/ A(k )/n
/ Seln ( A)
/ X( A)[n]
/ 0.
We will soon prove that Seln ( A) is finite, hence A(k )/n and X( A)[n] are finite.
2.4.2 Conjecture (Tate-Shafarevich). If A is an abelian variety over a number field k, the group
X( A) is finite.
This is currently wide open. A positive answer would show that for n sufficiently
large, A(k)/n ' Seln ( A). In particular, this would imply that for p large enough, rk( A) =
dimF p Sel p ( A), where as before rk( A) = rkZ A(k) is the algebraic rank of A.
The Selmer groups Seln ( A) are effectively computable. If X( A) were always finite, then
A(k) would be computable.
2.5
Crash course in algebraic number theory
If k is a number field, we already used (without defining it) the notion of a place of k. In this
section, we will see that k comes with a lot of extra structure which will be used later on.
Let’s start with places.
2.5.1 Definition. A local field is a topological field that is locally compact as a topological space.
21
Let k be a local field. Then the additive group of k is a locally compact group, so it has a
nontrivial Haar measure, i.e. a translation-invariant Borel measure µ. For α ∈ k, the measure
α∗ µ defined by α∗ µ(S) = µ(αS) is easily seen to be translation-invariant as well. It is known
that µ is unique up to scalar, so there is a real number, denoted |α|, such that α∗ µ = |α|µ.
It turns out that | · | induces the topology that k already has, and that k is complete with
respect to | · |. Unfortunately, | · | is not always an absolute value. That is, it may not satisfy
1. | x | = 0 if and only if x = 0
2. | xy| = | x | · |y|
3. | x + y| 6 | x | + |y|
However, the only exception is k = C, in which case | · | is the square of the usual absolute
value. For any other local field, | · | is an honest absolute value, so we will speak of the
“canonical absolute value” on a local field, even though it may not actually be an absolute
value.
Local fields can be completely classified. If the strict triangle inequality holds, i.e.
| x + y| 6 sup{| x |, |y|} for all x, y ∈ k, we say that k is non-archimedean. If k is not nonarchimedean, we say it is archimedean. If k is archimedean, one can prove that k has characteristic zero, so Q ⊂ k. The absolute value on k induces one on Q, and it is a theorem
that the only archimedean absolute value on Q is the usual one. Thus R ⊂ k. It is a general
theorem that k can only be locally compact if [k : R] < ∞, from which it follows that either
k = R or k = C.
If k is non-archimedean of characteristic p, then one can prove that k = Fq ((t)) for some
finite field Fq , where q = pr . If k is non-archimedean of characteristic zero, then once again
Q ⊂ k. It is known (Otrowski’s theorem) that the only non-archimedean absolute values on
Q are of the form | · | p for primes p, where | x | p = p−v p ( x) . Here v p : Q× → Z is the unique
homomorphism with v p ( p) = 1, v p (n) = 0 for p - n. If we write Q p for the completion of Q
with respect to | · | p , then k contains some Q p . Once again a general theorem shows that
[k : Q p ] < ∞. To summarize, local fields are one of the following:
• R or C
• κ ((t)) for some finite field κ
• a finite extension of Q p
If k is a non-archimedean local field, we write ok = { x ∈ k : | x | 6 1} and pk = { x ∈ k :
| x | < 1}. It turns out that ok is a complete discrete valuation ring with maximal ideal pk . We
denote the residue field by κk = ok /pk . When k is understood, we write o, p, κ instead of
ok , pk , κk . Choose a separable closure ks of k. Because the field k is henselian (| · | has a unique
extension to ks ), the integral closure oks of ok in ks is a local ring (no longer noetherian) and
elements of Gk preserve | · |. The field oks /pks is separably closed, so we get a map Gk → Gκ ,
given by σ 7→ σ¯ , where σ¯ ( x¯ ) = σx. The kernel is called the inertia group of k, and denoted Ik .
2.5.2 Definition. A global field is either a finite extension of either Q or F p (t) for some prime p.
A place of a global field k is an equivalence class of embeddings k ,→ K, where K is a
local field such that the image of k is dense. Two such embeddings are equivalent if they are
(topologically) isomorphic over k. We will use the letter v to denote places of k, and denote
22
a choice of representative by k v . If k v is non-archimedean, we call v finite, otherwise it is
infinite. For the remainder, let v be a finite place of k.
The absolute value on k v induces one on k, which we will denote by | · |v . If v is nonarchimedean (i.e. k v is non-archimedean) then we will also use v to denote the valuation on
k induced by the canonical valuation on k v . We will write ov , pv , κv instead of okv . . . . We can
choose ksv ⊃ ks , in which case restriction induces a continuous homomorphism Gkv → Gk .
This is injective by Krasner’s lemma. The image is often denoted Dv , and the image of Ikv
inside Dv will be written Iv . Note that Dv , as a subset of Gk , is only well defined up to
conjugacy.
A useful fact about global fields is the product formula. Given the canonical absolute
value | · |v associated with a place, we have
∏ | x |v = 1
v
This property actually characterizes global fields – see [AW45]. For more details on local
and global fields, see [Wei95].
Let k be a number field, i.e. a finite extension of Q. Let ok be the ring of integers of k, that
is, ok is the integral closure of Z in k. The ring ok is a dedekind domain. Thus if a ⊂ ok is
e
a nonzero ideal, we have a factorization a = p11 · · · prer where the pi ⊂ ok are prime (hence
maximal) ideals, and each ei > 1. This factorization is unique up to reordering if we require
that the pi be distinct.
For a prime p ⊂ ok , we write (as before) κp = ok /p for the residue field of p. The field κp
is finite because it is finitely generated (as a ring) over its prime field. There is a unique
homomorphism vp : k× Z such that for a ∈ ok r 0, we have ( a) = pvp (a) · b with (b, p) = 1.
This gives us an absolute value
(
0
if a = 0
| a |p =
−
v
(
a
)
p
(#κp )
otherwise
Completing k with respect to this absolute value, we get a local field kp . Our valuation
(and absolute value) extend by continuity to a valuation vp : k×
p → Z and absolute value
| · |p : kp → R>0 . This agrees with our previous definition of the canonical absolute value
on a local field.
Let L/k be a finite extension, p ⊂ ok a prime ideal. The ideal po L factors uniquely
e(q /p)
e(q /p)
as q1 1
· · · qr r , where the qi ⊂ p L are prime. For example, if L = Q(i ), then 2 =
−i (1 + i )2 and 5 = (1 + 2i )(1 − 2i ). One can show that
2o L = ((1 + i )o L )2
3o L = 3o L
5o L = ((1 + 2i )o L ) · ((1 − 2i )o L )
are prime factorizations in o L . One says that p is unramified in L if e(qi /p) = 1 for all i. It is
an easy theorem that only finitely many primes can ramify.
Now assume L/k is Galois. The action of the galois group G = Gal( L/k) preserves o L ,
fixing ok . This action does not fix ideals in o L . In fact, if q1 , . . . , qr are the primes lying above
p ⊂ ok , then G acts transitively on {p1 , . . . , pr }, from which we see that each e(qi /p) is the
same integer, denoted ep . Fix q = q1 . The decomposition group of q/p is
D (q/p) = {σ ∈ Gal( L/k) : σ (q) = q}
23
There is a canonical homomorphism D (q/p) → Gal(κq /κp ) given by “reduce σ modulo p.”
This gives us an exact sequence
/ I (q/p)
1
/ D (q/p)
/ Gal(κq /κp )
/1
(surjectivity on the right is non-trivial). So the inertia group of q/p, denoted I (q/p), is
the subgroup of D (q/p) consisting of automorphisms whose action is trivial modulo q.
Choosing q = qi for some i 6= 1 gives a D (qi /p) that is conjugate to D (q/p). We will often
write Dp and Ip , keeping in mind that they are only well-defined up to conjugacy. We will
use the fact that #Ip = ep . So p is unramified in L if and only if Ip = 1.
Choosing q lying over p, we can complete L and k to get an extension Lq /kp of local
fields. We have seen that restriction gives a map Gal( Lq /kp ) → Gal( L/k). It turns out that
this map is an isomorphism onto the image D (q/p) ⊂ Gal( L/k). We’ll write I ( Lq /kp ) for
the inverse image of I (q/p) in Gal( Lq /kp ). It can be defined directly in exactly the same
manner as I (q/p).
Passing to the algebraic closure of kp , we can consider the absolute Galois group
Gkp = lim Gal( Lq /kp )
←−
Lq ⊃ k p
where Lq ranges over all finite Galois extensions of kp . The group Gkp has a distinguished
subgroup Ip , which is the inverse limit
Ip = lim I ( Lq /kp ).
←−
Lq /kp
Our exact sequence
1
/ Ip
/ Gk
p
/ Gal(κp /κp ) = Zˆ
/1
extends to a filtration of Gkp by normal closed subgroups whose successive quotients are
abelian. Once again, recall that there is a canonical embedding Gkp ,→ Gk via an embedding
k¯ ,→ kp .
2.6
Reduction of abelian varieties
Let A be an abelian variety over a local field kp . Let op be the ring of integers of kp , i.e.
op = { x ∈ kp : | x |p 6 1}. The ideal pop is the unique maximal ideal of op , and it turns out
that pop = { x ∈ kp : | x |p < 1}. In an abuse of notation, we write p for pop .
2.6.1 Definition. We say that a variety X/kp has good reduction if there is a smooth proper model
X /op .
That is, there is a smooth proper scheme X over op such that the following diagram is
cartesian:
/X
X
Spec(kp )
/ Spec(ok )
24
We write Xp for X ×Spec(ok ) Spec(κp ), and call Xp the reduction of X modulo p.
One can show that if A is an abelian variety over kp with good reduction, then Ap is
independent of the choice of A, that A gets the structure of an abelian scheme over ok , and
that Ap /κp is an abelian variety.
There is in fact a canonical model for A over o p , and it exists in great generality. Let S be
a connected dedekind scheme with field of fractions k. If A is an abelian variety over k, one
calls a N´eron model of A a smooth model A for A over S for which any morphism Xk → A,
where X is a smooth scheme over S, has a unique extension to a morphism X → A. In
other words, A represents the functor X 7→ homk ( Xk , A) from smooth schemes over S to
k-varieties. It is clear that A (if it exists) is unique up to unique isomorphism. Fortunately, it
is a theorem (see [BLR90, 1.4.1]) that in our setting (S a connected dedekind scheme and A
an abelian variety over k) N´eron models always exist. Since the functor X 7→ homk ( Xk¯ , A)
that A represents is naturally a group-valued functor, A naturally has the structure of a
commutative group scheme [BLR90, 1.2.6].
So if S = Spec(ok ) for a number field k and p ⊂ o is a prime, we could have said that an
abelian variety A/k has good reduction at p if the N´eron model A for op (the localization of
o at p) is proper.
2.6.2 Example. Let E : y2 = x3 + ax + b, where a, b ∈ Z and ∆ = −16(4a3 + 27b2 ) 6= 0. If
p - ∆, then E/Q p has good reduction, and E p /F p is given by the reduction of our original
equation modulo p.
There is a reduction map A(kp ) → Ap (κp ) which is a group homomorphism. This is the
composite
A(kp ) = A(op ) → A(κp ) = Ap (κp )
To see that A(kp ) = A(op ), think of A as being a subset of some projective space P N . For
a point x = ( x0 : · · · : x N ) ∈ P N (kp ), we can scale x be the denominators of the xi to get a
model x = ( x0 : · · · : x N ) ∈ P N (op ). We can even get a model with some xi 6≡ 0 mod p,
and the image of x in P N (κp ) is in Ap (κp ). By Hensel’s lemma, the map A(kp ) → Ap (κp )
is a surjection. The kernel is a pro-p group, where p is the characteristic of κp . In fact, the
kernel is a p-adic Lie group.
We can extend our reduction map to algebraic closures, getting a homomorphism
A(kp ) → Ap (κp ), that has a pro-p kernel. Choose an integer n > 2 with p - n. The map
A(kp )[n] → Ap (κp )[n] is an isomorphism because both groups have the same cardinality
(n2 ) and the kernel is pro-p. In that isomorphism both groups have a Galois action – Gkp
on the left and Gκp = Zˆ on the right. The map is compatible with the Galois action, so in
particular, the inertia group Ip acts trivially on A(kp )[n].
Let A/kp be an abelian variety with good reduction. Recall we had a map δ : A(kp )/nA(kp ) ,→
H1 (kp , A[n]) defined as follows. For x ∈ A(kp ), choose y ∈ A(kp ) such that n · y = x. We
define the 1-cocycle ϕ = δ( x ) by σ 7→ σy − y. For σ ∈ Ip , the elements σy and y have the
same image in Ap (κp ), hence σy − y ∈ A(kp )[n] has trivial image modulo p. It follows that
σy − y = 0 since it is an n-torsion point that is 0 modulo p. This tells us that ϕ(σ ) = 0 for all
σ ∈ Ip , i.e. ϕ( Ip ) = 0.
2.6.3 Lemma. Let A be an abelian variety over a number field k. With the above notation, ϕ( Ip ) = 0
for all p ⊂ ok of good reduction for A.
25
If we let S be the finite set of primes for which A has bad reduction, we will show that
the group
H1S (k, A[n]) = {[ ϕ] ∈ H1 (k, A[n]) : ϕ( Ip ) = 0 for all p ∈
/ S}
is finite.
2.7
Restricted ramification
Let k be a number field, and let S be a finite set of primes of k. We defined, for a Gk -module
M, the “cohomology with restricted ramification”
H1S ( Gk , M) = {[ ϕ] ∈ H1 ( G, M ) : ϕ( Ip ) = 0 for all p ∈
/ S}
It is better to interpret H1S ( Gk , M) in terms of Galois cohomology. First a clarification:
ϕ( Ip ) = 0 actually means that ϕ( Ip ) = 0 for any choice of Ip . (Recall that the embeddings
Gkp ,→ Gk , and hence Ip ,→ Gk , are only well-defined up to conjugacy.) One easily checks
that, for any [ ϕ] ∈ H1 ( Gk , M ), the set ker( ϕ) = {σ : ϕ(σ ) = 0} is a subgroup of Gk . In other
words, if H ⊂ Gk denotes the (normal) subgroup generated by the Ip for p ∈
/ S, then
H1S ( Gk , M ) = ker H1 ( Gk , M ) → H1 ( H, M ) = H1 ( Gk /H, M H ),
the second equality coming from the inflation-restriction sequence.
S
The normal subgroup of Gk generated by p/
∈S Ip is the Galois group of an extension
k S /k. One can prove that whenever L/k is a finite extension unramified away from S, then
L ⊂ k S . We write Gk,S = Gal(k S /k) = Gk /GkS , and call Gk,S a Galois group with restricted
ramification. In the future, if M is a Gk -module for which the action of Gk on M is unramified
away from S, we will write H1 ( Gk,S , M ) instead of H1S ( Gk , M).
Just as one can interpret the absolute Galois group Gk as the e´ tale fundamental group
π1 (Spec k), the group Gk,S can be interpreted as an e´ tale fundamental group using the
following theorem.
2.7.1 Theorem. Let S be normal connected scheme with function field k. Then the group π1 (S)
is naturally isomorphic to Gal(k S /k), where k S is the composite of all finite extensions L ⊂ ks for
which the normalization of S in L is e´tale over S.
Proof. See [Sza09, 5.4.9].
Let S be a finite set of primes of k, and write ok,S = S−1 ok . A field extension K/k is
unramified outside S precisely when the integral closure of ok,S inside K is e´ tale over ok,S .
In other words, the above theorem shows that Gk,S = π1 (Spec oo,S ). In proving the weak
Mordell-Weil theorem, we will need following important finiteness result.
2.7.2 Theorem. Let S be a finite set of primes of a number field k. Then Gk,S has only finitely many
open subgroups of any given index.
Proof. This is just a rewriting of Theorem 2.7.3.
2.7.3 Theorem (Hermite). Let k be a number field, S a finite set of places of ok . For a fixed integer
N > 1, there is a finite extension K ⊂ k¯ such that if L ⊂ k¯ is an extension of k unramified outside of
S with [ L : k ] 6 N, then L ⊂ K.
26
Proof. This follows easily from [BG06, B.2.14].
Hermite’s theorem has a huge generalization. Call a profinite group G small if it has only
finitely many open subgroups of any given index. If X is a connected scheme of finite type
such that X → Spec(Z) has dense image, then π1 ( X ) is small [HH09, 2.8]. This is useful
because of the following theorem.
2.7.4 Theorem. Let G be a small group. Then H1 ( G, M) is finite for all finite continuous G-modules
M.
Proof. Since M is finite, there is an open normal subgroup H ⊂ G such that H acts trivially
on M. Recall the inflation-restriction exact sequence is
0
/ H1 ( G/H, M H )
inf
/ H1 ( G, M)
res
/ H1 ( H, M ),
where the first arrow is [ ϕ] 7→ [ ϕ ◦ ( G G/H )] on cocycles, and the restriction map is
[ ϕ] 7→ [ ϕ| H ]. The group H1 ( G/H, M H ) is finite because both G/H and M H = M are finite,
and H1 ( H, M) = homcts ( H, M) is finite because H is small.
If we are willing to consider H1 ( G, M ) for M non-abelian, then the converse is true.
2.8
Torsion and weak Mordell-Weil
Let A be an abelian variety over a number field k. Recall that we are trying to show that
A(k) is finitely generated. We have shown that it is sufficient to prove that some quotient
A(k)/n with n > 2 is finite. Once we know that A(k ) is finitely generated, we can write
A(k) = A(k)tors ⊕ Z · x1 ⊕ · · · ⊕ Zxr , where each xi is of infinite order. The algebraic rank
r = rk A is very difficult to compute in general, but A(k)tors is computable.
Choose a prime p ⊂ ok at which A has good reduction (this is true for all but finitely
many p). We have a reduction map A(kp ) → Ap (κp ), which has pro-p kernel. The group
A(k)tors is contained in A(kp ), so we can think about its image in Ap (κp ). Since A(k )tors is
finite, the kernel of A(k)tors → Ap (κp ) is a p-group. Pick another prime p0 of good reduction
for A, with residue characteristic ` 6= p. The kernel of the map A(k)tors → Ap (κp ) × Ap0 (κp0 )
is a p-group and a `-group, hence trivial, i.e. A(k)tors is a subgroup of Ap (κp ) × Ap0 (κp0 ).
(This gives us a way to compute A(k)tors , because Ap (κp ) and Ap0 (κp0 ) are computable.
(X (κ ) for X any projective variety over any finite field κ is computable, for stupid reasons.)
2.8.1 Example. Let E be the elliptic curve over Q given by y2 = x3 + 3. This has descriminant
∆ = −24 · 35 , so E has good reduction away from 2 and 3. We can compute
E5 (F5 ) = {O, (1, ±2), (2, ±1), (3, 0)}
| E7 (F7 )| = 13
The kernel of E(Q)tors → E5 (F5 ) is a 5-group, and the kernel of E(Q)tors → E7 (Q7 ) is a
7-group. From this, we know that E(Q)tors has no points of order 5 or 7. Thus E(Q)tors
embeds into groups of oder 6 and 13, so it is the trivial group. Since (1, 2) ∈ E(Q) and
E(Q)tors = 0, we know that E(Q) is infinite.
2.8.2 Example. Let E/Q be the curve defined by y2 + y = x3 − x2 − 10x − 20. One can
check that E has good reduction away from 11. Easy computations yield
27
p
2
3
5
7
13
#E p (F p )
5
5
5
10
10
This shows us that E(Q)tors is either 0 or Z/5. In fact, it is the latter with E(Q)tors = h(5, 5)i.
Let’s get back to the weak Mordell-Weil theorem. Let A be an abelian variety over a
number field k, and let S be the (finite) set of primes p for which A has bad reduction at
p. There are natural maps δ : A(kp )/n → H1 (kp , A[n]), so we get a commutative diagram
with exact rows:
δ
/ A(k)/n
/ H1 (k, A[n])
0
0
/ A(kp )/n
δ
/ H1 (kp , A[n])
Since A has good reduction outside S, the group Gk,S defined in 2.7 acts on A[n], so we can
consider the cohomology group H1 ( Gk,S , A[n]). Moreover, the n-Selmer group Seln ( A) sits
inside H1 ( Gk,S , A[n]), so A(k)/n ,→ Seln ( A) ⊂ H1S (k, A[n]). The group Seln ( A) is a “better
approximation” of A(k )/n than H1S (k, A[n]), but it is H1 ( Gk,S , A[n]) that we will actually
show is finite.
2.8.3 Theorem. Let A be an abelian variety over a number field k, and let S be the set of places of k
at which A has bad reduction. Then the group H1 ( Gk,S , A[n]) is finite.
Proof. This follows immediately from the fact that Gk,S is small, coupled with Theorem 2.7.4.
Explicitly, we first choose L = k( A[n]) ⊃ k. Then L/k is unramified outside S and GL acts
trivially on A[n]. Let S0 denote the set of places of L lying above S. The inflation-restriction
exact sequence
0
/ H1 (Gal( L/k), A[n])
/ H1 ( Gk,S , A[n])
/ H1 ( GL,S0 , A[n]),
reduces the problem to showing that H1 ( GL,S0 , A[n]) = hom( GL,S0 , A[n]) is finite. So we
may as well assume k ⊃ A[n] in the first place, and try to prove that H1 ( Gk,S , A[n]) ≈
hom( Gk,S , (Z/n)⊕2d ) is finite, where d = dim A. For any f : Gk,S → (Z/n)⊕2d , the image
of f is the Galois group of an extension L/k unramified outside S with [ L : k] 6 n2d .
By Hermite’s theorem, there is a fixed K/k such that all L ⊂ K. Thus H1 ( Gk,S , A[n]) ,→
hom(Gal(K/k), Z/n)⊕2d , a finite group.
There is a geometric interpretation of the Tate-Shafarevich group X( A), defined by the
exact sequence
/ X( A )
/ H1 (k, A)
/ ∏ H1 (k v , A)
0
v
Recall that a torsor of A over k is a nice variety X/k with a simply transitive group action
A × X → X which is a morphism of k-varieties. In other words, we require A( L) × X ( L) →
28
X ( L) to be a simply transitive group action whenever X ( L) 6= ∅. If x ∈ X ( L), we get an
isomorphism A L → X L which is not generally defined over k. Two torsors are equivalent if
they have compatible group actions (i.e. if they are A-equivariantly isomorphic over k). A
torsor is trivial if it is equivalent to A with the usual left action. It turns out that a torsor X is
trivial if and only if X (k) 6= ∅.
There is a natural bijection
{torsors over A}/equivalence ↔ H1 (k, A)
2.8.4 Example. Let C/k be a nice curve of genus 1. (It could be that C is not elliptic, for
instance if C (k) = ∅.) Let E = Jac C; this is an elliptic curve. There is a bijection C → Pic1C
given by x 7→ [ x ]. This induces a bijection E = Pic0C . There is an action Pic0C × Pic1C → Pic1C ,
induced by ( D1 , D2 ) 7→ D1 + D2 . This makes C a torsor over E.
2.9
Tate-Shafarevich groups
Let k be a number field, A an abelian variety over k. Recall the Tate-Shafarevich group of A
is
!
X( A) = ker H1 (k, A) →
∏ H1 ( k v , A )
.
v
The Weil-Chˆatelet group of A, written WC( A), is the group of torsors over A modulo
equivalence. As we have seen, there is a bijection WC( A) = H1 (k, A). If k is a number field, then WC( A) is infinite, but if k is finite then WC( A) = 0. We’ll define the map
WC( A) → H1 (k, A). Fix an A-torsor X. There exists some L/k with X ( L) 6= ∅. Choose
x ∈ X ( L), and define a cocycle ϕ : Gk → A(k¯ ) by σ 7→ σx − x, where σx − x is the unique
point a ∈ A(k¯ ) such that a + x = σ ( x ). The image of X in H1 (k, A) is the cocycle ϕ. It isn’t
too hard to check that this map is well-defined.
This tells us that there is a natural bijection between X( A) and the set of torsors X of
A/k, such that X (k v ) 6= ∅ for all places v of k. This can be generalized. Let X/k be a nice
variety. We say that X satisfies the Hasse principle if X (k v ) 6= ∅ for all v implies X (k ) 6= ∅. So
X( A) classifies torsors over A that do not satisfy the Hasse principle. Clearly, if X (k) 6= ∅,
then X satisfies the Hasse principle. Similarly, if X (k v ) = ∅ for some v, then X satisfies the
Hasse principle.
2.9.1 Example (Selmer). The plane curve C ⊂ P2Q given by 3x3 + 4y3 + 5z3 = 0 fails the
Hasse principle. In other words, C (Q p ) 6= ∅ for all p, C (R) 6= ∅, but C (Q) = ∅. If we let
E = Jac C, then we know that X( E) 6= 0.
It turns out that the Hasse principle can be checked. That is, for a nice variety X over
Q there is an algorithm to determine whether X (Qv ) 6= ∅ for all primes. This is because
X will have good reduction at all but a finite (computable!) set of primes. If X has good
reduction at p with integral model X , then X (F p ) 6= ∅ by the Weil conjectures. Since X is
smooth, Hensel’s lemma lets us lift an element of X (F p ) to X (Z p ) ⊂ X (Q p ). For a prime p
at which X has bad reduction, we can still pick an integral model X . Either there will be an
n for which X (Z/pn ) = ∅, in which case X (Q p ) = ∅, or elements of X (Z/pn ) for n 0
will lift to X (Z p ). Checking whether X (R) = ∅ is easy analysis.
If C is a nice curve of genus zero over a number field, then C satisfies the Hasse principle
[Cas67, 3.4]. This is essentially the Hasse-Minkowski theorem.
29
2.9.2 Example (descent). Let E/Q be the elliptic curve defined by y2 = ( x − e1 )( x − e2 )( x −
e3 ) where the ei ∈ Z are distinct. It’s easy to see that E[2] = {0, (e1 , 0), (e2 , 0), (e3 , 0)} '
(Z/2)⊕2 . We use {(e1 , 0), (e2 , 0)} as a F2 -basis for E[2]. We have
H1 (Q, E[2]) = hom( GQ , E[2])
= hom( GQ , Z/2)⊕2
⊕2
= Q× /2
Here, as always, “hom” denotes the group of continuous homomorphisms, and we write
Q× /2 for Q× /(Q× )2 . The isomorphism H1 (Q, E[2]) ' (Q× /2)⊕2 comes from Kummer
√
theory. Given a homomorphism ϕ : GQ → Z/2, the group ker( ϕ) fixes a field k = Q( d)
with d ∈ Q× . The equivalence class of d in Q× /2 is dependent only on ϕ. The fact that
ϕ 7→ d is a bijection is the content of Kummer theory.
The boundary morphism in group cohomology gives us a map δ : E(Q)/2 ,→ H1 (Q, E[2]).
∼
The composite of δ with the isomorphism H1 (Q, E[2]) −
→) = (Q× /2)⊕2 is quite explicit –
we have for x = ( x0 : x1 : 1):

(1, 1)




 ( x 0 − e1 , x 1 − e2 )
δ( x ) =
e1 − e3
,
e
−
e

2
1
 e1 − e2


 e − e , e2 − e3
2
1 e2 − e1
if x = 0
if x0 ∈
/ { e2 , e3 }
if x0 = e1
if x0 = e2
Let S be the set of primes that divide 2(e1 − e3 )(e2 − e3 )(e2 − e1 ); this contains the set of
primes at which E has bad reduction. We know that the group H1 (ZS , E[2]) = H1 ( GQ,S , E[2])
is finite. Even better, one can show that it is H⊕2 , where
H = b ∈ Q× /2 : v p (b) ≡ 0 (mod 2) for p ∈
/S
The group H is generated by S ∪ {−1}. This allows us to bound the rank of E. We know
that dimF2 (H) 6 #S + 1, so the fact that E(Q)/2 ,→ H⊕2 implies rk( E) 6 2(#S + 1).
We know that E(Q)/2 ,→ Sel2 ( E) ⊂ H⊕2 , so we could get a better bound on rk( E) if
we could compute the image of Sel2 ( E) inside of H⊕2 . Recall that there is a commutative
diagram:
/ H1 (Q, E)
H1 (ZS , E[2])
O
O
0
?
/ E(Q)/2
?
/ Sel2 ( E)
/ X( E)[2]
/0
For a pair b = (b1 , b2 ) ∈ H⊕2 , we get an element of H1 (Q, E). This gives us a torsor Xb of
E/Q. Observe that b ∈ Sel2 ( E) if and only if Xb (Q p ) 6= ∅ for all p and Xb (R) 6= ∅.
The curve Xb can be computed. We have Xb ⊂ P3Q , a nice curve of genus one defined by
b1 z21 − b2 z22 = (e2 − e1 )z20
b1 z21 − b1 b2 z23 = (e3 − e1 )z20
30
2.9.3 Example. Let E/Q be the elliptic curve y = x3 − x = x ( x − 1)( x + 1). We’ll pick e1 = 0,
e2 = 1, e3 = −1. For our curve we have S = {2}, so H is the subgroup of Q× /2 generated
by {−1, 2}. The group H⊕2 has representatives {(±1, ±1), (±2, ±1), (±1, ±2), (±2, ±2)}.
We know that for b = (b1 , b2 ) ∈ (Q× )2 , the curve Xb is
b1 z22 − b2 z22 = z20
b1 z21 − b1 b2 z23 = −z20
Thus Xb (R) = ∅ if b1 < 0 and b2 > 0, or if b > 0 and b2 < 0. That tells us that
Sel2 ( E) ⊂ {(b1 , b2 ) ∈ H ⊕2 : b1 b2 > 0}, a group of order eight with representatives
{±(1, 1), ±(2, 1), ±(1, 2), ±(2, 2)}.
We have
/ E(Q)/2 δ / Sel2 ( E) 
/ h−(1, 1), −(2, 1), −(1, 2), −(2, 2)i
0
Since E[2] ⊂ E(Q), this tells us that E(Q)/2 has order at least four. We have δ( E[2]) =
{(1, 1), (−1, −2), (−1, −1), (1, 2)}. Consider b = (2, 1). Then Xb is
2z21 − z22 = z20
2z21 − 2z23 = −z20
This curve has real points, so we only need to check Xb (Q2 ). As an exercise, show that
Xb (Z/4) = ∅, which implies Xb (Q2 ) = ∅, which shows that b ∈
/ Sel2 ( E), and thus
#E(Q)/2 = 4. Since 4 = #E[2], we know that rk( E) = 0.
2.10
Weil heights
Recall that to prove the Mordell-Weil theorem, we needed the weak Mordell-Weil theorem
and a good height function. We gave an extremely terse introduction to heights earlier –
here we will do things more carefully.
The idea is as follows. Let X be a nice variety over a number field k. We want a
function H : X (k) → R that measures the “arithmetic complexity” of a point. For example,
1
100001
2 and 200001 are very close in R, but we should think of the latter as being much more
“arithmetically complex.” We would want H to have properties arising from the geometry
of X, and (this is very important), be such that the sets { x ∈ X (k) : | H ( x )| 6 c} are finite for
all c.
Let’s start with heights on projective space over Q. Consider a point x = ( x0 : · · · :
xn ) ∈ Pn (Q). After scaling by a rational number, we can assume that the xi ∈ Z and
gcd( x0 , . . . , xn ) = 1. We set
HQ ( x ) = sup{| x0 |, . . . , | an |}
It is easy to see that this is well defined, but it doesn’t work very well over a general number
field. Returning to our example, we have HQ ( 12 : 1) = 2, while HQ ( 100001
200001 : 1) = 200001,
which is much larger.
Now let k be a number field. Each finite place v of k is associated with a prime pv ⊂ o = ok .
We set
(
#(o/pv )−vp (a) if a 6= 0
k a k v = | a | pv =
0
if a = 0
31
We call k · kv the canonical absolute value associated with v. If v is a real place of k, i.e. v
corresponds with i : k ,→ R, we set k akv = |i ( a)|. Similarly, if v is a place corresponding to
an embedding i : k ,→ C with dense image, we set k akv = |i ( a)|2 . Note that k · kv is not an
absolute value because it doesn’t satisfy the triangle inequality. Recall the following
2.10.1 Theorem (product formula). Let k be a global field. For a ∈ k× , we have
∏ k akv = 1
v
Proof. We’ll only look at k = Q. Take a = ± ∏ p pv p (a) ∈ Q× . We have k ak p = p−v p (a) for
each p, while k ak∞ = | a|. Thus
∏ k a kv = k a k∞ · ∏ p−v p ( a) = ∏ pv p ( a) · ∏ p−v p ( a) = 1
v
p
p
p
For general k/Q, one uses the norm map N : k → Q to prove the product formula for k.
For k a general number field, we define Hk : Pn (k) → [1, ∞) by
Hk ( a0 : · · · : an ) =
∏ sup{ka0 kv , . . . , kan kv }
v
If k = Q and the ai ∈ Z with gcd( a0 , . . . , an ) = 1, then max{| a0 | p , . . . , | an | p } = 1. Thus
HQ ( a0 : · · · : an ) = sup{| a0 |, . . . , | an |}. The height Hk is well-defined because of the product
formula. For c ∈ k× , we have
∏ sup{kca0 kv , . . . , kcan kv } = ∏ kckv ∏ sup{ka0 kv , . . . , kan kv }
v
v
v
= 1 · ∏ sup{k a0 kv , . . . , k an kv }
v
which implies Hk (ca0 : · · · : can ) = Hk ( a0 : · · · : an ). It turns out that for each c ∈ R, we
have
#{ a ∈ Pn (k ) : Hk ( a) 6 c} = O c( N +1)[k:Q]
In particular, the set on the left is finite.
¯ ) → [1, ∞) is defined by H ( a) = Hk ( a)[k:Q]−1 for
The absolute height is a map H : Pn (Q
n
¯ with a ∈ k. The sets { x ∈ P (Q
¯ ) : H ( a) 6 c} can be infinite, but the sets
any k ⊂ Q
¯ ) : H ( a) 6 c and [Q( x ) : Q] 6 d}
{ x ∈ Pn (Q
¯ ) → [0, ∞).
are finite. The logarithmic absolute height is the map h = log H : Pn (Q
Let X be a nice variety over k. We would like to define a reasonable height function on
X. One way to do this is to take an embedding φ : X → Pn (probably for some very large
φ
h
n) and let hφ : X (k¯ ) → [0, ∞) be the composite X (k) −
→ Pn (k¯ ) −
→ [0, ∞). We can rephrase
this. Let D ∈ Div( X ) be a very ample divisor. A choice of a generating set for the global
sections of L ( D ) gives an embedding φD : X ,→ Pn . We write h D (k¯ ) → [0, ∞) for the
φD
h
composite X (k¯ ) −→ Pn (k¯ ) −
→ [0, ∞). Note that the notation h D is a little bit misleading,
because h D actually depends on a choice of a generating set for L ( D ). If we chose another
0 of X into some projective space, then we have
generating set, getting an embedding φD
32
0 = O (1), i.e. the two heights differ by a (globally) bounded function X ( k¯ ) → R.
hφD − hφD
¯
So if we consider h D as an equivalence class in R X (k) , then h D is well-defined. If E ∼ D
is another very ample divisor, then h D − h E = O(1), so the equivalence class of h D only
depends on the class of D in the class group of X. Finally, if E, D ∈ Div( X ) are very ample,
then h D+ E = h D + h E + O(1).
2.10.2 Theorem (Weil’s “height machine”). Let X be a nice variety over a number field k. Then
¯
there exists a unique homomorphism h : Pic( X ) → R X (k) /O(1) such that
1. If D is very ample, then h D = h ◦ φD + O(1). (normalization)
2. For a morphism φ : X → Y of nice k-varieties, we have h X,φ∗ D = hY,D ◦ φ + O(1) for all
D ∈ Div(Y ). (functoriality)
3. For all D, h D > O(1) on the complement of the base locus of D. (positivity)
Proof. The existence and uniqueness of h follows easily from the above remarks. For a proof
of 2 and 3, see [BG06, §2.4].
2.11
N´eron-Tate heights
The case we are interested in is when X = A is an abelian variety over k. For D ∈ Div( A),
n ( n −1)
n ( n +1)
one can show (see [Mila, I.5.4]) that [n]∗ D ∼ 2 D + 2 [−1]∗ D. In particular, if D is
symmetric (that is, [−1]∗ D = D) then [n]∗ D ∼ n2 D. (There are plenty of symmetric divisors:
if D is arbitrary, D + [−1]∗ D is symmetric). By theorem 2.10.2, if D is symmetric, we have
h A,D ◦ [n] + O(1) = h A,[n]∗ D + O(1)
= h A,n2 D + O(1)
= n2 h A,D + O(1)
So for all x ∈ A(k¯ ), we have h A,D (n · x ) = n2 h A,D ( x ) + O(1). In other words, repeated
addition gives a quadratic growth in the “complexity” of a point x ∈ A(k¯ ).
Fix a symmetric divisor D ∈ Div( A). Define the canonical (or N´eron-Tate) height associated
to D, as a function b
h D : A(k¯ ) → R, by
1
b
h D ( a) = lim n h D (2n · a).
n→∞ 4
Let’s show that the limit defining b
h D exists. Our previous discussion shows that |h D (2 ·
x ) − h D ( x )| 6 C for some constant C depending only on A. We write this as h D (2 · x ) =
h D ( X ) + O(1), keeping in mind that this O(1) only depends on A. Note that
h D ( 2i · x )
h (2 · (2i−1 · x ))
4h (2i−1 · x ) + O(1)
h ( 2i − 1 · x ) O ( 1 )
= D
= D
= D i −1
+ i .
i
i
i
4
4
4
4
4
33
Thus, for any n > m, we have
n
i
i −1 · x ) h D (2n · x ) h D (2m · x ) 6 ∑ h D (2 · x ) − h D (2
−
4n
4m
4i
4i − 1
i = m +1
n
h ( 2i − 1 · x ) O ( 1 ) h D ( 2i − 1 · x ) = ∑ D i−1
+ i −
2
4
2i − 1
i = m +1
n
6
C
,
i
4
i = m +1
∑
∞
which shows that {4−n h D (2n · x )}n=1 is a Cauchy sequence. So the limit defining b
hD (x)
exists. If we set m = 0 and let n → ∞, we see that h D = b
h D + O (1).
If D is antisymmetric, we define b
h D ( x ) = limn→∞ 21n h D (2n · x ) The same proof shows
¯
that b
h D is well-defined. We can extend b
h to a function b
h : Div( A) → R A(k) by setting
1 b
b
hD =
h D+[−1]∗ D + h D−[−1]∗ D .
2
It turns out that b
h D depends only on the class of L ( D ) in Pic( A), so we will think of b
h as a
¯
b
function h : Pic( A) → R A(k) .
For c ∈ Pic( A), the N´eron-Tate height b
hc has many nice properties, for example
b
h c ( n · x ) = n2 b
hc ( x )
b
hc ( x + y) + b
hc ( x − y) = 2 b
hc ( x ) + b
hc (y)
These are easy corollaries of the following lemma.
2.11.1 Lemma. Let A be an abelian variety over a number field k, and let c ∈ Pic( A). Then the
function h·, ·i : A(k¯ ) × A(k¯ ) → R defined by
h x, yi = b
hc ( x + y) − b
hc ( x ) − b
hc (y)
is bilinear.
Proof. We are trying to prove that for x, y, z ∈ A(k¯ ), we have
b
hc ( x + y + z) − b
hc ( x + y) − b
hc (z) = b
hc ( x + z) − b
hc ( x ) − b
hc (z) + b
hc (y + z) − b
hc (y) − b
hc (z) .
This is equivalent to
−b
hc ( x + y + z) + b
hc ( x + y) + b
hc ( x + z) + b
hc (y + z) − b
hc ( x ) − b
hc (y) − b
hc (z) = 0.
(1)
Now we use the fact that for a morphism f : A → B of abelian varieties, one has b
h f ∗c = b
hc ◦ f .
S
For a finite set S and σ ⊂ S, write πσ : A = ∏S A → A for the map ( xs )s∈S 7→ ∑s∈σ xs .
With this, we can rewrite equation (1) as
∑
(−1)|σ| b
hc ◦ πσ = 0.
σ ⊂{1,2,3}
34
By the functoriality of b
h, it is enough to show that one has
∑
(−1)|σ| πσ∗ = 0.
σ ⊂{1,2,3}
This is just the “theorem of the cube” (theorem 2.11.2).
The basic idea in this proof comes from the proofs of theorems 8.6.11 and 9.2.8 in [BG06].
2.11.2 Theorem. Let A be an abelian variety over a field k. For any finite set S with #S > 3, one
has
∑ (−1)|σ| πσ∗ = 0
σ⊂S
as a map Pic( A) →
Pic( AS ).
Proof. For #S = 3 this is just [vdGM, 2.7]. The general case follows easily by induction.
hc > 0 and for all x ∈ A(k¯ ), b
hc ( x ) = 0
If c is the class of a very ample line bundle, then b
if and only if x is torsion. Indeed, if x is torsion, then for some n > 1, we have n2 b
hc ( x ) =
b
hc (nx ) = b
hc ( x ), which implies b
hc ( x ) = 0. On the other hand, if b
hc ( x ) = 0, then b
hc (n · x ) = 0
for all n. The set { x, 2 · x, 3 · x, . . . } is contained inside A( L), where L = k( x ) is the finite
extension of k generated by (the coordinates of) x. Thus N · x ⊂ { a ∈ A( L) : b
h c ( a ) 6 0},
which we know is a finite set. It follows that x is torsion.
2.11.3 Example. Let E/Q be an elliptic curve. Then a divisor on E is just a formal sum
∑ x∈E n x · x. Of course, E has a canonical point – the origin O, so for each n ∈ Z, we can
hnO = nb
hO , we don’t lose any
consider the N´eron-Tate height associated with nO. Since b
information by restricting ourselves to n = 1, but since O isn’t very ample, we can only
directly compute hnO starting with n = 2.
For n = 2, the Riemann-Roch theorem tells us that `(2O) = 2, i.e. H0 (L (2O)) = k ⊕ kx,
where x is a rational function on E. For E with chosen model y2 = x3 + ax + b, x the
rational function corresponds to the map E → P1 given by ( x : y : 1) 7→ ( x : 1). Thus
h2O ( x : y : 1) = hQ ( x ).
For n = 3, the divisor 3O is actually very ample. By the Riemann-Roch theorem,
`(3O) = 3, so H0 (L (3O)) = k ⊕ kx ⊕ ky, where ( x : y : 1) : E → P2 gives a Weierstrass
embedding. That is, there exist a, b such that y2 = x3 + ax + b, and φ3O : E → P2 induces
an isomorphism between E and the zero set of y2 = x3 + ax + b in P2 . One has b
h3O (e) =
h Q ( x ( e ) : y ( e ) : 1) + O (1).
Once again, let c be the class of a symmetric, very ample line bundle on an abelian
variety A/k. The function
h x, yic =
1 b
hc ( x + y) − b
hc ( x ) − b
hc (y)
2
is bilinear, and descends to A(k¯ )/A(k¯ )tors . We can even tensor with R to get a bilinear form
on A(k )R = A(k) ⊗ R. The form h·, ·ic is positive-definite on A(k )R . Thus A(k)R with the
pairing h·ic is isomorphic to Rn with the standard inner product, where n = rk A. The
35
group A(k)/A(k)tors is a lattice in A(k)R , and we can define the regulator of A with respect
to c to be the volume of the fundamental domain of A(k)/A(k)tors in A(k )R . That is,
Regc ( A) = det h xi , x j ic i,j where { xi } is a Z-basis for A(k )/A(k)tors .
If A is an arbitrary abelian variety, there is no natural “good choice” for a canonical
divisor on A. On the other hand, if A = J = Jac C for some nice curve C of genus g with
C (k ) 6= ∅, then we can use the induced embedding j : C → J to define a canonical divisor
on J. Let Θ = j(C ) + · · · + j(C ) be the ( g − 1)-fold sum of C in J, and consider Θ as a divisor
on J. We call Θ, and the induced class θ in Pic( J ), the theta-divisor of J. Note that if J = E
is an elliptic curve with origin O, the theta-divisor is just O. It turns out that θ is ample
(see [BG06, 8.10.22]), and thus h·, ·iθ is positive-semidefinite.
For any c ∈ Pic( J ) for which
p
1/2
b
h·, ·ic is positive-semidefinite, set | x |c = h x, x ic = hc ( x ) . The following theorem is very
deep.
2.11.4 Theorem (Vojta’s inequality). Let C be a nice curve of genus g > 2 over a number field k.
Let J = Jac C, and let θ be the theta-divisor on J. Then there are effectively computable constants
λ, λ0 depending only on C and a chosen x0 ∈ C (k¯ ) such that for x, y ∈ C ,→ J, if | x |θ > λ and
|y|θ > λ0 | x |θ , then
3
h x, yiθ 6 | x |θ · |y|θ .
4
Proof. This is hard – see [BG06, 11.9.1]. The constant
would hold for any constant in the interval ( g−1/2 , 1].
3
4
is not important, as the theorem
2.11.5 Theorem (Faltings). Let C be a nice curve of genus g > 2 over a number field k. Then C (k)
is finite.
Proof. We can assume C (k) 6= ∅, so the choice of x0 ∈ C (k) gives us an embedding
j : C ,→ J = Jac C defined over k. Let V = J (k ) ⊗Z R. By Mordell-Weil, this is a finitedimensional vector space, and its rank is just the (algebraic) rank of J. The induced pairing
h·, ·iθ on V is now a nondegenerate positive-definite bilinear form, so V is isomorphic as a
Hilbert space to Rn with the standard dot product. Note that by basic properties of heights,
S is a discrete subset of V.
For nonzero u, v ∈ V, the angle between u and v is the unique number ϕ(u, v) in [0, π ]
satisfying
hu, viθ
cos ϕ(u, v) =
.
|u|θ |v|θ
For a nonzero v and fixed α ∈ (0, π ], define the cone Γv,α = {u ∈ V r 0 : ϕ(u, v) < α}. This
is an open subset of V that is stable under scaling by R+ . Let S be the image of C (k) in
J (k )/J (k)tors ,→ V. Since J (k)tors is finite, we only need to show that S is finite.
Let ψ = 12 cos−1 ( 34 ), and for nonzero v ∈ V, let Uv = Γv,ψ . We claim that Uv ∩ S is finite.
If it is not, then there exist x, y ∈ S ∩ Uv such that | x |θ > λ and |y|θ > λ0 | x |θ . By Vojta’s
theorem, cos ϕ( x, y) 6 34 , so ϕ( x, y) > cos−1 ( 43 ). This forces ϕ( x, u) > ψ, a contradiction.
The sets {Uv : v ∈ V r 0} form an open cover of V. Since the unit sphere {v ∈ V : |v| =
1} is compact, there is a finite list v1 , . . . , vn ∈ V such that V = Uv1 ∪ · · · ∪ Uvn . Since each
Uvi ∩ S is finite, the set S is finite.
36
This proof essentially gives an effectively computable upper bound for #C (k ). Indeed,
since the dimension of V can be bounded above (for instance, by computing certain Selmer
groups) it is possible to give an effectively computable upper bound for #C (k ). On the other
hand, the set C (k ) is not known to be effectively computable, because our proof does not
give a bound for the heights of points in C (k).
3
3.1
Curves and abelian varieties over finite fields
Tate modules
Let’s start with some motivation. Let A be an abelian variety of dimension d over C. We
have seen that G = A(C) is a compact connected complex Lie group. Moreover, G is a torus,
i.e. is of the form V/Λ for some complex vector space V and lattice Λ ⊂ V. We can realize
G as a torus quite explicitly.
Let g = T0 G be the Lie algebra of G. Since G is commutative, the exponential map
exp : g → G is a group homomorphism, and so its image is open (since the map T0 g → G is
surjective). Since G is connected, the image of exp : g → G is all of G, and so we have an
exact sequence
0
/Λ
/g
exp
/G
/0
There is a natural isomorphism Λ ' H1 ( G, Z). Indeed, we can identify g with the set of
one-parameter subgroups of G, and such a subgroup has trivial exponential if and only if
it is of the form R · λ for some λ ∈ Λ. But those are exactly the one-parameter subgroups
that are also closed curves, hence Λ '
R H1 ( G, Z). Alternatively, we can use the pairing
H1 ( G, Z) × g∨ → C given by hσ, ω i = σ ω to define H1 ( G, Z) → g, and show that it is an
injection with image the kernel of the exponential map.
For any φ ∈ End( G ), we have an induced map φ∗ on H1 ( G, Z). Tensoring with Q, we
this gives a representation
End( G ) → EndQ H1 ( G, Q) ' M2d (Q)
This lets us take the characteristic polynomial Pφ ∈ Q[t] of φ for any φ ∈ End( G ). In fact,
the degree 2d polynomial Pφ is an element of Z[t] because φ fixes the lattice H1 ( G, Z) inside
H1 ( G, Q).
Since G = g/Λ, we have G [n] = Λ/n, and thus there are natural isomorphisms G [n] '
H1 ( G, Z/n) for all integers n > 2. We will think of G [n] as a kind of algebraic avatar
for H1 ( G, Z/n). Unfortunately, there is no algebraic analogue of H1 ( G, Z), but there is a
b = lim Z/n be the profinite completion of Z. The isomorphisms
good substitute. Let Z
←−
b
G [n] ' H1 ( G, Z/n) are compatible, so we get an isomorphism of Z-modules
b ).
TG = lim G [n] → lim H1 ( G, Z/n) = H1 ( G, Z
←−
←−
b = ∏ p Z p , we will consider TG only one p-part at a time. This is important because
Since Z
if A is an abelian variety over a field of characteristic p > 0, the groups A[ pe ] are not well
behaved, so the p-part of TA = lim A[n] should be excluded.
←−
37
Let A be an abelian variety of dimension d over an algebraically closed field k. For the
moment, we will think of A[n] as a scheme – namely the fiber product
/0
A[n]
A
n
/A
If n is invertible in k, then we will have A[n] ' (Z/n)⊕2d . However, if n = p and k has
characteristic p, then for some r 6 d, there will be an isomorphism of group schemes
2( d −r )
A[ p] ' (Z/p)r × α p
× µrp .
Here α p ( R) = {r ∈ R : r p = 0} and µ p ( R) = {r ∈ R : r p = 1} for any F p -algebra R. One
might hope that A[ pe ] can be described just as easily for e > 1, but this is false. The group
schemes A[ pe ] can exhibit very complicated behavior as e → ∞. Indeed, their “formal
inductive limit” A[ p∞ ], called the Barsotti-Tate group of A, carries quite a lot of information
about A.
For the rest of this section, fix a prime ` that is invertible in k. The groups A[`n ] come
with natural surjections A[`n+1 ] → A[`n ] given by x 7→ ` · x.
3.1.1 Definition. Let A be an abelian variety over a field k. The `-adic Tate module of A is
T` A = lim A[`n ] = {( a1 , a2 , . . . ) : an ∈ A[`n ], ` xn+1 = xn }
←−
The Tate module T` A is a free Z` -module of rank 2d. We write V` for T` ( A) ⊗Z` Q` ; this
is a Q` -vector space of dimension 2d. The main idea is that A[`n ], T` ( A) and V` ( A) are
algebraic analogues of H1 ( A, Z/`n ), H1 ( A, Z` ) and H1 ( A, Q` ).
Two things act on T` A. The ring End( A) of endomorphisms of A defined over k (as an
¯ )
abelian variety) acts on each A[`n ], and hence on T` and V` . Also, the group Gk = Gal(k/k
n
acts compatibly on each A[` ], so we get a representation
ρ A,` : Gk → GL( T` ) ' GL(2d, Z` ).
The action of End( A) and Gk commute, so we actually have a representation
ρ A,` : End( A)JGk K → M2d (Z` ).
3.1.2 Theorem (Faltings). Let A, B be abelian varieties over a finitely generated field k. Then the
natural map
hom( A, B) ⊗ Q` → homQ` JGk K (V` A, V` B)
is an isomorphism.
It follows that the functor V` : AbVark ⊗ Q` → RepQ` ( Gk ) is fully faithful.
3.1.3 Corollary. Abelian varieties A, B over a finitely generated field k are isogeneous if and only if
V` A and V` B are isomorphic as Q` JGk K-modules.
38
If k has characteristic p, set Tp A = 0. We define the “global Tate module” of A to be
TA =
∏ Tp A.
p
b
b 2d . It turns out that
This is a Z-module
that is isomorphic (if k has characteristic zero) to Z
TA, as a Gk -module, has a natural interpretation in terms of e´ tale fundamental groups. If
U → Aks is an e´ tale cover of Aks = A ⊗ ks , then it turns out that U can be given the structure
of an abelian variety in such a way that U → Aks is an isogeny. We can use a polarization of
Aks to majorize U → Aks by an isogeny Aks → Aks , which must be of the form [n] for some
n ∈ k× . It follows that π 1 ( Aks ) ' TA, though see [vdGM, 10.37] for an actual proof. The
standard exact sequence for fundamental groups:
/ π1 ( A k s )
1
/ π1 ( A )
/ Gk
/1
induces a representation ρ : Gk → Aut (π1 ( Aks )). It turns out that this is exactly the action
of Gk on TA, so that we get a canonical isomorphism
π1 ( A) ' Gk nρ TA.
In particular, for each prime ` invertible in k, we have an isomorphism π1 ( A)(`) ' T` A nρ A,`
Gk , where (−)(`) denotes taking pro-` completion.
It is possible to define the `-adic representation ρ A : Gk → GL(2d, Z` ) without introS
ducing Tate modules. Let A[`∞ ] = n>1 A[`n ]. As an abelian group, A[`∞ ] ' (Q/Z)[`∞ ],
which we denote by Z(`∞ ) (such groups are called quasi-cyclic). It is easy to show that
End Z(`∞ ) ' Z` as topological rings, where End Z(`∞ ) is given the subset topology in
∏Z(`∞ ) Z(`∞ ) and Z(`∞ ) has the discrete topology. Thus Aut A[`∞ ] ' GL(2d, Z` ). The
induced homomorphism Gk → GL(2d, Z` ) is continuous, and (after a change of basis) is the
same as ρ A,` .
3.2
Endomorphisms of abelian varieties
Let A be an abelian variety of dimension d > 1 over a field k, and fix a polarization
λ : A → A∨ = Pic◦A .
3.2.1 Definition. An abelian variety A is simple if the only abelian subvarieties of A defined over
k are 0 and A.
Note that this definition depends on k; it is possible for a simple variety to become
non-simple after base change. We say A is geometrically simple if A L is simple for all fields
L ⊃ k (equivalently, if Ak¯ is simple).
3.2.2 Theorem (Poincar´e). If B ⊂ A is a nontrivial abelian subvariety, then there is another
abelian subvariety C ⊂ A such that the morphism B × C → A, (b, c) 7→ b + c, is an isogeny.
Proof. This is taken from [Mila, I.10.1]. Let i : B ,→ A be the inclusion, and let i∨ : A∨ → B∨
be its dual map. Let C = ker(i∨ ◦ λ)◦ be the connected component of the identity in
i∨
the kernel of A −
→ A∨ −
→ B∨ . Since dimension is additive on exact sequences of abelian
varieties, we see that dim C > dim A − dim B∨ = dim A − dim B. It turns out that (λ ◦ i∨ )| B
is a polarization of B, so B ∩ C is finite, hence dimension zero. It follows that B × C → A is
an isogeny.
λ
39
There need not be a complement to B “one the nose.” That is, there may not exist C
such that A ' B × C. So the category of all abelian varieties is not semisimple, but the
category of “abelian varieties up to isogeny” is. The theorem of Poincar´e implies that for any
abelian variety A, there exist simple abelian varieties B1 , . . . , Br such that A is isogenous to
B1 × · · · × Br . Standard arguments show that this decomposition is unique up to ordering
and isogeny.
Denote by AbVariso
k the localisation of the category of abelian varieties over k by the
collection of all isogenies. Morphisms from A to B in AbVariso
k are (equivalence classes) of
formal factorizations
f
φ
/B
Ao
C
where φ is an isogeny. From [vdGM, 5.12], there exists an integer n and an isogeny ψ : A →
C such that φψ = n. The commutative diagram:
CO
φ
A _o
f
ψ

A
n
n
fψ
/B
?
fψ
A
shows that the factorization f ◦ φ−1 is equivalent to ( f ψ) ◦ n−1 . It follows that AbVariso
k is
n
equivalent to the localisation of AbVark at all isogenies of the form A −
→ A. Thus
homAbVariso ( A, B) = homAbVark ( A, B) ⊗ Q.
k
Write End( A) = homAbVark ( A, A) for ring of endomorphisms of A as an abelian variety
over k. If n ◦ f = 0 for some f : A → A, then the image of f is contained in the (discrete)
subvariety A[n] ⊂ A. Since f ( A) is irreducible, we have f = 0. Thus End A is torsion-free,
so it embeds into the ring
End◦ A = EndAbVariso ( A) = End( A) ⊗ Q.
k
Since End◦ A can be defined in terms of the category AbVariso
k , it only depends on the isogeny
∼
class of A. In particular, an isogeny f : A → B induces an isomorphism End◦ A −
→ End◦ B.
There is an obvious inclusion End◦ ( A)× ⊃ Aut( A), and f ∈ End◦ ( A)× if and only if n f is
an isogeny for some n.
The group End( A) has rank at most (2g)2 . Over C, this is obvious, given the faithful
action of End( A) on H1 ( A(C), Z) ' Z2g . The general case in characteristic zero follows
from the Lefschetz principle, or one can prove it directly as in [Mum08, IV.18.3].
Poincar´e’s reducibility theorem shows that an abelian variety A is isogenous to a product
e
B11 × · · · × Brer , where the Bi are simple and pairwise non-isogenous over k. A standard
argument (which works in any semisimple abelian category) shows that the rings Di =
End◦ ( Bi ) are division algebras, and moreover
End◦ ( A) '
r
∏ M e i ( Di ) .
i =1
40
Thus the ring End◦ ( A) is semisimple, so we know that both the ei and the Di are uniquely
determined by A. It follows that the decomposition type of A can be inferred from the ring
End◦ ( A).
3.2.3 Example. If A is two-dimensional, then A is either simple or a product of elliptic
curves. If A is simple, then End◦ ( A) is a division algebra. If A is isogenous to E × E0 where
E and E0 are not isogenous, then End◦ ( A) ' End◦ ( E) × End◦ ( E0 ). On the other hand, if A
is isogenous to E × E, then End◦ ( A) ' M2 (End◦ ( E)).
We have remarked several times that there is no good “algebraic definition” of H1 ( X, Q)
for varieties X over a general field. Serre constructed an example of a variety over F p which
shows not only that there isn’t a good definition, but that such an homology group cannot
exist.
First, we need to discuss Frobenius morphisms. Let p be a prime, q a power of p, and Fq
the field with q elements. Let X be a nice variety over Fq . Choose an embedding X ,→ PnFq .
Write Φ X : X → X for the map that on projective coordinates is
q
q
( a0 : · · · : a N ) 7 → ( a0 : · · · : a N ).
This is well-defined because x 7→ x q is a ring homomorphism on Fq -algebras. So if X is the
zero-set of homogeneous polynomials f i ∈ Fq [ x0 , . . . , xn ] and a ∈ X, then
q
q
f i ( a0 : · · · : a n ) = f i ( a0 : · · · : a n ) q = 0
so Φ X ( a) ∈ X. Note that we have to raise to the q-th power rather than the p-th power
because x 7→ x p does not fix all elements of Fq .
There is a scheme-theoretic definition of Φ X that allows us to talk about the Frobenius
on arbitrary schemes over Fq . For a scheme X/Fq , let Φ X : X → X be the the identity
on the underlying topological space of X, with Φ∗X : OX → OX the map x 7→ x q . If
X = V ( f 1 , . . . , f r ) ⊂ AnFq , then it is easy to see that these definitions are equivalent, for both
correspond to the usual Frobenius map on Fq [ x1 , . . . , xn ]/( f 1 , . . . , f r ).
It may seem confusing that the scheme-theoretic Frobenius is the identity on points,
while our coordinate-wise definition raises elements to the q-th power. Actually, there is
no contradiction. A point x ∈ X with coordinates in Fqn should be thought of as a map
x : Spec(Fqn ) → X. Raising those coordinates to the q-th power corresponds to composing
x with Φ X , or, equivalently, precomposing x with the automorphism x 7→ x q of Fqn .
3.2.4 Example (Serre). Choose a prime p ≡ 3 (mod 4). Let E/F p be the elliptic curve
y2 = x3 − x. It is possible to give an explicit description of the division ring D = End◦ ( E).
Fix α ∈ F p2 such that α2 = −1. Then φ : E → E, defined by ( x, y) 7→ (− x, αy), is a
well-defined endomorphism. One has φ2 ( x, y) = ( x, −y), so φ2 = −1 in D. Moreover,
Φ E φ( x, y) = (− x p , α p y p ) = −(− x p , α p y p ) = −φΦ E ( x, y),
so φΦ = −Φφ in D. It turns out that Φ2 = − p, so D is the standard quaternion algebra
with parameters (−1, − p), i.e.
D
E
D = Q i, j, k : i2 = −1, j2 = − p, ij = k, ji = −k .
41
Now can show that “V = H1 ( E, Q)” cannot exist. If it did, we would expect it to be
functorial in E, so there would be an action of D on V. But we would also want V to be
two-dimensional over Q. Since D is four-dimensional over Q, this would mean that V
has D-dimension 1/2, which is nonsense. Alternatively, we would have an (injective) ring
homomorphism D → End(V ) ' M2 (Q). Since both rings have the same dimension, this
would yield D ' M2 (Q). But M2 (Q) has zero-divisors and D does not, so this cannot be
the case.
Let A and B be abelian varieties over a field k. If f : A → B is an isogeny, the degree of f is
[k( A) : k( B)] via the embedding f ∗ : k( B) → k( A) induced by f . Equivalently, deg( f ) is the
order of ker( f ), considered as a group scheme. The extension k ( B)/k ( A) is separable if and
only if f is e´ tale, if and only if ker( f ) is e´ tale over k. If f is e´ tale, then the scheme-theoretic
order of ker( f ) is equal to # ker( f )(k¯ ).
3.2.5 Example. Let n 6= 0, and consider the isogeny [n] : A → A that is multiplication by n.
One can show that deg[n] = n2d , where d = dim A. To see this, note that A[n] = ker[n] '
(Z/n)⊕2d , at least if n ∈ k× .
3.2.6 Example. If A is defined over a finite field Fq , let Φ = Φ A : A → A be the Frobenius.
With some difficulty, one can show that Φ A is purely inseparable (i.e. k ( A)/Φ∗ k( A) is
purely inseparable) of degree qd . Alternatively, ker(Φ)(k¯ ) = 0, so ker( f ) is “as far from
e´ tale as possible.”
Define a map deg : End( A) → Z as follows. If f is an isogeny, let deg( f ) be as above. If f
is not an isogeny, set deg( f ) = 0. Given f , g ∈ End( A), one has deg( f g) = deg( f ) · deg( g).
If one of f , g is not an isogeny this is obvious, and if they are both isogenies, this follows from
the multiplicativity of the degree of field extensions. As a special case, deg(n f ) = n2d deg( f ).
Using the multiplicativity of the degree map, we can extend it to deg : End◦ ( A) → Q by
deg( f /n) = deg( f )/n2d . The degree map is not additive.
3.2.7 Theorem. Let A be an abelian variety of dimension d. Then the map deg : End◦ ( A) → Q is
a homogeneous polynomial of degree 2d.
By this we mean the following. Choose a Q-basis e1 , . . . , er of End◦ ( A). The theorem
claims that there is a homogeneous polynomial f ∈ Q[ x1 , . . . , xr ] of degree 2d such that
deg (c1 e1 + · · · + cr er ) = f (c1 , . . . , cr ).
It follows that for any α ∈ End( A), there exists a unique polynomial Pα ∈ Q[ x ], called the
characteristic polynomial of α, such that Pα (n) = deg(n − α) for all n ∈ Z. It turns out that the
polynomial Pα is monic, has degree 2d, and has integer coefficients.
3.2.8 Example. Let k be an imaginary quadratic field, o = ok , and let E = C/o. One has
End( E) = o and End◦ ( E) = k. It turns out that the degree map deg : k → Q is the classical
norm map Nk/Q . More generally, let T be a complex torus, written Cd /Λ. Let α be an
endomorphism of T. We can lift α to a linear map e
α : Cd → Cd . One has
End( A) = {φ ∈ EndC (Cd ) : φ(Λ) ⊂ Λ}.
Note that Λ = H1 ( T, Z). It is straightforward to check that
deg(α) = # ker(α) = #(Λ/e
αΛ) = det(e
α, H1 ( T, Z)) = det (α∗ , H1 ( T, Z)) .
Thus deg(n − α) = det(n · 1 − α∗ , H1 ( T, Z)), so Pα = det (t · 1 − α∗ , H1 ( T, Z)).
42
Fix a prime ` ∈ k× . Recall that we have an inverse system of Gk -modules
A[`] o
`
A[`2 ] o
`
A[`3 ] o
···
where here we think of A[`n ] as A(k¯ )[`n ]. The inverse limit is T` A = lim A[`n ], and we
←−
set V` A = T` A ⊗ Q. We call T` A and V` A the `-adic Tate modules associated with A. We
will use V` as an `-adic avatar for H1 ( A, Q` ). In fact, T` A ' H1 ( Ae´ t , Z` )∨ as a Gk -module.
(This is essentially a tautology. For lisse `-adic sheaves F on A, one has H• ( A, F ) =
H• (π1 ( A), F0 ), where F0 is the stalk of F at 0. For F = Z` , this gives H1 ( Ae´ t , Z` ) =
hom( TA, Z` ) = ( T` A)∨ .)
3.2.9 Theorem. Let ` be a prime invertible in k. For any α ∈ End( A), we have Pα = det(t · 1 −
α∗ , T` A).
Proof. This is nontrivial. See [Mila, I.10.20].
3.2.10 Corollary. For α ∈ End◦ ( A), we have Pα (α) = 0.
Proof. We use the fact (see [Mila, I.10.15]) that the natural map
End( A) ⊗ Z` → EndZ` ( T` A)
is injective. In End( T` A), one has Pα (α) = 0 by the Cayley-Hamilton theorem. Injectivity
tells us that Pα (α) = 0 in End( A) ⊗ Z` , and since End( A) is torsion-free, we see that
Pα (α) = 0 in End( A).
From the theorem, we see that the characteristic polynomial of α acting on T` A is in Z[t],
and is independent of `. This is highly non-obvious. Note that A 7→ T` A is a functor from
abelian varieties over k to Z` -representations of Gk . If k has characteristic p and we want
to assign a p-adic representation of Gk to an abelian variety, we have to work harder. Of
course, Tp A is a Z p -module with a Gk -action, but there examples when rkZ p Tp A < 2 dim A.
In this case, instead of studying Tp A, (essentially the p-adic e´ tale cohomology of A) one
should study the crystalline or rigid cohomology of A. If the base field k is perfect, these
yield W (k)-modules with the “correct” rank. For a brief survey of crystalline cohomology,
see [Ill94], and for an introduction to rigid cohomology, see [LS07].
3.3
Abelian varieties over finite fields
Let A be an abelian variety over Fq , where q is a power of p. For example, A could be the
jacobian of a smooth curve. Recall that there is a distinguished endomorphism Φ = Φ A of A,
called the Frobenius of A. If we embed A into some projective space Pn , then in coordinates
q
q
we have Φ( x0 : · · · : xn ) = ( x0 : · · · : xn ). As a morphism of schemes, Φ is the identity on
the underlying space of A, and is the q-th power map on O A . Let d > 1 be the dimension of
A.
3.4
Characteristic polynomial of Frobenius
Recall that there is a polynomial PA = PΦ A ∈ Z[t] of degree 2d such that PA (n) = deg(n −
Φ A ) for all integers n. Also, if ` is a prime not dividing q, PA is the characteristic polynomial
of Φ∗ as a Z` -linear map Φ∗ : T` A → T` A. Thus we have a map P : AbVarFq → Z[t].
43
3.4.1 Theorem. If A and B are isogenous abelian varieties over Fq , then PA = PB .
Proof. Let f : A → B be an isogeny. For each e, we have an exact sequence
0
/C
/ A[`e ]
f
/ A[`e ]
/ 0,
where C is independant of e if e 1 (since C ⊂ ker( f ), a finite group). At the level of
Tate modules, if f ∗ ( x ) = 0, then f ( xi ) = 0 in B[`i ] for all i, so each xi ∈ C. But C is a finite
group, so it contains no nonzero sequences ( xi ) with ` xi+1 = xi for all i. Thus x = 0, so
f ∗ : T` A → T` B is injective. Since T` A and T` B have the same rank, the map f ∗ : V` A → V` B
is an isomorphism. (So far we have shown that V` A is isogeny-invariant over any field.) It
is easy to check that f ◦ Φ A = Φ B ◦ f . It follows that f ∗ ◦ (Φ A )∗ = (Φ A )∗ ◦ f , and from this
we see that PA = PB .
Alternatively, we could note that T` : AbVark → RepZ` ( Gk ) naturally descends to a
functor V` : AbVark ⊗ Q → RepZ` ( Gk ) ⊗ Q. Since AbVark ⊗ Q = AbVariso
k and RepZ` ( Gk ) ⊗
Q = RepQ` ( Gk ), this gives a functor
V` : AbVariso
k → RepQ` ( Gk ).
Here and elsewhere, if A is an additive category, we write A ⊗ Q for the localization of A
at the class of morphisms n · 1 A , n ∈ Z r 0.
It turns out that PA (0) = deg(−Φ A ) = deg(Φ A ) = q g ; this can be verified by a direct
(but messy) computation. A more interesting quantity is PA (1) = deg(1 − Φ). Let f = 1 − Φ.
We claim that f : A → A is e´ tale. Essentially, the map d f : Lie( A) → Lie( A) is the
identity, and this comes down to the fact that the differential of Frobenius is zero (morally,
d q
q−1 = 0). Thus P (1) = # ker( f ). Note that f ( x ) = 0 if and only if Φ ( x ) = x, i.e.
A
dx x = qx
x ∈ A(Fq ). So PA (1) = #A(Fq ). This has a surprising consequence: the quantity #A(Fq ) is
isogeny-invariant! This fails badly over number fields.
Let’s specialize to elliptic curves. Let E be an elliptic curve over Fq . We have PE (t) = t2 −
at + q, for some integer a = a( E). One calls a the trace of Frobenius. Indeed, a = tr(Φ E,∗ , T` )
for all ` 6= p. Since #E(F p ) = PA (1) = 1 − a + q, we could have defined a = 1 + q − #E(Fq ).
Since #P1 (Fq ) = q + 1, we can think of a as an arithmetically interesting “error term”
measuring how much E(Fq ) and P1 (Fq ) differ.
√
3.4.2 Theorem (Hasse bound). We have | a| 6 2 q, i.e.
√
|#E(Fq ) − (q + 1)| 6 2 q.
Proof. Let n ∈ Q, and take PE (n) = n2 − an + q = deg(n − π E ) > 0. Thus t2 − at + q > 0
√
for all t ∈ R. Basic calculus tells us that a2 − 4q 6 0, i.e. | a| 6 2 q.
The following graph of a p ( E) for E : y2 = x3 + x shows that while for “most” primes
√
one has a p ( E) = 0, there are still a lot with a p ( E) = b2 pc.
44
Hasse bound for y2 = x3 + x
80
y =2x1/2
60
40
20
500
1000
1500
2000
An equivalent way of formulating the Hasse bound is to claim that the complex roots of
√
PE have absolute value q. Indeed, one computes
r
a ± p a2 − 4q a ± p4q − a2 i √
4q − a2
a2
+
= q.
=
=
2
2
4
4
If 6 is a unit in Fq , then we can write E as y2 = x3 + ax + b. It follows that
#E(Fq ) = q + 1 +
∑
x ∈Fq
x3 + ax + b
q
,
where qc = 0 if c = 0, 1 if c is a square in Fq , and −1 otherwise.
√
A special case of the Honda-Tate theorem says that for each integer a with | a| 6 2 q,
there is an elliptic curve E over Fq with a( E) = a. So the Hasse bound is sharp. Moreover,
we will see that a( E), and hence PE , determines E up to isogeny.
3.4.3 Example. Fix a prime p ≡ 3 (mod 4), and consider the elliptic curve E over F p given
by the equation y2 = x3 − x. Recall that End◦ ( EF p2 ) was a quaternion algebra over Q, of type
(−1, − p). We used the fact that Φ2E = − p, but didn’t prove this. We know that PE (Φ E ) = 0,
i.e. Φ2E − aΦ E + p, so all we need to do is show that a = 0. Since a = p + 1 − #E(F p ), we
just need to show that #E(F p ) = p + 1. The two sets
2
{ x ∈ F p : x3 − x ∈ (F×
p) }
{ x ∈ F p : x3 − x ∈
/ ( F p )2 }
are bijective via x 7→ − x. Call their common cardinality m. Since #{ x ∈ F p : x3 − x = 0} = 3,
we have p = 2m + 3, so #E(F p ) = 2m + 3 + 1 = p + 1.
45
3.5
Zeta functions
Let A be an abelian variety over Fq . We can consider its base extension AFqn , which is an
abelian variety over Fqn . It has a characteristic polynomial PAF n . Write
p
2d
PA (t) =
∏ ( x − ωi ),
i =1
where the ωi ∈ C.
n
3.5.1 Theorem. We have PAF n (t) = ∏2d
i =1 ( x − ωi ).
q
A0
= AFqn , and fix a prime ` - q. We have equality V` A = V` A0 ; write V for this
Proof. Let
vector space. The Frobenius on A induces a linear map Φ A,∗ : V → V with characteristic
polynomial PA . We have ΦnA = ΦnA0 as morphisms on A0 , so Φ A0 ,∗ = ΦnA,∗ , whence the
result.
Let’s look at the case where E/Fq is an elliptic curve. We have PE (t) = t2 − at + q,
where a = a( E) ∈ Z is the trace of Frobenius acting on T` E for any ` - q. Recall that
√
since PE (1) = #E(Fq ), we get a = q + 1 − #E(Fq ). Moreover, we showed that | a| 6 2 q.
Equivalently, the roots of PE have absolute value q1/2 .
3.5.2 Theorem (Weil). Let A be an abelian variety over Fq . Then the roots of PA in C have absolute
value q1/2 .
Back to the case of elliptic curves. We can factor PE (t) = (t − ω1 )(t − ω2 ) where
ω1 + ω2 = a and ω1 ω2 = q. We know that
PEF n (t) = ( x − ω1n )( x − ω2n ) = x2 − (ω1n + ω2n )t + qn ,
q
so #E(Fqn ) = qn + 1 − (ω1n + ω2n ). Recall that the zeta function of E is the formal power series
!
Z
dt
Z ( E, t) = exp
∑ #E(Fqn )tn t .
n>1
We can compute:
Z
∑ #E(Fqn )tn
n>1
dt
tn
= ∑ #E(Fqn )
t
n
n>1
(qt)n
tn
(ω t)n
(ω t)n
+∑ −∑ 1
−∑ 2
n
n
n
n
= − log(1 − qt) − log(1 − t) + log(1 − ω1 t) + log(1 − ω2 t)
(1 − ω1 t)(1 − ω2 t)
= log
.
(1 − t)(1 − qt)
=∑
It follows that
Z ( E, t) =
1 − at + qt2
.
(1 − t)(1 − qt)
Note that the numerator is t2 PE (1/t).
46
3.5.3 Example. Let E/F5 be the elliptic curve y2 = x3 + x + 2. One can check directly
that E(F5 ) = {0, (1, ±2), (4, 0)}. Thus #E(F5 ) = 4 = 5 + 1 − a, so a = 2. It follows that
+5t2 . In this case,
PE (t) = t2 − 2t + 5 and Z ( E, t) = (11−−t2t
)(1−5t)
d
∑ #E(F5n )tn = t dt log Z(E, t) = 4t + 32t2 + 148t3 + 640t4 + 3044t5 + O(t6 ).
n>1
One puts ζ ( E, s) = Z ( E, q−s ). The fact that the zeros of PE have absolute value q1/2
implies that the zeros of ζ ( E, s) have real part 12 . So the “Riemann hypothesis” for ζ ( E, s) is
a theorem!
3.6
Honda-Tate theory
¯ such that under any embedding
3.6.1 Definition. A q-Weil number is an algebraic integer ω ∈ Q
1/2
Q(ω ) ,→ C, the image of ω has absolute value q .
Two q-Weil numbers ω, ω 0 are conjugate if there exists a field isomorphism σ : Q(ω ) →
¯ Equivalently, ω
Q(ω 0 ) such that σ (ω ) = ω 0 , i.e. ω and ω 0 lie in the same GQ -orbit in Q.
0
and ω are conjugate if they have the same minimal polynomial over Q.
It is easy to check if a given algebraic integer ω is q-Weil. Let k = Q(ω ), and classify
embeddings k ,→ C. The number ω is q-Weil if and only if ω ω¯ = q in each such embedding.
3.6.2 Theorem (Honda-Tate). Let A be an abelian variety over Fq .
1. If A is simple, then PA (t) = h(t)e for an irreducible polynomial h ∈ Z[t] and e > 1.
2. The map
{isogeny classes of simple abelian varieties over Fq } → {conjugacy classes of q-Weil numbers}
¯ of PA , is a bijection.
sending A to the set of roots in Q
3. Fix h ∈ Z[t] the minimal polynomial of a q-Weil number. Then there exists a unique e > 1
such that he is PA for a simple A/Fq . In fact, it is the smallest e > 1 such that
• h (0) e > 0
• for every monic Q p -irreducible factor g ∈ Q p [t] of h, we have v p ( g(0)e ) ∈ rZ, where
q = pr .
The injectivity of the map is due to Tate, and the surjectivity is due to Honda.
3.6.3 Corollary. The map
√
{isogeny classes of elliptic curves over Fq } → { a ∈ Z : | a| 6 2 q}
given by E 7→ a( E) = tr(Φ E , T` E) = q + 1 − #E(Fq ) is a bijection.
√
3.6.4 Example. The number 5i is 5-Weil, with minimal polynomial t2 + 5. The conditions
of part 3 of the Honda-Tate theorem show that there exists an abelian variety E over F5 such
that PE = t2 + 5. As an exercise, check that E : y2 = x3 + 1 works.
47
√
3.6.5 Example. The number 5 is 5-Weil with minimal polynomial t2 − 5. It is easy to check
that e = 2, so there is a (simple) abelian variety A over F5 such that PA (t) = (t2 − 5)2 . One
can show that PAF (t) = (t − 5)4 . Since 5 is a 25-Weil number, there exists an elliptic curve
25
E over F25 such that PE (t) = (t − 5)2 . We will find that AF25 is isogenous to E × E.
3.6.6 Lemma. Let A and B be abelian varieties over Fq . Then PA× B = PA × PB .
Proof. On A × B, we have Φ A× B = Φ A × Φ B . Fix ` - q, and consider V` ( A × B) = V` A ⊕ V` B.
On V` ( A × B), we have Φ A× B,∗ = Φ A,∗ × Φ B,∗ . So it comes down to showing that the
characteristic polynomial of f ⊕ g is the product of the respective characteristic polynomials.
But this is obvious.
For any abelian variety A over Fq , the Poincar´e reducibility theorem implies A is isogen
nous to ∏ Bi i , where the Bi are simple and pairwise non-isogenous. The lemma gives
n
PA = P∏ Bni = ∏ PB i , where the PBi are powers of distinct irreducible polynomials. If we
i
i
m
started with PA , we could factor it as PA = ∏ hi i , where the hi are distinct monic irreducible
integral polyomials. For each hi , there is a unique (and computable) ei > 1 such that
e
n
hi i = PBi . Thus PA = ∏ PB i , where ni = mi /ei .
i
3.6.7 Theorem. Let A and B be abelian varieties over Fq . Then A and B are isogenous if and only
if PA = PB .
We can rephrase this as: A and B are isogenous if and only if V` A and V` B are isomorphic
b
GFq = Z-modules.
Let A be a simple abelian variety over Fq . Let D = End◦ ( A) = EndFq ( A) ⊗ Q; this is a
division algebra over Q. Then the field F = Q(Φ A ) is contained in the center of D.
3.6.8 Theorem. Let A be a simple abelian variety over Fq . Then
1. D = End◦ ( A) has center F = Q(Φ A )
2. dimF D = e2 , where PA = he with h irreducible
3. 2 dim A = e · [ F : Q].
The division algebra D can be described explicitly from Φ A . You can give local invariants
that classify it over each Fv , where v ranges over the places of F.
The previous remark merits some explanation. Let k be an arbitrary field. Let Br(k), be
the set of isomorphism classes of division algebras with center k. The set Br(k) has a highly
non-obvious group structure. For D, D 0 ∈ Br(k), there is a unique D 00 ∈ Br(k) such that
D ⊗k D 0 ' Mn ( D 00 ) for some n. Set D 00 = D + D 0 . Under this operation, Br(k ) is an abelian
group, where the inverse of D is opposite algebra D ◦ . It turns out (see [Ser79, X.5]) that
Br(k) is naturally isomorphic to the Galois cohomology group H2 ( Gk , (ks )× ) = H2 (k, Gm ).
If k is a nonarchimedean local field, there is a canonical isomorphism Br(k) → Q/Z, so to
specify a division algebra with center k is the same as giving an element of Q/Z.
Suppose k is a number field. If v is a place of k, then the operation D 7→ D ⊗k k v defines
a homomorphism Br(k) → Br(k v ). It turns out that the image of D under this map is 0 for
all but finitely many v. In fact, we have an exact sequence
0
/ Br(k)
/
M
Br(k v )
v
48
/ Q/Z
/ 0.
This is very deep – at the level of the main theorems of class field theory. For a proof of this
(and the rest of class field theory) see [Sha92, 2.86]. In any case, to give an element of Br(k ),
it is sufficient to give an element of Br(k vi ) = Q/Z for finitely many finite places v1 , . . . , vr
and an element of Br(R) = Z/2 for each real place of k.
3.6.9 Example. Let E be an elliptic curve over Fq with PE = x2 + q. Then Φ2E = −q, so
√
F = Q(Φ E ) ' Q( −q). It follows that e = 1, so End◦ ( E) = Q(Φ E ).
3.6.10 Example. Let E be an elliptic curve over Fq such that PE is reducble. Then PE (t) =
(t − ω )2 , where ω ∈ Z with |ω | = q1/2 . It follows that q is a square and ω = ±q1/2 . Then
Φ E = ω ∈ Z, so F = Q(Φ E ) = Q and e = 2. The division ring End◦ ( E) is a quaternion
algebra (we saw an example earlier).
3.7
Curves and their jacobians
Here and elsewhere we will consider alternating sums
∑(−1)i tr( fi , Hi )
where f • : H• → H• is a graded linear endomorphism of a locally finite-dimensional graded
vector space. We will write tr( f • , H• ), or sometimes just tr( f , H ), for this alternating sum.
Let’s start with some motivation. Let X and Y be smooth oriented manifolds of dimension n. Let f : X → Y be a smooth map that is finite-to-one. For x ∈ X, we can choose
small neighborhoods U, V of x and y = f ( x ) such that U and V are n-balls, and for which
f : U → V is injective. The relative homology Hn (U, U r x ) and Hn (V, V r y) are both
isomorphic to Z with a canonical basis coming from the orientation. With respect to these
bases, f ∗ is multiplication by an integer, called the degree of f at x, and denoted degx f .
3.7.1 Theorem (Lefschetz). Let f : X → X be smooth with finitely many nondegenerate fixed
points. Then we have
∑ degx f = tr( f ∗ , H• (X, Q))
x∈X f
If f is “well-behaved” around each fixed point, the local degrees are all one, so the
theorem reduces to #X f = tr( f ∗ , H• ( X, Q)). If X is a Riemann surface, the alternating sum
is
tr( f ∗ , H0 ( X, Q)) − tr( f ∗ , H1 ( X, Q)) + tr( f ∗ , H2 ( X, Q)).
There is an analogue of the Lefschetz fixed point theorem for e´ tale cohomology. Let X be
a nice variety over a field k, and let f : X → X be an arbitrary morphism. Let ` be a prime
invertible in k. It is proven in [Del77, IV.3.3] that one has
(Γ f · ∆ X ) = tr( f ∗ , H• ( X, Q` )).
Here (Γ f · ∆) is the intersection number of the graph of f and the diagonal as cycles in
X × X. If k = Fq and f = Φ is the Frobenius on X, then ΓΦ and ∆ X intersect transversally
with multiplicity one, so (ΓΦ · ∆ X ) = #X Φ = #X (Fq ). More generally (see [Del77, IV.3.7])
we have
#X (Fqn ) = ∑(−1)i tr(Φn ∗ , Hi ( X, Q` )).
49
If X = C is a nice curve over Fq , we can be much more concrete. The alternating sum
has only three terms:
#C (Fqn ) = tr(Φn , H0 (C, Q` )) − tr1 (Φn , H0 (C, Q` )) + tr(Φn , H2 (C, Q` ))
One can show that the trace on H0 (C, Q` ) is 1, while the trace on H2 (C, Q` ) is qn . Let J be
the jacobian of C. Since H1 (C, Q` )∨ ' V` J as GFq -modules, we have
#C (Fqn ) = qn + 1 − tr(Φn∗ , V` J ).
The following theorem is a straightforward consequence of the Lefschetz fixed point theorem
and Theorem 3.5.1.
3.7.2 Theorem. Let C be a nice curve over Fq of genus g. Let J be its jacobian, and write PJ (t) =
2g
∏i=1 ( x − ωi ). Then
n
#C (Fqn ) = qn + 1 − (ω1n + · · · + ω2g
).
3.7.3 Corollary. |#C (Fqn ) − (qn + 1)| 6 2gqn/2 .
3.7.4 Example. If C is a nice curve of genus zero over Fq , then we know that |#C (Fq ) − (q +
1)| 6 0, so #C (Fq ) = q + 1. In particular, C has a Fq -rational point, whence C ' P1Fq .
3.7.5 Example. If C is a nice curve of genus one over Fq , then we know that #C (Fq ) >
√
√
q + 1 − 2 q = ( q − 1)2 > 0. So once again C has a Fq -rational point, hence C is an elliptic
curve.
It is a good exercise to find a curve C of genus g = 2 for which C (Fq ) = ∅. (do this)
3.7.6 Example.
√ Let C be a nice curve of genus two over F4 . We know that #C (F4 ) 6
4 + 1 + 2 · 2 · 4 = 13. We claim that #C (F4 ) 6= 13, i.e. the Hasse bound is not sharp. To see
this, note that #C (F4 ) = 5 − (ω1 + ω2 + ω3 + ω4 ), where each ωi has absolute value 2. Thus
#C (F4 ) 6 5 + (2 + 2 + 2 + 2), with equality if and only if each ωi = −2. If each ωi = −2,
then we would have PJ = (t + 2)4 . This would imply
#C (F16 ) = 16 + 1 − (ω12 + ω22 + ω32 + ω42 ) = 16 + 1 − (4 + 4 + 4 + 4) = 1
which contradicts C (F4 ) ⊂ C (F16 ).
3.7.7 Example (Hermite curve). Let C be the curve over F p defined by the equation x p+1 +
y p+1 + z p+1 = 0. Then C ⊂ P2 is a nice curve, and #C (F p ) = p + 1. This is because for
( x, y, z) ∈ F3p , we have x p+1 + y p+1 + z p+1 = x2 + y2 + z2 . This is the equation of a genus
zero curve. It turns out that #C (F p2 ) = p3 + 1. One way to prove this is to use the fact that
p+1 is surjective. The curve C has degree p + 1, so its
the map F×
→ F×
p given by a 7 → a
p2
genus is g =
(d−1)(d−2)
,
2
which in our case is
p ( p −1)
.
2
We know that
q
p3 + 1 = #C (F p2 ) 6 p2 + 2g p2 = p3 + 1.
In particular, our bound on #C (F p2 ) is sharp in this case. The fact that #C (F p2 ) = p2 + 1 +
p
2 p2 tells us that PJF (t) = ∏(t − ωi2 ) with each ωi2 = − p. In other words, PJF (t) =
p2
p2
50
√
(t + p)2g . Each ωi = ± pi, so p + 1 = #C (F p ) = ( p + 1) − ∑ ωi tells us that half of the
√
√
ωi = pi, and half are − pi. In other words,
PJ (t) = (t2 −
√
pi ) g (t +
√
pi ) g = (t2 + p) g
Thus J is isogenous to E g , where E is an elliptic curve over F p with a( E) = 0.
3.8
The Weil conjectures
Recall that for a curve C of genus g over Fq , the zeta function of C is the formal power series
!
!
Z
dt
tn
n
Z (C, t) = exp
∑ #C(Fqn )t t = exp ∑ #C(Fqn ) n .
n>1
n>1
By Theorem 3.7.2, there exist q-Weil numbers ω1 , . . . , ω2g such that for all n,
n
#C (Fqn ) = qn + 1 − (ω1n + · · · + ω2g
).
This allows us to compute:
Z (C, t) = exp(− log(1 − qt) − log(1 − t) + log(1 − ω1 t) + · · · + log(1 − ω2g t))
=
(1 − ω1 t) · · · (1 − ω2g t)
.
(1 − t)(1 − qt)
Let J be the jacobian of C. Then the numerator is the “reverse” t2g PJ (1/t) of PJ (t). For any
nice variety X of dimension d over Fq , one can define Z ( X, t) in the same way, and one has
the Weil conjectures. These state that
i +1
(−1) , where the P ( X, t ) are
1. Z ( X, t) is a rational function of the form ∏2d
i
i =0 Pi ( X, t )
integral polynomials with P0 ( X, t) = 1 − t and P2d ( X, t) = 1 − qd t.
2. Z ( X, 1/qn t) = ±qdχ/2 tχ Z ( X, t), where χ = (∆ X · ∆ X ) is the Euler characteristic of X
3. Each Pi ( X, t) = ∏ j (1 − ω j t), where the ω j are qi -Weil.
See [Wei49] for the original (and surprisingly modern) statement of the Weil conjectures. For
a proof, see [Del74], or [Milb] for a proof in English. Note that our Pi ( X, t) are not the same as
PJac C (t) for a curve C; P1 (C, t) is the “reverse polynomial” of PJ (t). Also, note that because
i +1
of conjecture 3, each Pi is relatively prime to the others, i.e. in Z ( X, t) = ∏ Pi ( X, t)(−1)
there is no cancellation.
Let’s relate our approach to the Weil conjectures for a curve to the general proof. The
fact that
det(1 − Φ∗ t, V` J )
Z (C, t) =
(1 − t)(1 − qt)
has a general analogue. For a graded vector space H • and a graded endomorphism f • :
H • → H • , write
i
det( f , H • ) = ∏ det( f i , H i )(−1) .
i
51
There is a general theorem (see [Del74, 1.5.4]) that
Z ( X, t) =
1
.
det(1 − Φ∗ t, H• ( X, Q` ))
For X = C a nice curve of genus g, the map Φ∗ on H0 is the identity, and on H2 is multiplication by qd . For any endomorphism θ of a d-dimensional vector space V, one has
det(t · 1 − θ ) = td det(1 − θt−1 )
In other words, det(1 − θt) is the “reverse” of the characteristic polynomial of θ. Thus PJ (t)
is the reverse of the polynomial P1 (C, t), and the roots of PJ (t) are q-Weil if and only for
P1 (C, t) = ∏i (1 − ωi t), the ωi are q-Weil.
Next, we check that the functional equation holds for Z (C, t). We know that PJ (t) =
∏(t − ωi ), where the ωi are q-Weil. For each root ω of PJ , ω¯ is also a root that is q-Weil, so
ω ω¯ = q, i.e. ω¯ = q/ω. It follows that the polynomials t2g PJ (q/t) and PJ (t) have the same
roots, so they are equal up to a constant (which turns out to be q g ). This yields the functional
equation.
Requirement 3 in the statement of the Weil conjectures is often called the Riemann
hypothesis for the variety. For X = C a nice curve of genus g, it is easy to motivate this.
Define ζ (C, s) = Z ( X, q−s ). This is a holomorphic function of s on some {s ∈ C : <s > c},
and has a meromorphic continuation to the complex plane. By definition, ζ (C, s) = 0 if and
only if Z (C, q−s ) = 0, which happens if and only if q−s = ωi for some i. Since the ωi are
q-Weil, this tells us that |s| = logq (q1/2 ) = 12 . The requirement that zeros of ζ (C, s) have
absolute value 12 is commonly called the Riemann hypothesis for C. Note that the Riemann
hypothesis for curves is a theorem, unlike the (much more difficult) Riemann hypothesis for
ζ (Spec(Z), s).
3.9
Generalizing the Weil conjectures
Proving that Z ( X, t) is rational is the easiest part of the Weil conjectures. Indeed, Dwork
proved it before Deligne, using p-adic analytic methods. The cohomological proof is almost
a triviality, following from basic properties of e´ tale cohomology. Let X be a nice variety
over Fq , and fix a prime ` not dividing q. Write H( X ) for H• ( XF¯ q , Q` ). This is a graded
Q` -vector space with an action of Gk . In fact, H( X ) is a graded-commutative Q` -algebra,
with multiplication the cup product. We have the following trace theorem [Del77, II.3.1].
#X (Fqn ) = tr(Φ∗ n , H( X ))
where Φ = Φ X is the Frobenius of X. Now we use the following theorem from linear algebra
3.9.1 Theorem. Let V be a finite-dimensional graded vector space over a field k of characteristic
zero, and let θ : V → V be a k-linear map. There is an equality of formal power series
d
1
n
n
∑ tr(θ , V )t = t dt log det(1 − θt) .
n>1
Proof. See [Del77, II.3.3]. Essentially, one reduces to the case where V is concentrated in
degree zero and k is algebraically closed. Then use the fact that θ can be diagonalized.
52
Summing the trace formula over n gives
∑ #X (Fqn )tn = ∑ tr(Φ∗ , H(X )).
n>1
n>1
We can apply Theorem 3.9.1 to conclude that
∑
#X (Fqn )tn = t
n>1
Applying exp
R
dt
t
d
log
dt
1
det(1 − Φ∗ t, H( X ))
.
to both sides yields the formula Z ( X, t) = det(1 − Φ∗ t, H( X ))−1 . So
we know that Z ( X, t), a priori a power series over Q, is a rational function over Q` . That
is, Z ( X, t) ∈ QJtK ∩ Q` (t). The theory of Hankel determinants (see exercise one from §4 in
[Bou90, A.IV]) shows that Z ( X, t) ∈ Q(t).
This proof can be easily generalized. Let k be an arbitrary field, X a nice variety over
k. Fix a prime ` invertible in k, and write H( X ) for H• ( Xk¯ , Q` ). For a surjective morphism
f : X → X, let Γ f ⊂ X × X be the graph of f , and let ∆ X ⊂ X × X be the diagonal. There is
the relative zeta function of X with respect to f :
!
Z
n dt
Z ( X, f , t) = exp
∑ (Γ f n · ∆ X )t t .
n>1
One has a generalized trace formula ([dJ, ex.11] and [Del77, IV.3.3]):
(Γ f n · ∆ X ) = tr( f ∗ n , H( X ))
Exactly as before, it follows that Z ( X, f , t) = det(1 − f ∗ t, H( X ))−1 .
A more fruitful (but more difficult) generalization of the Weil conjectures involves the
Grothendieck ring of varieties. Let VarFq denote the category of all varieties over Fq . Write
K = K0 (VarFq ) for the quotient of the free abelian group on isomorphism classes of varieties
over Fq by relations of the form [ X ] = [Y ] + [U ] whenever Y ⊂ X is a closed subvariety and
U = X r Y. The ring K has a natural product operation induced by [ X ] · [Y ] = [ X × Y ]. If we
had started with an algebraically closed field of characteristic zero, K would be generated
by smooth projective varieties – see [Bit04]. For a variety X over Fq , it is a theorem that
#X (Fqn ) = # Symn ( X )(Fq ) (see [Del77, III.2.11] for a colossal generalization). This motivates
our definition of the motivic zeta function of X as
ζ ( X, t) =
∑ [Symn (X )]tn ∈ Λ(K) = 1 + tKJtK.
n>0
One can ask whether ζ ( X, t) is a rational function over K, but this turns out to be false in
general.
3.10
Computing zeta functions
Let C be a nice curve of genus g over Fq , J the jacobian of C. We have
PJ (t) = t2g + a1 t2g−1 + · · · + a g t g + a g−1 qt g−1 + a g−2 q2 t g−2 + · · · + at q g−1 t + q g .
53
In particular, t2g PJ (1/t) is congruent to 1 + · · · + a g t g modulo t g+1 . Thus
1 + a1 t + · · · + a g t g ≡ (1 − t)(1 − qt) Z (C, t)
tn
≡ (1 − t)(1 − qt) exp ∑ #C (Fqn )
n
n>1
!
(mod t g+1 )
The values #C (Fq ), . . . , #C (Fq g ) determine a1 , . . . , a g , and hence PJ (t) and Z (C, t) are determined by #C (Fqr ) for r 6 g.
3.10.1 Example. Let C be the nice curve over F p arising from y2 = x6 − x3 + x + 1. (The
projective closure of the zero set of this polynomial has a singularity, so we have to blow it
up once.) Let J be the jacobian of C. We have
PJ (t) = t4 + a1 t3 + a2 t2 + qa1 t + p2
and
t2
1 + a1 t + a2 t ≡ (1 − t)(1 − pt) exp #C (F p )t + #C (F p2 )
2
2
(mod t3 )
For p = 3, one can compute #C (F3 ) = 7, #C (F9 ) = 13. Thus
t2
1 + a1 t + a2 t ≡ (1 − t)(1 − 3t) exp 7t + 13
2
2
≡ 1 + 3t + 6t2
(mod t3 )
It follows that PJ (t) = t4 + 3t3 + 6t2 + 9t + 9. This factors as (t2 + 3)(t2 + 3t + 3).
From this, we see that J is isogenous to a product of non-isogenous elliptic curves.
If p = 5, one can brute-force #C (F5 ) = 9, #C (F25 ) = 19. The same process yields the
irreducible polynomial
PJ (t) = t4 + 3t3 + t2 + 15t + 25.
Thus J is a simple abelian variety of dimension 2. As a consequence, there is no nonconstant
morphism C → E, where E is an elliptic curve. If there was, we would get a nonconstant
morphism J → E, the kernel of which would be a nontrivial abelian subvariety of J (at least
after taking its connected component).
We claim that J is geometrically simple. We want PF5n to be irreducible for all n. For, this
would imply that JF5n is simple, for each nand hence JF¯ 5 is simple. Let ω be a root of PJ (t).
We know that Q(ω )/Q is a degree-four
Q(ω n ) = Q(ω ). We claim
√ extension. We want
n
that Q(ω ) has only the subfields√Q( 5) and Q. So if Q(ω ) 6= Q(ω, then ω n is real, i.e.
ω n = ±5n/2 . This implies ω = ζ 5 for ζ a root of unity,
√ whence ζ ∈ Q(ω ). One can show
√
that the extension Q(ω )/Q is not Galois, so ζ ∈ Q( 5), so ζ = ±1, whence ω = ± 5,
which cannot be the case.
4
Birch and Swinnerton-Dyer conjecture
Consider an elliptic curve E over Q. It has a model of the form y2 = x3 + ax + b with
a, b ∈ Z and the discriminant ∆ = −16(4a3 + 27b2 ) 6= 0. The Mordell-Weil theorem (proved
in this case by Mordell) tells us that E(Q) is finitely generated. By basic algebra, we can
write E(Q) = E(Q)tors ⊕ Zr where E(Q)tors is finite, and r = rk E(Q) is the algebraic
54
rank of E (over Q). The group E(Q)tors can be computed. One way is to look at the map
E(Q)tors → E(F p ) × E(F p0 ), which is injective when p and p0 are distinct primes of good
reduction. For example, we could choose p 6= p0 , with both not dividing ∆.
The torsion subgroup E(Q) is one of a finite list. In fact, there is the following deep
theorem of Mazur (conjectured by Ogg).
4.0.2 Theorem. Let E be an elliptic curve over Q. Then E(Q)tors is isomorphic to one of the
following:
Z/n
with n = 1, . . . , 10 or n = 12
Z/n ⊕ Z/2 with n = 1, . . . , 4
Proof. See [Maz77, III.5.1]. Essentially, Mazur classifies rational points on the modular
curves X0 ( N ).
Each of these possibilities occur infinitely often. An analogous theorem holds over any
number field. More precisely, for each integer d, there is a global bound B(d) such that for
any elliptic curve E over a number field k with [k : Q] = d, one has #E(k )tors 6 B(d). This is
proven in [Mer96]. For k quadratic over Q, the possible groups E(k)tors have been classified.
Unlike the situation with E(Q)tors , where everything is computable and well-understood,
there is no known algorithm to compute the rank of an elliptic curve. If X( E) is finite, then
there is an algorithm, but the finiteness of X( E) is not known in general.
The vague idea of the Birch and Swinnerton-Dyer conjecture (henceforth BSD) is as
follows. Suppose E(Q) has high rank. Then for p a prime of good reduction, we have a map
E(Q) → E(F p ). (Scheme-theoretically, E(F p ) doesn’t make sense. We set E(F p ) = E (F p ),
where E is the N´eron model of E over Z( p) . Since E(Q) = E(Z), we are really looking at
the map E(Z) → E (F p ).) The group E(F p ) has order p + 1 − a p ( E), where a p ( E) = a(EF p ),
√
and | a p ( E)| 6 2 p. The expectation is for E(F p ) to be “slightly larger” than average, given
that the map E(Q) → E(F p ) gives us some points “for free.” This effect turns out to be very
subtle.
4.1
L-functions of elliptic curves
Define the function π E on R>0 by
πE (x) =
#E(F p )
.
p
p6x good
∏
We hope that r = rk E can be detected from the growth of π E . In the early 60s, SwinnertonDyer used a computer to get numerical data on these types of problem. He and Birch
conjectured the following [BSD65, A]:
lim
x →∞
log π E ( x )
=r
log log x
One could also conjecture that π E ( X ) ∼ C (log x )r as x → ∞ for some constant C. The
function π E is hard to work with. Instead, we will follow standard practice in number
theory and introduce an L-function.
55
4.1.1 Definition. Let E be an elliptic curve over Q with discriminant ∆. The partial L-function
of E is
1
L∆ ( E, s) = ∏
.
−s + p1−2s
1
−
a
(
E
)
p
p
p-∆
Here, s is a complex variable, but this product does converge on the whole complex plane.
Recall that if E is an elliptic curve over F p , the characteristic polynomial of the Frobenius
Φ E acting on T` E is t2 − a p t + p. The reverse of this is det(1 − Φ∗ t, V` E) = 1 − a p t + pt2 . It
follows that we could have defined
L∆ ( E, s) =
∏ det(1 − Φ∗
p-∆
E,p
1
p−s , H1 ( E
p , Q` ))
for some prime ` | ∆. There is a way of adding factors at the “bad primes” dividing the
discriminant of E. We will come back to that later.
4.1.2 Lemma. The product defining L∆ ( E, s) converges absolutely for all s ∈ C with <s >
Moreover, L∆ ( E, s) is holomorphic on that region.
3
2.
Proof. The basic idea is to use the factorization 1 − a p t + pt2 = (1 − ω p,1 t)(1 − ω p,2 t), where
|ω p,i | = p1/2 by the Weil conjectures. We can write
L∆ ( E, s) =
∏
1−
p-∆
ω p,1
p1/2
p
1/2−s
−1 1−
ω p,2
p1/2
p
1/2−s
−1
Standard arguments (using only the fact that the ω p,i are p-Weil) yield the result.
Following Euler, we look at L∆ ( E, 1), (which does not exist unless L∆ ( E, s) has some
kind of analytic continuation, which we have not proved). At least formally (ignoring
convergence) we have
L∆ ( E, 1) =
∏ (1 − a p ( E ) p −1 + p −1 ) −1
p-∆
=∏
p
p − a p ( E) + 1
=∏
p
#E(F p )
p-∆
p-∆
This looks very similar to our definition of the function π E . At least intuitively, larger r
should lead to “quicker vanishing” at s = 1.
4.1.3 Conjecture (Birch and Swinnerton-Dyer). The function L∆ ( E, s) has an analytic continuation to a neighborhood of s = 1. Moreover,
ords=1 L∆ ( E, s) = rk E.
In other words, the conjecture says that L∆ ( E, s) ∼ c(s − 1)r near s = 1 for some constant
c. Modern formulations of BSD include a formula for c.
56
Let’s define the factors in L( E, s) for primes p | ∆. The following seems quite ad-hoc
without motivation:

1


if E has good reduction at p

1 − a p p−s + p1−2s
L p ( E, s) =
1


if E has bad reduction at p

1 − a p p−s
Here, a p = −1 if E has split multiplicative reduction at p, a p = 1 if E has nonsplit multiplicative reduction, and a p = 0 if E has additive reduction. This terminology deserves some
explanation. Fix a prime p. A minimal model for E at p is an equation y2 = x3 + ax + b, where
a, b are integral over Z( p) , and such that the descriminant ∆ = −16(4a3 + 27b2 ) has minimal
e be a minimal model for E. Then E
e is a scheme
p-adic valuation among such models. Let E
e
e
over Z( p) , so we can consider its reduction E p = EF modulo p. If E has bad reduction at
p
ep will be singular, but its nonsingular locus E
ep,ns will be a smooth group scheme over
p, E
e
F p . There are three possibilities for E p,ns . Either it will be the additive group G a,F p , in which
case we say E has additive reduction at p, or it will be a one-dimensional torus (isomorphic
to GF¯ p after base change). In the second case we say E has split multiplicative reduction at p
ep,ns ' Gm,F , and we say E has nonsplit multiplicative reduction at p if E
ep,ns 6' Gm,F (it
if E
p
p
turns out that we get isomorphism after base change to a quadratic extension of F p ). See
[Sil09, III.2.5, 2.6] for a proof and details.
4.2
Conductors
Let E be an elliptic curve over Q with rank r. The conductor of E is an integer measuring the
how badly E reduces over various primes. Before we can define the conductor, we need to
recall some basic facts about Galois groups of local fields. Let k be a local field, L/k a finite
Galois extension. Let v be a valuation on K normalized by the condition v(π L ) = 1. Let
G = Gal( L/k ). Then the higher ramification groups of the extension L/k are:
Gi = {σ ∈ G : v(σ ( x ) − x ) > i for all x ∈ o L }.
For example, G−1 = G, G0 is the inertia subgroup of L/k, and G1 is the “wild inertia”
subgroup of G. The Gi form an exhaustive descending filtration of G. For a careful study of
the higher ramification groups, see [Ser79, IV].
The conductor of an elliptic curve E over Q is defined as a product of local data: N =
∏ p p f p . Each of the f p only depend on the base change EQ p . So, treat E is an elliptic curve
over Q p , and choose a prime ` 6= p. Let V = E[`]; this is a two-dimensional vector space
over F` with an action of GQ p . Since GL(V ) is finite, the action of GQ p factors through a
finite quotient G = Gal( L/Q p ). We define
f p ( E) =
1
∑ [G0 : Gi ] dimF` V/V Gi .
i>0
Note that G0 = I p , the inertia group at p, and G1 = Pp , the Sylow p-subgroup of G
(consisting of “wild inertia”), see [Ser79, IV] for a proof. So we can split up the sum as
f p ( E) = dimF` V/V I p +
1
∑ [G0 : Gi ] dimF` V/V Gi .
i>1
57
The quantity dim V/V I p is called the tame part of f p , sometimes written ε p , and the rest of
the sum is called the wild part, written δp . If p > 5, the wild part is zero and there is an easy
formula for f p :

0 if E has good reduction at p

f p ( E) = 1 if E has multiplicative reduction at p


2 if E has additive reduction at p
See [Sil94, IV.10.2] for a proof of this. Sometimes f p is called the Swan conductor of E[`].
(mention `-independence of Swan conductor)
4.2.1 Theorem. The function L( E, s) has an analytic continuation to all of C. Moreover, it satisfies
a functional equation: there exists ω = ±1 such that
Λ(s) = ωΛ(2 − s)
where Λ(s) = N s/2 (2π )−s Γ(s) L( E, s).
The constants N and ω are easily computed. They are a product of local factors, so in
particular no analysis is needed to find their values.
We can give a sketch of a proof of this theorem. Expand the product formula for L( E, s)
to get a Dirichlet series
an ( E)
.
L( E, s) = ∑
ns
n>1
It is not immediately obvious that these an = an ( E) agree with our earlier definition of the
a p = p + 1 − #E(F p ). It follows from an argument similar to the one Euler used to prove
the product formula for ζ (s). Moreover, we have an ∈ Z for all n. The function a 7→ an is
multiplicative in the number-theoretic sense: if (m, n) = 1, then amn = am an . If n > 2, then
a pn = a p a pn−1 − pa pn−2 .
This allows us to compute an for any n just using the a p . Define the Fourier series
f (z) =
∑ an qn
(q = e2πiz )
n>1
where z ranges over the upper half-plane h = {z ∈ C : =z > 0}.
4.2.2 Theorem (Fermat-Wiles). Let E be an elliptic curve over Q with conductor E. Then the
function f = f E defined above is a cusp form of weight 2 and level N. Moreover, f is an eigenform
(eigenvector for all Hecke operators).
Proof. See [BCDT01] for a proof which relies heavily on previous work by Wiles (with help
from Taylor).
This is one version of the Taniyama-Shimura conjecture. We’ll take a moment to explain
everything in the theorem.
58
4.3
Modularity
Recall that SL(2, R) acts on h via fractional linear transformations:
az + b
a b
·z =
.
c d
cz + d
There is nothing ad-hoc about this: the group scheme GL(n + 1) acts on Pn in the obvious
way, and our action is just a restriction of the action of GL(2, C) on P1 (C) to an action of
SL(2, R) on h ⊂ P1 (C). The space Ω1 (h) of (complex) holomorphic one-forms on h consists
of differentials η = f (z)dz. For γ = ac db ∈ SL(2, R), one easily computes
γ∗ η = f (γz)d(γz) = (cz + d)−2 f (γz)dz.
Let N > 1 be an integer, and let Γ be one of the following groups:
∗ ∗
Γ0 ( N ) = γ ∈ SL(2, Z) : γ ≡
mod N
0 ∗
1 ∗
Γ1 ( N ) = γ ∈ SL(2, Z) : γ ≡
mod N .
0 1
We say a function f : h → C is a weak modular form of weight 2 for Γ if it is holomorphic
and f (z) dz is invariant under the action of Γ. Explicitly, for for all matrices ac db ∈ Γ, we
require:
az + b
f
= (cz + d)2 f (z).
cz + d
The quotient YΓ = h/Γ may be a non-compact orbifold, but there is a canonical way of
removing the non-smooth points and compactifying (by adding finitely many “cusps”)
to arrive at a compact Riemann surface XΓ . We say f is a modular cusp form of level N
and weight two if it descends to a holomorphic differential on XΓ . For weight-two forms,
the differential f dz automatically vanishes on the cusps of XΓ . For details, see [DS05], or
[Kna92, XI.11] for the case Γ = Γ0 ( N ). One writes X0 ( N ), and X1 ( N ) for XΓ0 ( N ) and XΓ1 ( N ) ,
and calls these modular curves. If we write S2 (Γ) for the space of weight-two modular forms
of level Γ, then by definition we have equality
S 2 ( Γ ) = H0 ( X Γ , Ω 1 ) .
Modular curves have a moduli-theoretic interpretation. There are canonical bijections
[DS05, 1.5]:
X0 ( N ) ' {( E, C ) : E a complex torus and C ⊂ E cyclic of order N }/ ∼
X1 ( N ) ' {( E, P) : E a complex torus and P ∈ E of order N }/ ∼ .
This allows us to define Hecke operators which are correspondences on X0 ( N ). Suppose
p - N. Then there are canonical maps
X0 ( N ) o
α
X0 ( pN )
β
/ X0 ( N ) .
If we think X0 ( N ) and X0 ( pN ) as modular spaces, these maps are easy to define. An
element of X0 ( pN ) is a pair ( E, C ) where C is cyclic of order pN; we may decompose C
59
uniquely as C p ⊕ CN , where C p and CN are cyclic of orders p and N. The map α sends ( E, C )
to ( E, CN ) and β sends ( E, C ) to ( E/CN , C/CN ). Write Tp for the image of α × β; this is a
subvariety of X0 ( N ) × X0 ( N ). In fact, it is a correspondence, so it induces an endomorphism
of H( X0 ( N )) for H any Weil cohomology theory. In particular, we have maps called Hecke
operators (and also denoted Tp ):
Tp : S2 (Γ0 ( N )) = H0 ( X0 ( N ), Ω1 ) → H0 ( X0 ( N ), Ω1 ) = S2 (Γ0 ( N )).
See chapter 12 of [RS] for a careful discussion. A cusp form f ∈ S2 (Γ0 ( N )) is an eigenform if
it is an eigenfunction for each Tp where p - N.
To see how Theorem 4.2.1 follows from Theorem 4.2.2, we need to briefly introduce
the Mellin transform. Recall that for a locally compact abelian group G, the Pontryagin
b of G is the group hom( G, S1 ) of one-dimensional unitary representations of G, with
dual G
b is also locally compact. If G = R, then G
b=R
the compact-open topology. The group G
2πisx
b
as well, where we interpret s ∈ R as a character of R via x 7→ e
. For x ∈ G, y ∈ G,
write h x, yi instead of y( x ). The Fourier transform of a function f : G → C is the function
b → Cdefined by
fb : G
Z
fb(y) =
f ( x )h x, yi dx,
G
where dx is a Haar measure on G.
The locally compact group R× (which is isomorphic to R) has dual iR via h x, si = x −s .
A Haar measure on R× is d× x = dx/x. The Mellin transform of a function f : R× → C is
just its Fourier transform:
(M f )(s) =
Z
R×
f (t)ht, si d× t =
Z ∞
0
f (t)ts−1 dt
(This is not quite the standard version of the Mellin transform, but it is the most natural in
our setting.) For a modular form f of level N, we can look at what is essentially its Mellin
transform:
Z ∞
Z ∞
Λ( f , s) = N s/2
f (it)ts−1 dt = ∑ an
e−2πnt ts−1 dt.
0
n>1
0
Making the substitution u = 2πnt, we get
Λ( f , s) = N s/2
an
∑ (2πn)s
n>1
Z ∞
0
e−u us−1 du = N s/2 (2π )−s Γ(s)
∑
n>1
an
.
ns
Note that if f = f E for an elliptic curve, we have Λ( f , s) = Λ( E, s). By work of Hecke,
the function Λ( f , s) has an analytic continuation to all of C and a functional equation
Λ( f , s) = ±Λ( f , 2 − s). See [Kna92, VII.9.8] for a modern proof.
The function ∑ an n−s (coming from a modular form f ) has an analytic continuation
and a functional equation by a theorem of Hecke. It follows from Theorem 4.2.2 that if E
is an elliptic curve over Q, the function L( E, s) has an analytic continuation to C and the
prescribed functional equation.
The modularity theorem has a sort of converse.
4.3.1 Theorem (Eichler-Shimura). Let f = ∑ an ( f )qn be a newform 2 and level N with a1 = 1
and an ∈ Q for all n. Suppose further that Tn f = f for all n relatively prime to N. Then there exists
an elliptic curve E f over Q such that an ( E f ) = an ( f ) for all n.
60
Proof. See [Kna92, XI.11] for a proof and a definition of newforms. The elliptic curve E f
can be constructed explicitly in two ways. First, the Riemann surface X0 ( N ) actually has a
canonical model (also denoted X0 ( N ) over Q; we denote its jacobian by J0 ( N ). There is a
subring T N of End J0 ( N ) called the Hecke algebra which contains all of the Tn . If one puts
I f = { T ∈ T N : T f = 0}, then E f = J0 ( N )/I f . Complex-analytically, E is the quotient of C
by the lattice
Z γ ·i
i
4.4
f (z) dz : γ ∈ Γ0 ( N ) .
Small analytic rank
Back to BSD. Let E be an elliptic curve over Q with rank r, and analytic rank
ran = ords=1 L( E, s)
By the previous discussion, L( E, s) has an analytic continuation to s = 1, so this is welldefined. As stated earlier, a weak version of BSD is that r = ran . From the functional
equation, we get (−1)ran = ω. The parity conjecture is that r ≡ ran (mod 2). The better
variant is that (−1)r = ω. Since ω can be computed without dealing with L( E, s), the parity
can be approached computationally. The parity conjecture is a theorem whenever X( E) is
finite. See [Dok10] for discussion and a proof.
4.4.1 Theorem. Let E be an elliptic curve over Q with ran 6 1. Then r = ran and X( E) is finite.
Proof. Kolyvagin [Kol88] proved the theorem if ran = 0 and E is modular (which is known
over Q). If ran = 1, this is follows from work of Gross and Zagier [GZ86].
One can compute L( E, s) to arbitrary precision. In particular, we can look at L( E, 1). It
turns out that
an −2πn/√ N
L( E, 1) = (1 + ε) ∑
e
n
n>1
If ε = −1, then L( E, 1) = 0, and it turns out that
L0 ( E, 1) = 2
∑
n>1
where
E1 ( x ) =
an
E
n 1
Z ∞
x
e−t
2πn
√
N
dt
.
t
y2
4.4.2 Example. Let E be the elliptic curve = x3 + 875x over Q. We can compute L( E, 1) =
−3.9 . . .. In particular, L( E, 1) 6= 0, which implies ran = 0. The above theorem tells us that
r = 0, so E(Q) = E(Q)tors = {O, (0, 0)}.
4.4.3 Example. Let E be the elliptic curve y2 = x3 + 877x over Q. It turns out that L( E, 1) =
0.000 . . .. This does not imply L( E, 1) = 0. We can compute ω = −1, which means ran is
odd. Thus we cannot have L( E, 1) = 0. The BSD conjecture would imply E has rank at least
one. One can compute L0 ( E, 1) = 32.7 · · · 6= 0. In particular, ran 6 1, so the above theorem
tells us r = 1. In other words, E(Q) ' Z/2 ⊕ Z, hence E(Q) is infinite.
61
4.5
The strong Birch and Swinnerton-Dyer conjecture
The standard version of BSD predicts the order of vanishing of the L-function L( E, s) of
an elliptic curve at 1 in terms of the algebraic rank of E. There is a stronger version of the
conjecture that also predicts the leading term in the Laurent expansion of L( E, s) at 1.
4.5.1 Conjecture (strong BSD). Let E be an elliptic curve over Q. Then rk( E) = ran ( E), X( E)
is finite, and
Ω E RegE #X( E) ∏ p c p
L( E, s)
lim
=
.
r
s →1 ( s − 1 )
#E(Q)2tors
We need to explain the quantities appearing in the conjecture. First we define the real
period Ω E of E. Choose a minimal Weierstrass model
y2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
for E, with the ai ∈ Z and ∆ = −16(43 + 28b2 ) minimal with respect to divisibility. It is not
obvious that this is possible. In fact, it can only be done for elliptic curves over global fields
with class number one [Sil09, VIII.8.3]. The differential
ω=
dx
2y + a1 x + a3
is called the invariant differential associated to E. It is unique up to a unit in Z [Sil09, VII.1.3].
We define
Z
ΩE =
| ω |,
E(R)
where we choose the orientation of E(R) necessary to make the integral positive. The real
period of E is easily computable using a trick that takes advantage of the fast convergence
of the algebro-geometric mean.
Next we define the regulator RegE of E. Our minimal model for E induces an embedding
E ,→ P2 , and we write h for the induced Weil height (see Section 2.10 for a general definition).
We can write any x ∈ E(Q) as ( x0 , x1 , x2 ) with the xi ∈ Z and gcd( x0 , x1 , x2 ) = 1. Explicitly,
h( x ) = log max{| x0 |, | x1 |, | x2 |}.
Recall that the N´eron-Tate height b
h associated with our embedding can be computed as
h (2n · x )
b
h( x ) = lim
.
n→∞
4n
Using this we define a pairing on E(Q):
h x, yi =
1 b
h( x + y) − b
h( x ) − b
h(y) .
2
This induces a nondegenerate bilinear pairing (hence the structure of a Hilbert space)
on V = E(Q) ⊗ R. There is a canonical Haar measure on any Hilbert space. It is the
unique Haar measure such that, given an orthonormal basis {v1 , . . . , vr } of V, assigns to the
fundamental domain
(
)
r
∑ λi vi : 0 6 λi 6 1
i =1
62
the volume 1. For a lattice (discrete, cocompact subgroup) Λ ⊂ V, put vol(Λ) = vol(V/Λ).
Then
RegE = vol( E(Q)/E(Q)tors )2 ,
where E(Q)/E(Q)tors is a thought of as a lattice in E(Q) ⊗ R. More explicitly, for any basis
{ x1 , . . . , xr } of E(Q), one has
RegE = det h xi , x j ii,j .
The Tate-Shafarevich group of E is
1
X( E) = ker H (Q, E) →
∏H
!
1
(Qv , E)
v
where as usual we write H1 (k, A) for H1 ( Gk , A(k¯ )) when k is a field and A is a commutative
group variety over k.
The c p are Tamagawa numbers of E. Let E0 (Q p ) be the group of points in E(Q p ) whose
reduction modulo p is is nonsingular. We define
c p = [ E(Q p ) : E0 (Q p )]
Clearly c p = 1 if E has good reduction at p. The c p are easy to compute using Tate’s
algorithm.
4.6
Predicting the order of X
4.6.1 Example. Let E be the rational elliptic curve y2 + xy = x3 + x2 − 1154x − 15345.
Using Sage, we can predict the order of X( E). First, Ω E ≈ 0.819917869389698. Next, we
compute L( E, 1) ≈ 1.84481520612682 6= 0, so ran ( E) = 0. By Theorem 4.4.1, E(Q) and X( E)
are both finite. It follows that the regulator of E is 1. Moreover, the discriminant of E is
310 · 2272 , so we only need to compute c3 = 2 and c227 = 2. Finally, it is possible to compute
#E(Q)tors = 4. We now know everything in the BSD formula except for the cardinality of
X( E). Solving for it, we get (conjecturally):
#X( E) =
L( E, 1) · #E(Q)2tors
≈ 9.00000000000000
Ω E · c3 · c227
so we could expect X( E) = 9.
4.6.2 Example. Let E be the rational elliptic curve y2 + y = x3 − x2 − 64403x − 6220110
(this is the quadratic twist of y2 = x3 + 4x2 − 48x + 80 by −139). We can compute L( E, 1) ≈
0.000000000000000, which may or may not be zero. Similarly, L0 ( E, 1) and L00 ( E, 1) appear
to be zero, but L000 ( E, 1) ≈ 110.576624177222 6= 0. So we would guess that ran ( E) = 3. We
know that (−1)ran = ω = −1, so ran is odd. This tells us L( E, 1) = 0 on the nose. One can
do a “2-descent” to show that rk E = 3 and E(Q)tors = 1. In fact, E(Q) has as a basis
[(−156 : 77 : 1) , (510 : −9661 : 1) , (1344 : 48302 : 1)] .
63
We can now compute
RegE = 15.3653647904235
Ω E = 0.299853560495299
#E(Q)tors = 1
∆ = 37 · 1396
c37 = 1
c139 = 4
Since ran > 1, we don’t know if X( E) is finite, but assuming BSD its order is
L(3) ( E, 1)/3!
≈ 1.00000000000000.
Ω E · RegE ·c139
Recall that if ran = 1, then work of Gross-Zagier and Kolyvagin implies r = 1. Since r > 1,
we have ran > 1, and since ran is odd it must be 3. So L0 ( E, 1) = L00 ( E, 1) = 0.
In general, if X( E) is finite, then its cardinality must be a square. This is because there is
a canonical alternating pairing X( E) × X( E) → Q/Z, which is nondegenerate if E is finite.
See [Mil06, I.6] for a definition of this pairing in much greater generality. Note that the parity
conjecture holds over Q by the functional equation, which is proved via the modularity
theorem.
In his paper [Gol85] Goldfeld, drawing on work of Gross and Zagier, used this curve
(in particular, the fact that it has analytic rank 3) to solve a very old problem of Gauss. For
every ε > 0, there is√an effectively computable constant c > 0 such that the class number
h( D ) of the field Q( − D ) for D > 0 square-free satisfies the bound
h( D ) > c (log D )1−ε .
It follows that for any constant C, it is possible to enumerate the (finite) list of imaginary
quadratic fields with class number
h 6 C. For example, the only imaginary quadratic fields
√
with class number 1 are Q( − D ) for D one of
1, 2, 3, 7, 11, 19, 43, 67, 163.
4.7
Average orders of Selmer groups
For a, b ∈ Z with 4a3 + 27b2 6= 0, let Ea,b be the rational elliptic curve y2 = x3 + ax + b. Set
E = { Ea,b : a, b ∈ Z : 4a3 + 27b2 6= 0 and p6 - b whenever p4 | a}.
Every elliptic curve over Q is isomorphic to a unique element of E . We define a “naive
height” on E by H ( Ea,b ) = max{|4a3 |, 27b2 }. For any x > 0, set
E x = { E ∈ E : H ( E ) 6 x }.
Given a map φ : E → R, we define (if the limit exists)
avg(φ) = lim
x →∞
and let avg(φ) be the corresponding lim sup.
64
∑ E∈Ex φ( E)
,
#E x
4.7.1 Theorem (Bhargava-Shankar). We have avg(# Sel2 ) = 3 and avg(# Sel3 ) 6 4.
Proof. See [BS10b] for a proof of the first equality, and [BS10a] for a proof of the second
inequality.
As a corollary, avg(rk) 6 67 . Many mathematicians (including Zywina) expect the average to be 12 . To see that the average rank is at most 76 , note that since since E(Q)/3E(Q) ,→
s
Sel3 ( E), we have 3rk E 6 # Sel3 ( E) = 3s . It follows that rk E 6 s 6 3 6+3 , so
avg(rk) 6
4.8
avg(3s ) 1
4 1
7
+ 6 + = .
6
2
6 2
6
The congruent number problem
Fix a squarefree integer d > 1. We say that d is a congruent number if there is a right triangle
with rational side lengths, with area d. The requirement that n is squarefree is not significant:
we can always scale the triangle by a rational number to make its area squarefree. From
the well-known right triangle with sides (3, 4, 5), we see that 6 is a congruent number. Our
general equations are:
a2 + b2 = c2
ab/2 = d.
These equations describe a curve Cd ⊂ A3Q = Spec Q[ a, b, c]. The integer d is congruent
precisely when Cd (Q) 6= ∅. It is natural to ask whether a given integer (for example
d = 157) is congruent. The curve Cd is not “nice” in the technical sense, because it is
not projective. Let Ed be the curve y2 = x3 − d2 x. This is birational to Cd via the map
∼
f : Cd −
→ En r {O, (0 : 0 : 1), (0 : ±d : 1)} defined by
( a, b, c) 7→ dbc( a + c) : d2 (( a + c)2 + b2 ) : b2 c .
This map has birational inverse
( x : y : z) 7→
x2 − n2 2nx x2 + n2
,
,
yz
y
yz
.
The only obvious rational points on Ed are the origin, (0 : 0 : 1) and (0 : ±n : 1), which
comprise Ed (Q)tors = Ed [2]. Thus Cd (Q) 6= ∅ if and only if rk Ed > 0. Since d is congruent
if and only if Cd has rational points, we conclude that d is congruent precisely when Ed has
positive rank. Assuming BSD, this occurs if and only if L( Ed , 1) = 0, i.e. ran ( Ed ) > 1. If
L( Ed , 1) 6= 0, which is something that can be computationally verified, then ran ( Ed ) = 0, so
by Theorem 4.4.1, we know that Ed has rank zero. In other words, it is easy to show that a
given integer d is not congruent, simply by checking that L( Ed , 1) 6= 0.
The fact that Cd and Ed are birational gives a way of constructing many different rational
triangles with area d, using the group law on Ed . For instance, in our example d = 6, the
triple ( a, b, c) = (3, 4, 5) maps to the point x = (12 : 36 : 1) in Ed (
Q). One can compute
25
35
7
1201
2 · x = 4 : − 8 : 1 , which corresponds to a triangle with sides − 10
, − 120
,
−
7
70 , at
least up to sign. Each multiple n · x gives a different triangle, with rapidly increasing
complexity. We can use Sage to compute the first few:
65
n
2
3
4
5
6
( a, b, c)
7
1201
− 10
, − 120
7 , − 70
3404
7776485
− 4653
851 , − 1551 , − 1319901
1437599 2017680 2094350404801
168140 , 1437599 , 241717895860
3122541453 8518220204 18428872963986767525
,
,
2129555051 1040847151 2216541307731009701
43690772126393
5405257799550679424342410801
− 20528380655970
, − 246340567871640
43690772126393 , − 896900801363839325090016210
It is possible to determine algorithmically whether any given d is congruent, but only
under the assumption that BSD holds. For an integer t, let χt : GQ → {±1} be the Dirichlet
√
character associated to the extension Q( t)/Q. We define
θt (q) =
∑
qtn
2
n>−∞
g(q) = q
∏ (1 − q8n )(1 − q16n ) = ∑
(−1)n q(4m+1)
2 +8n2
.
m,n∈Z
n>1
For each t, the function θt is a modular form of weight 1/2, level 4t and character χt . The
function g is the unique normalized newform of level 1, weight 128 and character χ−2 . We
can write formal expansions θ2 g = ∑ a(n)qn and θ4 g = ∑ b(n)qn . The functions θ2 g and
θ4 g are modular forms of weight 3/2, level 128, and trivial (resp. χ8 ) character. It turns out
that the coefficients a(n) and b(n) have a straightforward interpretation in terms of sums of
squares. Now let E = E1 be the curve y2 = x3 − x, and let Ω = Ω E be its real period; one
has
Z ∞
dx
√
≈ 2.62205755429212.
Ω=
1
x3 − x
4.8.1 Theorem (Tunnell). If d > 1 is a squarefree integer, then
(
a(d)2 Ωd−1/2 /2 if d is odd
L( Ed , 1) =
b(d/2)2 Ωd−1/2 if d is even.
Proof. This is Theorem 3 of [Tun83]. Tunnell uses a result of Waldspurger giving a functional
equation for certain types of L-functions.
4.8.2 Corollary. Assume the weak BSD conjecture. Then an integer d > 1 is congruent if and only
if a(d) = 0 in the case where d is odd, or b(d/2) = 0 in the case where d is even.
If we define
A(n) = #{( x, y, x ) ∈ Z3 : x2 + 2y2 + 8z2 = n}
B(n) = #{( x, y, z) ∈ Z3 : x2 + 2y2 + 32z2 = n}
C (n) = #{( x, y, z) ∈ Z3 : x2 + 4y2 + 8z2 = n/2}
D (n) = #{( x, y, z) ∈ Z3 : x2 + 4y2 + 32z2 = n/2},
Then for d odd, a(d) = 0 if and only if A(d) = 2B(d), and for d even, b(d/2) = 0 if and only
if C (d) = 2D (d). These equalities are easy to check via a brute search. An easy example is
d = 2. One has C (2) = D (2) = 2, so b(1) 6= 0. Since this implies L( E2 , 1) 6= 0, we can use
Theorem 4.4.1, to see that n = 2 is not a congruent number (unconditionally).
66
4.9
The Sato-Tate conjecture
Let E be an elliptic curve over Q with conductor N. For primes p not dividing N, recall
√
we defined a p ( E) = p + 1 − #E(F p ), and proved the Hasse bound | a p ( E)| 6 2 p. Thus if
√
we define b p ( E) = a p ( E)/2 p, the numbers b p ( E) lie in the interval [−1, 1]. It is natural to
ask, for fixed E, how the b p ( E) are distributed in that interval. It is useful to set up some
terminology, which we do following the appendix to Chapter 1 in [Ser68].
For a topological space, let C0 ( X ) denote the Banach space of continuous complex-valued
functions vanishing at infinity. This is the completion of the space of continuous functions
with compact support. If X is locally compact, it is a theorem (one could take this as a
definition of Borel measures) that the dual of C0 ( X ) is isomorphic as a Banach space to the
space of Borel measures on X. A measure µ corresponds to the functional
f 7→
Z
X
f dµ.
R
See [Rud87, 6.19] for a proof. Given x ∈ X, write δx for the point mass f dδx = f ( x ). For
discrete S ⊂ R, we call a sequence { xs }s∈S of points in X equidistributed with respect to µ if
µ = lim
c→∞
1
#{ s ∈ S : s 6 c }
∑ δxs
s6c
in the weak topology. By definition, this is the same as requiring, for each continuous f with
compact support, the equality
1
lim
c → ∞ #{ s ∈ S : s 6 c }
∑
f ( xs ) =
s6c
Z
X
f dµ.
If X is a smooth oriented n-dimensional manifold, then any n-form ω induces a Borel
measure via
Z
f 7→
f ω.
X
We will often identify ω with the measure it induces. In particular, we will often abuse
terminology by calling a sequence equidistributed with respect to ω.
√
Let E be an elliptic curve over Q. Recall that we defined b p ( E) = a p ( E)/2 p. The SatoTate conjecture predicts the distribution of the b p in the interval [−1, 1]. If E has complex
◦
multiplication, then things were worked out quite a while
√ ago. Let k = End ( E); this is
an imaginary quadratic field (we could write k = Q( − D ) for D a positive squarefree
integer). If p > 5 is inert in k, then a p ( E) = 0. The prime p is inert exactly when − D is not a
quadratic residue modulo p, and this happens for half of the primes.
4.9.1 Theorem (Deuring, Hecke). Let E be an elliptic curve over Q with complex multiplication.
Then the sequence {b p ( E)} p is equidistributed in [−1, 1] with respect to the measure
1
dt
√
δ0 +
.
2
2π 1 − t2
Proof. Do this for an arbitrary abelian variety with complex multiplication, using the associated `-adic representation.
67
In other words, if we restrict ourselves to primes that split in k, the b p ( E) are distributed
according to the following graph, corresponding to the curve y2 = x3 + 1. The image was
taken from Andrew Sutherland’s web page http://math.mit.edu/˜drew, which has many
more examples.
The Sato-Tate conjecture is an analogous prediction for elliptic curves without complex
multiplication.
4.9.2 Conjecture (Sato-Tate). Let E be a non-CM elliptic curve over Q. Then the sequence
{b p ( E)} p is equidistributed in [−1, 1] with respect to the measure
2p
1 − t2 dt.
π
Here is the example y2 = x3 + x + 1, also taken from Sutherland’s web page.
68
4.9.3 Theorem (Barnet-Lamb, Geraghty, Harris, Taylor). The Sato-Tate conjecture is true.
Proof. See [BLGHT, 8.3] for a (as yet unrefereed) proof.
There is a refined version of the Sato-Tate conjecture. Let ρ = {ρ` : GQ → GL(2, Z` )} be
a strictly compatible family of `-adic representations in the sense of [Ser68, ch.1]. For almost
all primes p, the characteristic polynomial of ρ` (Frob p ) will be of the form t2 − a p t + p.
Assume ρ is pure in the sense that the roots α p , α¯ p of t2 − a p t + p are q-Weil.
4.9.4 Conjecture (Lang-Trotter). For any integer n and imaginary quadratic field k, there are
constants C (n, ρ) and C (k, ρ) such that
√
x
#{ p 6 x : Q(α p ) = k} ∼ C (k, ρ)
log x
√
x
#{ p 6 x : a p = n} ∼ C (n, ρ)
.
log x
See the introduction to [LT76] for the original statement and some motivation.
4.10
Some computations
Let d = 157. We hope to show that d is congruent, i.e. that is is the area of a right triangle
with rational side lengths. In other words, the curve Cd defined as the solution set to
a2 + b2 = c2
ab/2 = d
has a rational point. We have seen that this is equivalent to the elliptic curve Ed : y2 =
x3 − d2 x having positive rank.
We’ll use Sage to do this. Sage can be accessed online at http://sagenb.com/, or just
type sage in the command line of a computer that has Sage installed.
In Sage, the constructor EllipticCurve([ a1 , a2 , a3 , a4 , a6 ]) returns the elliptic curve
y2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 .
The simpler constructor EllipticCurve([ a4 , a6 ]) returns the elliptic curve y2 = x3 + a4 x +
a6 . Sage will return an error if the curve you try to construct is singular. Since we are
interested in the curve E157 : y2 = x3 − 1572 x, we define
E = EllipticCurve([-157ˆ2, 0])
If we had wanted to define an elliptic curve over a finite field Fq , we would make sure
some of the coefficients were elements of Fq , as in EllipticCurve([GF(5)(1), 1]). We
can compute the conductor of by E.conductor(); in our case E157 has conductor 25 · 1572 .
Usually one can compute the rank E.rank() and generators for the Mordell-Weil group
E.gens(). In our case, these return a warning. Sage’s usual algorithm was not able to
determine the rank of E157 , and it asks you to do a two-descent. We do this:
E.two_descent(second_limit=13)
69
As Sage does the 2-descent, it outputs a bunch of text describing what it does (essentially a
computation of E(Q)/2 and X( E)[2]). Once the 2-descent is complete, we can compute the
rank to be 1 and a generator to be
43565582610691407250551997
562653616877773225244609387368307126580
−
:−
:1 .
609760250665615167250729
476144382506163554005382044222449067
This corresponds to a triangle with the shorter two sides being
6803298487826435051217540 411340519227716149383203
,
.
411340519227716149383203 21666555693714761309610
There are a number of other things that Sage can do with elliptic curves. To make
computations faster, let’s try the elliptic curve
E = EllipticCurve([4,6])
described by the equation y2 = x3 + 4x + 6. This curve has rank 1 and generator (−1 : 1 : 1).
We can do computations with points on our curve:
P = E.gens()[0]
5*P # as an element of E(Q)
P.height() # Neron-Tate height of P
A lot of analytic data can be computed:
E.torsion_subgroup() # trivial for this curve
L = E.lseries().dokchitser() # the L-function of E
L(1) # looks like zero
L.derivative(1,2) # non-zero, so r_an<=1
E.root_number() # sign in function equation (=-1, so r_an is odd)
E.regulator()
E.sha().an() # predicted order assuming BSD
A fantastic place to learn more about Sage is its documentation page at http://www.
sagemath.org/doc/.
4.11
The Sato-Tate conjecture and Haar measures
√
Let E be an elliptic curve over Q. Recall we defined b p ( E) = a p ( E)/2 p. If E has CM, then
the b p are uniformly distributed in [−1, 1] with respect to the measure
1
dt
√
δ0 +
,
2
2π 1 − t2
and if E is not CM, then the b p are uniformly distributed with respect to
2p
1 − t2 dt.
π
This has a natural reformulation which makes generalization easier. The characteristic
polynomial of ρ E,` (Frob p ) is t2 − a p t + p. If we normalize to have roots with absolute value
70
1, we get ϕ p = t2 −
a
√p t + 1.
p
This is the characteristic polynomial of a unique conjugacy class
in SU(2). If we write X for the space SU(2)\ of conjugacy classes in SU(2), then p 7→ ϕ p
can be thought of as a map {primes} → X. Embed U(1) in SU(2) by the diagonal, and let
K = N (U(1)) be its normalizer. The group K is compact, so it has a unique normalized
Haar measure.
4.11.1 Theorem. If E is an elliptic curve with complex multiplication, then the set { ϕ p ( E)} ⊂
SU(2)\ is equidistributed with respect to the pushforward of the normalized Haar measure on
N (U(1)).
Proof. This is a restatement of Theorem 4.9.1. To see this, note that the trace map induces an
isomorphism
∼ /
tr : SU(2)\
[−2, 2].
The group K = N (U(1)) has two connected components, both isomorphic to S1 :
z
−z
K=
: |z| = 1 ∪
: |z| = 1 .
z¯
z¯
The first connected component is mapped to [−2, 2] via z 7→ 2<(z), and the second is
mapped via z 7→ 0. It is easy to check that the pushforward of the Haar measure on K is
exactly 12 δ0 + √dt 2 .
2π 4−t
For non-CM elliptic curves, let K = SU(2). Then the Sato-Tate conjecture states that
{ ϕ p } ⊂ GL(2, C)\ is uniformly distributed with respect to the pushforward of the normalized Haar measure on K. To see this, check
√ that the pushforward by the trace of the
1
normalized Haar measure on K to [−2, 2] is 2π
4 − t2 dt.
More generally, let A be a d-dimensional abelian variety over Q, and let ` be a prime at
which A has good reduction. For any prime p of good reduction, we have the characteristic
polynomial PA p of the Frobenius at p acting on T` A. The roots of this polynomial are
ωi
p-Weil, so if we write PA p (t) = ∏(t − ωi ), then the polynomial ϕ p ( A) = ∏ t − √
p , has
roots with absolute value 1. Then by [Kat88, 13.1], ϕ p ( A) determines a conjugacy class in
SU(2d, C). As before, we think of ϕ( A) as a map {good primes} → GL(2d, C)\ .
4.11.2 Conjecture (Serre). There exists a compact real Lie group K in GL(2d, C) such that
{ ϕ p ( A)} ⊂ GL(2d, C)\ is equidistributed with respect to the pushforward of the normalized
Haar measure on K.
There is a conjectural prediction of the group K, which we will treat in the next section.
4.12
Motives and the refined Sato-Tate conjecture
The following mostly follows Serre’s original paper [Ser94], but see [Ser12] for a more
elementary and explicit approach.
Let k be a field, and let X be an n-dimensional smooth variety over k. Write A( X ) =
A• ( X ) for the Chow ring of X, consisting of algebraic cycles modulo rational equivalence.
The intersection product makes A( X ) into a commutative unital ring – for details, see [Ful98,
8.3]. There is a natural “composition” map
A (Y × Z ) ⊗ A ( X × Y ) → A ( X × Z ) ,
71
(∗)
∗
defined by g ◦ f = π X × Z,∗ (πY∗ × Z g · π X
×Y f ). This satisfies all of the natural linearity and
functoriality properties one would expect [Ful98, 16.1].
There is a canonical “degree map” deg : An → Z, and we say a cycle α ∈ Ar ( X ) is
numerically equivalent to zero if deg(α · β) = 0 for all β ∈ An−r ( X ). Write Anum ( X ) for the
quotient of A( X ) by the (graded) ideal generated by
{α ∈ A( X ) : α is numerically equivalent to zero}.
4.12.1 Definition. Let k be a field. A (pure) motive over k is a triple ( X, e, r ), where X is a smooth
projective variety over k, e ∈ Anum ( X × X )Q is an idempotent, and r ∈ Z.
See [And04, 4.1.3] for details. One defines a morphism ( X, e, r ) → (Y, f , s) to be an
element of
f · Adim X −r−s ( X × Y )Q · e.
Morphisms are composed via the “composition map” (∗).
With this, write M(k) for the category of (numerical) motives over k. Let SmProjk be the
category of smooth projective varieties over k, and let h : SmProjk → M(k) be the functor
X 7→ ( X, ∆ X , 0). The category M(k) is obviously Q-linear, and has a Tannakian structure
induced by h( X ) ⊗ h(Y ) = h( X × Y ). In fact, M(k) is a semisimple abelian category [Jan92].
One should think of a Weil cohomology theory as a functor H : M(k) → grAlg L for some
field L (c.f. [And04, 4.2.5.1]).
Write 1 = H(A0 ) for the trivial motive. Since every variety has a unique morphism
X → A0 , there is a unique morphism 1 → M for every motive M. The rational point ∞ ∈ P1
determines a splitting of 1 → h(P1 ), hence a direct sum decomposition h(P1 ) = 1 ⊕ 1(−1),
where 1(−1) is the motive (P1 , [∞] × P1 , 0). We define 1(r ) = 1(−1)⊗(−r) ; these are called
Tate motives. In general, put M (r ) = M ⊗ 1(r ). There is a decomposition h(Pn ) = 1 ⊕ · · · ⊕
1(−n). If A is a d-dimensional abelian variety over k, then there is a unique decomposition
h( A) = h0 ( A) ⊕ · · · ⊕ h2d ( A) in M(k) such that [n] acts as multiplication by ni on each hi ( A)
V
[vdGM, 13.29]. What is more,
there are canonical isomorphisms hi ( A) ' i h1 ( A), inducing
V• 1
an isomorphism h( A) '
h ( A) (13.47, loc. cit.).
Let k be a number field, and choose an embedding σ : k ,→ C. We have a Betti realization
functor Hσ : M(k) → grAlgQ , assigning to a motive M = ( X, e, r ) the vector space
•
Hσ ( M ) = e∗ · Hsing
( X (C), Q) ⊗ H2sing (P1 )⊗(−r)
Assuming Grothendieck’s standard conjectures, the functor Hσ is a fiber functor, so M(k ) is
equivalent to the category of representations of the (pro-reductive) motivic Galois group




Gmot (k ) = Aut⊗ (Hσ ) = ( x M ) ∈ ∏ GL(Hσ M ) : x M ◦ f ∗ = f ∗ ◦ x N for f : M → N .


M ∈M( k )
For a motive M, let G M be the automorphism group of the restriction of the fiber functor
to the largest Tannakian subcategory of M(k) containing M. There are obvious projections
from Gmot (k) to the groups G M .
Let M be a motive. There is a map w : Gm → G M , induced by the grading Hσ M =
L d
Hσ ( M). The action of a ∈ G M on the d-th piece is by a−d . For example, if E is an
elliptic curve without complex multiplication, then for M = h1 ( E), G M = GL(2) and
w : Gm → GL(2) is the inverse of the canonical injection. (this is not correct!)
72
For a motive M, there are `-adic realizations H` ( M ) coming from e´ tale cohomology.
After we fix a prime `, the representation ρ M,` is unramified at all but finitely many places.
For those unramified places v, put ϕv ( M ) = w( Nv1/2 )ρ M,` (Frobv ).
Let T = 1(−1) be the Tate motive, and let t : Gmot (k) → GT be the canonical projection.
For any motive M, let G1M be the image of ker(t) under the projection Gmot (k) → G M .
4.12.2 Conjecture (Serre). Let M be a motive over a number field k. Let K be a maximal compact
¯ and determine a unique conjugacy
subgroup of G1M . The elements ϕv ( M) have eigenvalues in Q,
1
\
class (independent of `) ϕv ( M ) ∈ G M (C) . The set { ϕv ( M )} ⊂ G1M (C)\ is equidistributed with
respect to the pushforward of the normalized Haar measure on K.
4.13
The Bloch-Kato conjecture
Let E be an elliptic curve over Q. Recall that the (strong) Birch and Swinnerton-Dyer
conjecture is the formula
lim
s →1
Ω E RegE #X( E) ∏ p c p
L( E, s)
=
,
(s − 1)rk E
#E(Q)2tors
together with the claim that everything involved is well-defined and finite. For a number
field k, recall that the Dedekind zeta function of k is the series
ζ k (s) =
∑
a⊂ok
1
,
( Na)s
where Na = [ok : a] for an ideal a. It is a theorem that ζ k has an analytic continuation to
C r 1. The pole at s = 1, and we have the following analytic class number formula. Let r be
the order of the pole of ζ k at 1, and let r1 , r2 be the number of real (resp. complex) places of
k. Let hk be the class number of k, dk be the discriminant of k. Then
2r1 (2π )r2 |dk |−1/2 Regk hk
ζ k (s)
=
.
#µ(k)
s →1 ( s − 1 ) r
lim
The Birch and Swinnerton-Dyer conjecture as well as the class number formula are both
special cases of a very far-reaching generalization called the Bloch-Kato conjecture.
(add BSD for abelian varieties, brief statement of Bloch-Kato)
Follow [Lan91, III] for definition of regulator of abelian variety, general BSD.
5
5.1
Some theorems of Faltings
Background and Tate’s conjecture
The goal of this section is to describe the relationships between a web of conjectures that
Faltings proved in his groundbreaking paper [Fal86].
Let A be a d-dimensional abelian variety over a field k. As usual, we write k¯ for the
¯ ) for the absolute Galois group of k. Fix a prime
algebraic closure of k and Gk = Gal(k/k
` invertible in k. The groups A[`n ] = { x ∈ A(k¯ ) : `n x = 0} are abstractly isomorphic to
(Z/`n )⊕2d , and carry a continuous action of Gk . They fit into an inverse system
A[`] o
`
A[`2 ] o
`
73
A[`3 ] o
···
Put
T` A = lim A[`n ] = ( xn ) ∈
←−
∏ A[`n ] : `xn+1 = xn
.
2d
This is the `-adic Tate module of A. As a Z` -module, T` A ' Z⊕
` . What makes T` A interesting
is that it carries a continuous action of Gk , induced by the action of Gk on the A[`n ]. In other
words, after choosing a basis of T` A, we have a continuous representation
ρ A,` : Gk → GL(2d, Z` ).
The action of Gk on T` A factors through the smaller group GSp(2n, Z` ) of symplectic
simlitudes. One sees this via the Weil pairing. There is, for any n invertible in k, a natural
perfect pairing A[n] × A∨ [n] → µn , defined at the level of schemes. For a prime ` invertible
in k, these pairings patch together to give a perfect Gk -equivariant pairing T` A × T` A∨ →
Z` (1). After a choice of polarization λ : A → A∨ , we get an (alternating) pairing T` A ×
T` A → Z` (1). If ` is relatively prime to the degree of λ, then this pairing is perfect. In what
follows, we will always assume this to be the case. For a proof of these facts in a pretty
general setting, see [vdGM, 11].
It is natural to ask how much ρ A,` “knows about” A, especially if k is a number field, or
more generally, a finitely generated field.
Let X be a nice variety over a finitely generated field k. For each i, there is a canonical
homomorphism
cl : Ai ( X ) → H2i ( Xks , Q` )(i ),
defined in [Del77, VI 2.2.10]. One calls cl( Z ) the cohomology class associated with a cycle Z.
∼
5.1.1 Conjecture (Tate). The cycle map induces an isomorphism Ai ( X ) ⊗ Z` −
→ H2i ( Xks , Q` )(i )Gk .
This is essentially Conjecture 1 in [Tat65]. Often, “the Tate conjecture” means the following special case.
5.1.2 Conjecture (Tate). Let A, B be abelian varieties over a finitely generated field k. For any
prime ` invertible in k, the natural map
homk ( A, B) ⊗ Q` → homGk (V` A, V` B)
is a bijection.
See the remarks after Conjecture 1 in Tate’s paper, or [FWG+ 84, IV.1.4], for a proof that
the second version of the conjecture follows from the first. Another way of stating the
(second version of the) Tate conjecture is that for any finitely generated field k, the functor
V` : AbVariso
k → RepGk ( Q` ) is fully faithful.
b the
5.1.3 Example. Let k = Fq be a finite field. Then Gk is naturally isomorphic to Z,
b corresponds to the arithmetic Frobenius Frobq ∈ GF ,
profinite completion of Z. Here 1 ∈ Z
q
q
given by x 7→ x . Representations ρ : GFq → GL(n, Q` ) are determined by ρ(Frobq ). If such
a representation is semisimple, the Brauer-Nesbitt theorem tells us that ρ is determined by the
characteristic polynomial of ρ(Frobq ). For an abelian variety A over Fq , we know that the
characteristic polynomial of ρ A,` (Frobq ) is PA (t) ∈ Z[t], which determines A up to isogeny
by Honda-Tate theory.
74
Since we will be using characteristic polynomials quite a lot, let us state a suitably
general version of the Brauer-Nesbitt theorem. Fix a field k, and for an arbitrary group G,
let K0 ( G ) denote the Grothendieck group of finite-dimensional k-representations of G. By
the “characteristic polynomial” of a representation ρ : G → GLk (V ), we mean the map
χρ : G → Λ(k) = 1 + tkJtK defined by
χρ ( g) =
1
.
det(1 − ρ( g) · t, V )
d
Alternatively, t dt
log χρ ( g) = ∑ tr(ρ( g)n ).
5.1.4 Theorem (Brauer-Nesbitt). If S spans k[ G ] as a k-vector space, then the map χ : K0 ( G ) →
Λ(k)S given by [ρ] 7→ χρ is an injection.
Proof. This is Theorem 5.21 of [Egg11].
5.1.5 Corollary. If k has characteristic zero and ρ1 , ρ2 : G → GLk (V ) are two semisimple representations with identical characters, then ρ1 ' ρ2 .
5.1.6 Theorem (Faltings’ isogeny theorem). Let A and B be abelian varieties over a number field
k. For any prime `, we have ρ A,` ' ρ B,` as Gk -modules if and only if A and B are isogenous over k.
From this, we see that we can fruitfully study A via ρ A,` . For example, the rank of an
abelian variety only depends on its isogeny class, so rk A only depends on ρ A,` .
If k is either finite or a global field, the representation ρ A,` is semisimple, so ρ A,` is
ˇ
determined by the characteristic polynomial of ρ A,` (Frobq ). For this, one needs the Cebotarev
density theorem.
5.2
Image of Frobenius for number fields
Fix a number field k, and a finite place v of k. Let p ⊂ o = ok be the corresponding maximal
ideal. Let k v be the completion of k at v. We choose k¯ ⊂ k v ; this gives a map Gkv → Gk ,
defined by σ 7→ σ |k¯ . This map is only well-defined up to conjugation. By Krasner’s lemma,
b where
the map is an injection. Reduction modulo p gives a homomorphism Gkv → Gκv = Z,
κv = ov /p is the residue field of p. This map is surjective, so we have an exact sequence
(where we write Dv for the image of Gkv in Gk ):
/ Iv
1
/ Dv
/Z
b
/ 1.
The group Gκv is procyclic, with generator Frob Nv , where as usual Nv = #κv . Write Frobv
for a lift of Frob Nv to Dv . The element Frobv ∈ Gk is only well-defined up to conjugacy and
multiplication by Iv .
As before, let A be an abelian variety over k with good reduction at v. Then (by definition)
there exists an abelian scheme A over ov whose generic fiber is Akv . The scheme A fits into
a commutative diagram with cartesian squares:
Av
/Ao
Akv
Spec(κv )
/ Spec(ov ) o
Spec(k v )
75
We have define Av = Aκv . The property of being abelian is stable under base change, so Av
is an abelian variety over κv , and we have a reduction map
A(k v ) = A(ov ) → A(κv ) = Av (κv ).
Extending to algebraic closures, we get a map A(k v ) → Av (κv ). This is a homomorphism
with pro-p kernel. Let ` - v (i.e. ` is relatively prime to the residue characteristic of v). At the
level of torsion, we have isomorphisms
A(k v )[`n ] → Av (κv )[`n ].
The map is injective because its kernel is pro-p, and it is surjective by cardinality considerations – both groups have cardinality (`n )2d ). This gives us an isomorphism A(k¯ )[`n ] =
∼
A(k v )[`n ] −
→ Av (κv )[`n ]. These groups have (compatible) actions of Gk , Gkv and Gκv . In
particular, the inertia group Iv acts trivially on A(k¯ )[`n ].
It follows that Frobv , a priori only well-defined up to conjugacy and multiplication by
Iv , has a well-defined action on A[`n ], and hence on T` A. That is, we have the following
theorem.
5.2.1 Theorem. Let A be an abelian variety over k with good reduction at v. Then for v - `, we have
1. ρ A,` is unramified at v (i.e. ρ A,` ( Iv ) = 1)
2. ρ A,` (Frobv ) is well-defined up to conjugacy and has characteristic polynomial PAv (t) with
integer coefficients that do not depend on `.
Proof. See 3.4 for a definition of PAv (t). This is Theorem 1, paired with the corollary to
Theorem 3 in [ST68].
ˇ
Recall the Cebotarev
density theorem. Let k be a number field, K/k a finite Galois extension. For v unramified in K/k, there is a well-defined conjugacy class Frobv ∈ Gal(K/k)\ .
ˇ
Cebotarev’s
density theorem is essentially the Sato-Tate conjecture for the motive h(Spec K ),
i.e. it predicts equidistribution of Frobenii, in the appropriate sense.
ˇ
5.2.2 Theorem (Cebotarev).
Let K/k be a finite Galois extension of number fields with Galois
group G. Then {Frobv } ⊂ G \ is equidistributed with respect to the Haar measure on G.
Proof. See [Ser68, 1.2.2] for a beautiful proof using the representation theory of compact
groups.
Recall that the statement “{Frobv } ⊂ G \ is equidistributed” means that for any conjugacy
class C ⊂ G, we have
lim
x →∞
{#v : Nv 6 x and Frobv ∈ C }
#C
=
#{v : Nv 6 x }
#G
It follows that each conjugacy class in G is Frobenius for infinitely many primes.
For example, if k = Q and K = Q(ζ n ), then Gal(K/Q) is naturally isomorphic to
(Z/n)× . For p - n, the Frobenius Frob p corresponds to p ∈ (Z/n)× . Dirichlet’s theorem
says that for a ∈ (Z/n)× , there exist infinitely many p such that p ≡ a (mod n), i.e. the
ˇ
Cebotarev
density theorem holds for cyclotomic extensions.
5.2.3 Theorem (N´eron-Ogg-Shafarevich). Let A be an abelian variety over a number field k. Let
v be a place of k, and let ` be a prime with v - `. Then A has good reduction at v if and only if ρ A,` is
unramified at v.
Proof. This is the main theorem of [ST68].
76
5.3
L-function of an abelian variety
Let’s define the L-function of an arbitrary abelian variety over a number field k. For a place
v of k, choose a prime ` with v - `. The action of Frobv on T` A is only well-defined up to the
action of Iv , but ( T` A) Iv = T` A/{σx − x : x ∈ Iv } has a well-defined action of Frobv . Define
Lv ( A, t) = det (1 − ρ A,` (Frobv ) · t, ( T` A) Iv )
−1
L( A, s) = ∏ Lv A, ( Nv)−s
.
v-∞
This is well-defined by the following theorem.
5.3.1 Theorem. Let A be an abelian variety over a number field k. For any finite place v, the local
factor Lv ( A, t) is an element of Z[t] that does not depend on `.
Proof. For v a place of good reduction, this is Theorem 5.2.1. The general case is a bit more
subtle. First, note that
1
∨ Iv
det(1 − ρ A,` (Frobv ) · t, ( T` A) Iv ) = det(1 − ρ A,` (Frob−
v ), (( T` A ) ) ).
The Weil pairing gives us an isomorphism ( T` A)∨ = T` A(−1), and because the `-adic
cyclotomic character is unramified at v - `, we get (( T` A)∨ ) Iv = ( T` A) Iv (−1).
Let A be the N´eron model for A over ov , and let Av be the connected component
of the identity in Aκv . By Lemma 2 of [ST68], there is a Dv -equivariant isomorphism
( T` A) Iv → T` Av .
Chevalley’s theorem (see [Con02] for a modern proof) gives us a linear algebraic group
G ⊂ Av such that B = Av /G is an abelian variety. In other words, we have a short exact
sequence
/G
/ Av
/B
/ 0.
1
The group G splits into a product G = T × U, where T is a (possibly non-split) torus and U
is unipotent. The group U will be an iterated extension of copies of G a , so T` U = 0.
b = hom ¯ ( T¯ , G ¯ ) be the group of characters of T. This has an obvious continuous
Let T
k k
m,k
Gk -action, and there is a Gk -equivariant pairing
b → Z ` (1),
T` T ⊗ T
¯ that this pairing is
given by ( xn )n ⊗ χ 7→ (χ( xn ))n . It is easy to see (by base-change to k)
∨
b
nondegenerate, so we have a Gk -isomorphism T` T ' T ⊗ Z` (1). We obtain
b ⊗ Z ` ).
det(1 − ρ Av ,` (Frobv ) · t) = det(1 − ρ B,` (Frobv ) · t, T` B) · det(1 − Frobv−1 · t, T
b ⊗ Z` via its action on T,
b the characteristic
Since B does not depend on ` and Gk acts on T
polynomial of Frobenius is an element of Z[t] independent of `.
Let’s check that our definition of L( A, s) agrees with our previous definition in the case
that A = E is an elliptic curve over k. If E has good reduction at v, ( T` E) Iv = T` E and
there is nothing to prove. If E has bad reduction at v, then as before let Ev be the connected
77
component of the special fiber of the N´eron model at v. Recall that E has multiplicative
reduction at v if Ev is a torus, and additive reduction if Ev is unipotent. Clearly


2 good reduction
rkZ` ( T` E) I p = 1 mult. reduction


0 add. reduction
If E has split multiplicative reduction at v, then the local factor Lv ( E, t) is the (reverse of) the
characteristic polynomial of Frobenius acting on T` Gm (−1) = Z` . On the other hand, if E
has nonsplit multiplicative reduction at v, then by [Sil09, III,2.5], Ev splits after a quadratic
bv as −1, whence the following:
base-change, from which we see that Frobv acts on E

1
additive reduction



1 − t
split multiplicative reduction
Lv ( E, t) =

1+t
nonsplit multiplicative reduction



1 − a p t + pt2 good reduction
The function L( A, s) should have an analytic continuation, functional equation, it should
satisfy BSD (ords=1 L( A, s) = rkZ A(Q)), and a “fancy BSD” with a precise prediction of the
coefficient in Taylor series. By the Weil conjectures, the function L( A, s) converges on some
region {=s > c}.
5.4
Tate conjectures and isogenies
Much of the rest of this section follows Lang’s excellent survey [Lan91] and the more
technical [FWG+ 84]. We begin with a useful fact.
5.4.1 Theorem. Let k be a number field and ρ1 , ρ2 : Gk → GL(n, Z` ) continuous semisimple
representations. If tr ρ1 (Frobv ) = tr ρ2 (Frobv ) for all v in a density-one set of places, then ρ1 ' ρ2 .
ˇ
Proof. By the Cebotarev
density theorem, the set {Frobv } ⊂ Gk\ is dense. It follows that
tr ρ1 = tr ρ2 , so the conclusion follows from the Brauer-Nesbitt theorem.
We are mainly interested in the case when ρ1 = ρ A,` and ρ2 = ρ B,` for A, B abelian
varieties over k. The theorem tells us that if PAv = PBv for almost all primes, then ρ A,` ' ρ B,` .
A morphism f : A → B of abelian varieties induces a Gk -equivariant morphism f ∗ : T` A →
T` B. If f is an isogeny, then f ∗ is an isomorphism after tensoring with Q. In particular, if we
think of ρ A,` as a Q` -representation, then ρ A,` ' ρ B,` . It follows that A and B have the same
bad primes.
The following theorem was conjectured by Tate, and proved when k is a finite field.
5.4.2 Theorem (Faltings). Let k be a finitely generated field and A, B abelian varieties over k. Then
1. (Semisimplicity) V` A is a semisimple Gk -module.
2. (Tate conjecture) The natural map
hom( A, B) ⊗ Z` → homGk ( T` A, T` B)
is an isomorphism.
78
Proof. See [FWG+ 84] for a proof when k has characteristic zero. Alternatively, see [Mila,
IV.2.5] for a proof that semisimplicity and the Tate conjecture follow from Theorem 5.5.4.
5.4.3 Corollary (Isogeny theorem). Abelian varieties A and B over a number field k are isogenous
if and only if ρ A,` and ρ B,` are isomorphic.
Proof. We’ve already seen that if A and B are isogenous, then ρ A,` ' ρ B,` . The Tate conjecture
gives us an isomorphism
hom( A, B) ⊗ Z`
∼
/ homG ( T` A, T` B).
k
Assuming ρ A,` and ρ B,` are isomorphic, we can choose a specific isomorphism f : ρ A,` →
ρ B,` . Since homGk ( T` A, T` B) is a finite rank Z` -module isomorphic to hom( A, B) ⊗ Z` ,
the space hom( A, B) is dense in homGk ( T` A, T` B). The property “ f : T` A → T` B is an
isomorphism” is open, so there exists ϕ : A → B such that ϕ∗ is an isomorphism. We claim
that ϕ is an isogeny. Indeed, let C = (ker ϕ)◦ ⊂ A. If C 6= 0, then rk T` C > 0. This cannot
be, because ϕ∗ ( T` C ) = 0 and ϕ∗ is an isomorphism. Thus C = 0, so ker ϕ is finite. By
dimension considerations, ϕ is an isogeny.
By [FWG+ 84, V.3.2], if ρ A,` ' ρ B,` , there actually exists an isogeny f : A → B with
` - deg f .
Choose a finite place v of k at which A has good reduction. The polynomial PAv (t) is
integral, monic, and has degree 2d. So we can write
PAv (t) = t2d − av ( A)t2d−1 + · · ·
√
where av ( A) = tr ρ A,` (Frobv ) ∈ Z. We have | av ( A)| 6 2g Nv, since the roots of PAv have
√
absolute value Nv.
If A = E is an elliptic curve over Q, then this definition of av ( E) agrees with the the
standard definition a p ( E) = p + 1 − #E(F p ). If C is a nice curve over Q of genus g, then for
J = Jac C, we have #C (F p ) = p + 1 − a p ( J ).
5.4.4 Theorem. Let A and B be abelian varieties over a number field k. Let S be a density-zero set of
places of k, containing the infinite places, as well as the bad places for A and B. Then A is isogenous
to B if and only if av ( A) = av ( B) for all v ∈
/ S.
Proof. Fix a prime `. If A and B are isogenous, then ρ A,` ' ρ B,` , so av ( A) = tr(ρ A,` (Frobv )) =
tr(ρ B,` (Frobv )) = av ( B) for all places v ∈
/ S ∪ {`}. The converse is an immediate corollary
of 5.4.1. and the isogeny theorem (5.4.3).
This theorem is not especially useful, because it requires checking av ( A) = av ( B) at an
infinite set of places. The following lemma and its corollary give us a way of capturing the
isogeny class of an abelian variety using a finite amount of data.
5.5
Finiteness theorems
5.5.1 Lemma (Faltings). Let k be a number field, S a finite set of places, and n > 1 an integer.
Then there is a finite set T of places, disjoint from S and depending only on (k, S, n), such that if
ρ1 , ρ2 : Gk,S → GL(n, Z` ) are continuous representations with tr ρ1 (Frobv ) = tr ρ2 (Frobv ) for all
v ∈ T, then ρ1 ' ρ2 .
79
Proof. Without loss of generality, we can assume S contains all places dividing `. Let d =
2
`2n = #Mn (F` ) × Mn (F` ). By Hermite’s theorem, there are only finitely many extensions
of k unramified outside S with degree 6 d. Let K/k be a Galois extension containing all
ˇ
these and let G = Gal(K/k). The Cebotarev
density theorem tells us that there is a finite set
T (disjoint from S) of places of k such that G \ = {Frobv : v ∈ T }.
Now let ρ1 , ρ2 be as in the statement of the lemma. Set ρ = ρ1 × ρ2 : Gk,S → GL(n, Z` ) ×
GL(n, Z` ). Let R be the Z` -subalgebra of Mn (Z` ) × Mn (Z` ) generated by the image of ρ.
Note that R is a free Z` -module of rank at most 2n2 . We can consider the reduction of ρ
modulo `, i.e. ρ¯ : Gk → ( R/`)× . We know that #( R/`)× 6 d, so ρ¯ factors through G as in
the commutative diagram:
Gk
ρ¯
/ ( R/`)×
;
G
It follows that R/` is generated (as a group) by the images of {ρ([Frobv ]) :∈ T }. By
Nakayama’s lemma, R is generated as a Z` -module by {ρ([Frobv ]) : v ∈ T }.
Let ϕ : R → Z/` be the map ( g, h) 7→ tr g − tr h. This is a homomorphism of Z` -modules.
Assume av ( A) = av ( B) for all v ∈ T. Then for v ∈ T, we have
ϕ(ρ(Frobv )) = tr ρ A,` (Frobv ) − tr ρ B,` (Frobv ) = av ( A) − av ( B) = 0
This implies ϕ = 0 since ϕ vanishes on a set of generators of R. It follows that tr ρ A,` = tr ρ B,` .
Since ρ A,` and ρ B,` are semisimple, this implies ρ A,` ' ρ B,` , and the isogeny theorem tells us
that A and B are isogenous.
5.5.2 Corollary. Let k be a number field, S a finite set of places of k and d > 1 an integer. Then
there is a finite set T of places of k, disjoint from S and depending only on (k, S, d), such that if A
and B are d-dimensional abelian varieties with good reduction outside S, then A and B are isogenous
if and only if av ( A) = av ( B) for all v ∈ T.
In the next section, we will prove the Mordell conjecture for number fields using Faltings
proof of a finiteness result for abelian varieties.
5.5.3 Conjecture (Shafarevich, for abelian varieties). Fix a number field k and a finite set S of
places, and an integer d > 1. Then there are only finitely many isomorphism classes of d-dimensional
abelian varieties over k with good reduction outside S.
Since isogenous abelian varieties have the same dimension and bad primes, the conjecture breaks up into two pieces.
5.5.4 Theorem (Finiteness I). Let A be an abelian variety over a number field k. Then there are
only finitely many isomorphism classes of abelian varieties over k which are isogenous to A.
Proof. This is a very brief sketch of the proof in [FWG+ 84, V.3], which has several parts. First,
one reduces to the case of principally polarized abelian varieties with semistable reduction
everywhere.
Next, one constructs the Faltings height h( A) of an arbitrary d-dimensional abelian variety
A over k as follows. Let A be the N´eron model A of A over o = ok , let s : Spec(o) → A be
the identity section, and define
∨
ωA/o = s∗ ΩdA/o .
80
This is a projective o-module of rank one. If v is an infinite place of k corresponding to
σ : k ,→ C, the vector space ωA/o ⊗o,σ C has an inner product defined by
d Z
i
¯
hη, ξ iv =
η ∧ ξ.
2
A(C)
This inner product induces a natural norm. For any place v, put ε v = 1 if v is real, and
ε v = 2 if v is complex. We define the Faltings height of A as h( A) = [k : Q]−1 deg(ωA/o ),
where
deg (ωA/o ) = log #(ωA/o /x ) − ∑ ε v log k x kv .
v|∞
for any nonzero x ∈ ωA/o . See [Mila, VI.6] for a proof that this is independent of x, and a
more detailed construction of h( A).
By relating the Faltings height to a natural Arakelov height on the moduli space of
principally polarized abelian varieties, Faltings proved that the set
{semistable principally polarized A/k with dim A = d and h( A) 6 c}
is finite for any c [FWG+ 84, II.4.3]. Moreover, for any A/k principally polarized with
semistable reduction, there exists an integer N > 1 such that if f : B → A is an isogeny
with (deg f , N ) = 1, then h( B) = h( A) [FWG+ 84, V.3.5]. Finally, there exists a finite set
A1 , . . . , An of abelian varieties isogenous to A such that if B is any abelian variety isogenous
to A, then there is an isogeny f : B → Ai with ( N, deg f ) = 1 [FWG+ 84, V.3.4].
¨
There is an alternative approach to Theorem 5.5.4 due to Masser and Wulsthoz
[MW93].
5.5.5 Theorem (Finiteness II). Let d > 1 be an integer, k a number field and S a finite set of places
of S. Then there are only finitely many isogeny classes of d-dimensional abelian varieties over k with
good reduction outside S.
Proof. Take A over k of dimension d > 1, with good reduction outside S. Lemma 5.5.2 gives
us a finite set T of places for which the isogeny class of A is determined
by { av ( A) : v ∈ T }.
√
Recall that the av ( A) are integers with absolute value 6 2g Nv. It follows that there are
only finitely many possibilities for the av , and hence only finitely many isogeny classes of
d-dimensional abelian varieties over k with good reduction outside S.
5.5.6 Conjecture (Shafarevich, for curves). Fix a number field k, an integer g > 1, and a finite set
S of places of k. Then there are only finitely many nice curves over k of genus g with good reduction
outside S.
Proof. Let J be the jacobian of C. Then J is an abelian variety over k of dimension g, with good
reduction outside S. There are only finitely many possibilities for J (up to isomorphism).
Recall that C is determined by ( J, θ ). By [NN81], abelian varieties over algebraically closed
fields have only finitely many isomorphism classes of principal polarizations. Thus there
can be only finitely many C corresponding to J.
5.6
Proof of the Mordell conjecture
Following Parshin [Par68], and the more expository accounts in [Lan91, IV.2] and [FWG+ 84,
V.4], we show that the Shafarevich conjecture implies the Mordell conjecture.
81
5.6.1 Conjecture (Mordell). Let C be a nice curve of genus g > 2 over a number field k. Then
C (k ) is finite.
A key ingredient is the following technical lemma.
5.6.2 Lemma. Let C be a nice curve of genus g > 2 over a number field k. Then there is a finite
extension k0 /k and a finite set S0 of places of k0 satisfying the following. For every x ∈ C (k ) there is
a nice curve Wk over k0 with good reduction outside S0 , and a morphism vx : Wx → Ck0 ramified
exactly at x (with ramification index 6 2) such that deg ϕ x 6 2 · 4g .
Proof. Assume C (k) 6= ∅, and let j : C ,→ J = Jac C be the embedding induced by some
e be
fixed x ∈ C (k ). The map “multiplication by two” is an e´ tale self-covering of J; we let C
its pullback:
/J
e
C
ϕ
C
j
/J
2
From the Chevalley-Weil theorem [BG06, 10.3.11], we obtain a finite extension L/k such that
e( L) for all x ∈ C (k). For any x ∈ C (k ), choose distinct x1 , x2 ∈ C
e( L) such that
ϕ −1 ( x ) ⊂ C
e
ϕ( xi ) = x. There exists a divisor D ∈ Div(C ) defined over some finite extension k0 /k (which
e for some rational function f .
does not depend on x) such that x1 − x2 + 2D = ( f ) in Jac C
p
0
0
e
e
e)[ f ] of function fields. It’s
Let ϕ x : Wx → Ck0 correspond to the inclusion k (C ) ,→ k (C
not to hard to show that Wx has the desired properties. See [Lan91, IV.2.1] for details.
5.6.3 Theorem. The Mordell conjecture is true.
Proof. Let C be a nice curve of genus g > 2 over a number field k. Suppose that C (k) is
infinite. By Lemma 5.6.2, there is a finite extension k0 /k and morphisms ϕ x : Wx → Ck0 for
each x ∈ C (k).
The genus of Wx is bounded (using Riemann-Hurwitz) since the deg ϕ x is bounded and
ϕ x is only ramified only at x. The Shafarevich conjecture tells us there are only finitely many
possibilities for the Wx . In particular, there exists W/k0 that is isomorphic to infinitely many
∼
ϕx
Wx . We have maps W −
→ Wx −→ Ck , unramified only at x. Choose k0 ,→ C; this gives
a morphism ϕ x : W (C) → C (C) of compact Riemann surfaces, ramified only at x. This
contradicts Theorem 5.6.4.
5.6.4 Theorem (de Franchis). Let X and Y be nice curves of genus > 2 over a field of characteristic
zero. Then there are only finitely many non-constant morphisms X → Y.
Proof. The theorem was originally proved by de Franchis for Riemann surfaces. See [Lan60,
p.29] for a general proof.
A
Brief introduction to e´ tale cohomology
Our definition of e´ tale morphisms follows [Gro67, 17.1]. For more detailed introductions on
e´ tale cohomology, see [Milb] and [Del77].
82
A.0.5 Definition. A morphism f : X → Y of schemes is formally e´ tale if whenever V is an affine
scheme over Y and V0 ⊂ V is defined by a nilpotent ideal, then for every v0 : V0 → X over Y, there
exists a unique v : V → X making the following diagram commute.
/V
V0
v0

X
v
f
/Y
We say f : X → Y is e´ tale if it is formally e´tale and locally of finite presentation.
For a scheme X, let Xe´ t be the category of e´ tale morphisms U → X. Every morphism in
Xe´ t is e´ tale. We say that a collection {Ui → V } of morphisms in Xe´ t is an e´ tale cover if each
Vi → X is e´ tale, and moreover the map ä Ui → V is surjective. With this notion of coverage,
the category Xe´ t is a site, the so-called small e´tale site of X. Let Sh( Xe´ t ) denote the category of
sheaves of abelian groups on Xe´ t . The global sections functor Γ : Sh( Xe´ t ) → Ab, F 7→ F ( X )
is left-exact, so we can put as usual
Hei´ t ( X, −) = Hi ( Xe´ t , −) = Ri Γ : Sh( Xe´ t ) → Ab
for the i-th right derived functor of Γ.
If X is a variety over a non-algebraically closed field and F is an e´ tale sheaf on X, then
one often considers Hei´ t ( Xks , F ) instead of Hei´ t ( X, F ). The former has a continuous action
of Gk = Gal(ks /k ). Alternatively, we can let f : X → Spec(k ) be the structure morphism
and consider the higher pushforward sheaves Ri f ∗ F on Spec(k)e´ t ' Gk -Mod. By [Del77, I
4.1.1], these are naturally isomorphic if X is proper.
Suppose X is a scheme on which n is invertible. Let Sh( Xe´ t , Z/n) be the category of
sheaves of Z/n-modules on Xe´ t . Since the e´ tale topology is subcanonical, the group µn of
i
n-th roots of unity is in Sh( Xe´ t , Z/n). Put Z/n(i ) = µ⊗
n , and for any F in shea f ( Xe´ t , Z/n ),
define F (i ) = F ⊗Z/n Z/n(i ). This gives functors (i ) : Sh( Xe´ t , Z/n) → Sh( Xe´ t , Z/n). If
f : X → Y is a morphism, then (i ) ◦ f ∗ = f ∗ ◦ (i ) as functors Sh( Xe´ t , Z/n) → Sh(Ye´ t , Z/n).
Since (i ) is exact, we compute
R f ∗ ◦ (i ) = R f ∗ ◦ R(i ) ' R( f ∗ ◦ (i )) = R((i ) ◦ f ∗ ) ' R(i ) ◦ R f ∗ = (i ) ◦ R f ∗ ,
as functors D( Xe´ t , Z/n) → D(Ye´ t , Z/n). It follows that for X proper over a field k in which
n is invertible, we have H• ( Xks , F (i )) ' H• ( Xks , F )(i ) for all i.
For simplicity, suppose X is proper over an algebraically closed field k in which a prime
` is invertible. Let Z/`n denote the constant sheaf associated to the group Z/`n . The
functoriality of e´ tale cohomology yields maps
Hi ( Xe´ t , Z/`) ← Hi ( Xe´ t , Z/`2 ) ← · · ·
and we put Hi ( Xe´ t , Z` ) = lim Hi ( Xe´ t , Z/`n ) and Hi ( Xe´ t , Q` ) = Hi ( Xe´ t , Z` ) ⊗ Z. This
←−
definition fits into the general framework of `-adic sheaves set up in [Del80].
For X a variety over a finite field Fq , the assignment X 7→ He•´ t ( XFq , Z` ) constitutes a Weil
cohomology theory in the sense of [And04, 3.3].
83
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¨
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