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ON EISENSTEIN IDEALS OF PRIME POWER LEVEL
HWAJONG YOO
A BSTRACT. Let p be a prime and let r ≥ 2 be an integer. Consider the modular curve
X0 ( pr ) and its Jacobian J0 ( pr ). For a prime ` not dividing 2p, we study an Eisenstein
maximal ideal m of level pr containing `. We show that the dimension of J0 ( pr )[m] is 2.
C ONTENTS
1.
Introduction
1
2.
Properties of degeneracy maps
2
3.
Eisenstein maximal ideals
3
4.
The rational cuspidal group and Shimura subgroup
4
5.
Eisenstein series
5
6.
The proof of the main theorem
6
References
7
1. I NTRODUCTION
Fix an integer N and let X0 ( N ) denote the modular curve over Q associated to Γ0 ( N ).
Let J0 ( N ) denote the Jacobian of X0 ( N ). Let T := T( N ) denote the Z-subalgebra of
End( J0 ( N )) generated by the Hecke operators Tn for all n ≥ 1. We say an ideal of T
Eisenstein if it contains I0 := ( Ts − s − 1 : for primes s - N ).
In his landmark paper [7], Mazur studied the Eisenstein ideal when N is prime. The
author generalized some part of his work to the case when N is square-free [12, 13].
More specifically, we computed the index of an Eisenstein ideal and the dimension of
J0 ( N )[m] := { x ∈ J0 ( N )(Q) : Tx = 0 for all T ∈ m},
where m is an Eisenstein maximal ideal containing ` not dividing 6N. In the forthcoming work [10], Ribet and the author study the structure of J0 ( N )[m] when N is
square-free.
Date: April 11, 2015.
2010 Mathematics Subject Classification. 11F33, 11F80 (Primary); 11G18(Secondary).
Key words and phrases. Eisenstein ideals, prime power level, multiplicity one.
1
2
HWAJONG YOO
In contrast to the previous works, we study Eisenstein ideals on prime power level
in this article. Let N := pr for a prime p and an integer r ≥ 2. Let ` be a prime not
dividing 2p and let m be an ideal of T containing ` and I0 . From now on, we define
two integer a and b as follows:
a :=
p−1
(12, p − 1)
and
b :=
p+1
.
(12, p + 1)
Theorem 1.1 (Main theorem). Let m0 = (`, U p , I0 ) and m1 := (`, U p − 1, I0 ).
(1) If m is maximal, then either m = m0 or m = m1 .
(2) If m = m0 (resp. m = m1 ) is maximal, then ` divides ab (resp. a).
(3) Conversely, if ` divides ab (resp. a), then m0 (resp. m1 ) is maximal.
(4) Assume that m is maximal. Then for each case, we have
J0 ( pr )[m0 ] ' E p
and
J0 ( pr )[m1 ] ' Z/`Z ⊕ µ` ,
where E p is the unique extension of µ` by Z/`Z (up to isomorphism), which is only
ramified at p and finite at `. In particular, dim J0 ( pr )[m] = 2.
The organization of this article is as follows. In §2, we discuss degeneracy maps between modular curves. This study will help us to understand eigenvectors of Hecke
operators of Jacobians from lower level. In §3, we study Eisenstein maximal ideals of
level pr and prove the first part of the main theorem. In §4, we review the properties
of the rational cuspdial subgroups and Shimura subgroups of J0 ( pr ), and find eigenvectors for the Hecke operators in these groups. In §5, we study the constant terms
of Eisenstein series (at various cusps) annihilated by U p and I0 . Finally, we prove our
main theorem in §6.
2. P ROPERTIES OF DEGENERACY MAPS
From now on, fix a prime p. Most results in this section were discussed in [8, §13].
For each r ≥ 2, let αr−1 : X0 ( pr ) → X0 ( pr−1 ) be the degeneracy covering with the
modular interpretation ( E, C ) 7→ ( E, C [ pr−1 ]), where C denotes a cyclic subgroup of
order pr in an elliptic curve E. Let β r−1 be the “other” degeneracy covering X0 ( pr ) →
X0 ( pr−1 ); it has the modular interpretation ( E, C ) 7→ ( E/C [ p], C/C [ p]). The degree of
degeneracy coverings αr−1 , β r−1 have degree p. They induce maps
α(r−1)∗ , β (r−1)∗ : J0 ( pr ) ⇒ J0 ( pr−1 ),
via the two functorialities of the Jacobian.
αr∗−1 , β∗r−1 : J0 ( pr−1 ) ⇒ J0 ( pr )
ON EISENSTEIN IDEALS OF PRIME POWER LEVEL
3
Let αr0 −1 and β0r−1 denote the transposes of αr−1 and β r−1 viewed as correspondences.
By direct computation of modular interpretations, we get
β0r−1 ◦ αr−1 = αr ◦ β0r ,
which is the Hecke correspondence Tp on X0 ( pr ).
Let U p (resp. τp ) denote the p-th Hecke operator in the Hecke ring T( pr ) (resp.
T( pr−1 )). In other words, U p is the pullback of the correspondence Tp to J0 ( pr ). Thus,
we have the following formula:
U p = β r∗ ◦ αr∗ = αr∗−1 ◦ β (r−1)∗ ;
U p ◦ αr∗−1 = αr∗−1 ◦ τp ,
U p · β∗r−1 = p · αr∗−1 .
Let γr denote the following map induced by degeneracy maps:
γr : J0 ( pr−1 ) × J0 ( pr−1 ) → J0 ( pr )
via the formula γr ( x, y) = αr∗−1 ( x ) + β∗r−1 (y).
Let J := J0 ( pr ) and T := T( pr ). The image of γr is called the old subvariety of J,
which is denoted by Jold . The quotient of J by Jold is called the new quotient of J, which
τ p
is denoted by J new . By the matrix relation U p = 0p 0 , γr is Hecke-equivariant and
hence the image of T in End( Jold ) (resp. End( J new )) is called the old (resp. new) quotient
of T, which is denoted by Told (resp. Tnew ). A maximal ideal of T is called old (resp.
new) if its image in Told (resp. Tnew ) is maximal.
From the description above, we get U p2 − τp U p = 0 in Told and U p = 0 in Tnew .
3. E ISENSTEIN MAXIMAL IDEALS
Fix a prime ` not dividing 2p. In this section, we study Eisenstein maximal ideals
containing `. Recall that a maximal ideal of T( N ) is called Eisenstein if it contains I0 :=
( Ts − s − 1 : for primes s - N ).
Lemma 3.1. Let m be an Eisenstein maximal ideal of T( pr ) containing ` and U p − e( p). Then,
e( p) is either 0 or 1 modulo m.
Proof. When r = 1, we get e( p) = 1 by Mazur (cf. [3] or [13, Proposition 5.5]).
Assume that n ≥ 2 and the assertion is true for r = n − 1. If m is new, e( p) = 0 from
the result in the previous section. (For more general cases, see [4, Proposition 3.28].)
Thus, we may assume that m is old. By abuse of notation, m denotes the corresponding
maximal ideal of T( pn−1 ) to m. Then, U p2 − τp U p = 0 in T/m ' Told /m ' T( pn−1 )/m.
Since τp ≡ 0, 1 (mod m) by induction hypothesis, e( p) is either 0 or 1 modulo m. Thus,
the result follows by induction.
4
HWAJONG YOO
We define Eisenstein ideals of T( pr ) as follows:
I0 := (U p , I0 ),
I1 := (U p − 1, I0 ),
mi := (`, Ii ).
Let Ii0 (resp. mi0 ) be the ideal of T( pr−1 ) corresponding to Ii (resp. mi ) of T( pr ).
Lemma 3.2. The map αr∗−1 induces an embedding of J0 ( pr−1 )[ Ii0 ] into J0 ( pr )[ Ii ], and
J0 ( pr−1 )[mi0 ] ' αr∗−1 ( J0 ( pr−1 )[mi0 ]) = J0 ( pr )old [mi ]
unless r = 2 and i = 0.
Proof. Note that αr∗−1 is injective and we have the matrix representation U p =
τp p 0 0
on
J0 ( pr )old , where U p ∈ T( pr ) and τp ∈ T( pr−1 ). Therefore the results follows.
4. T HE RATIONAL CUSPIDAL GROUP AND S HIMURA SUBGROUP
In this section, we assume that p ≥ 3.
4.1. The rational cuspidal group. Let
x
pi
denote a cusp of level pi on X0 ( pr ), where
0 ≤ i ≤ r and x such that ( x, p) = 1 and x taken modulo ( pi , pr−i ). In this notation,
the cusp ∞ (resp. 0) corresponds to p1r (resp. p10 ). Thus, the number of cusps of
X0 ( pr ) is pk−1 ( p + 1) (resp. 2pk ) if r = 2k (resp. r = 2k + 1).
We denote by ( Pi )r the divisor of the sum of all the cusps of level pi on X0 ( pr ) and
C1 := [0 − ∞] ∈ J0 ( p)
and
Cr := [( p − 1)( P0 )r − ( P1 )r ] ∈ J0 ( pr ) for r ≥ 2.
Let C2 := C2 and Cr := αr∗−1 (Cr−1 ); let D1 := C1 and Dr := αr∗−1 (Dr−1 ).
Lemma 4.1. The order of Cr (resp. Dr ) is ab (resp. a). Furthermore, Cr (resp. Dr ) is annihilated
by I0 (resp. I1 ).
Proof. Since αr∗ is injective, the orders of Cr and C2 (resp. Dr and D1 ) coincide. Thus, by
Ling [5] (resp. Ogg [9]), the first assertion follows.
For the second one, by the same argument as in Lemma 3.2, it suffices to prove that
C2 (resp. C1 ) is annihilated by I0 (resp. I1 ).
• By Mazur [7, §II, Proposition 11.1], C1 is annihilated by I1 .
• By direct computation, we get ( Ts − s − 1)(C2 ) = 0 for any prime s 6= p. (It is
well-known that the cuspidal group is “Eisenstein”.) For U p operators, we recall
the formula in §2: U p = α1∗ ◦ β 1∗ . Since β 1∗ (C2 ) = 0 (cf. [5, p. 43]), U p (C2 ) = 0.
ON EISENSTEIN IDEALS OF PRIME POWER LEVEL
5
Remark 4.2. By direct computation (cf. [5, p. 42]), we get
Cr = p
r −2
Cr
and
Dr = p
r −1
r
( P1 ) − ∑ pm(i) ( Pi )r ,
r
i =1
where m(i ) = max{r − 2i, 0}.
4.2. The Shimura subgroup. Let Σ(r ) denote the Shimura subgroup of J0 ( pr ), which
is the kernel of the natural map J0 ( pr ) → J1 ( pr ).
Proposition 4.3. The following hold.
(1) Σ(r ) is cyclic of order pn × a, where n = k − 1 if r = 2k; and n = k if r = 2k + 1.
(2) Σ(r ) is annihilated by I1 .
(3) If µ` ⊆ J0 ( pr )[`] and ` 6= p, then µ` ⊆ Σ(r )[`].
(4) For a prime ` 6= p, Σ(r )[`] 6= 0 if and only if ` divides a.
Proof. The first, second and last assertions follow by Ling-Oesterl´e [6, Corollary 1 and
Theorem 6]. The third follows by Vatsal [11, Theorem 1.1].
5. E ISENSTEIN SERIES
Let E be the unique normalized Eisenstein series of weight 2 and level p. The qexpansion of E at ∞ is
p−1
+ ∑ σ0 (n)qn ,
24
n ≥1
where σ0 (n) = ∑d|n, (d, p)=1 d. Note that E is annihilated by I1 ⊆ T( p).
Definition 5.1. We define an Eisenstein series of level p2 as follows:
E2 (z) := α1∗ (E (z)) − β∗1 (E (z))/p = E (z) − E ( pz).
For each r ≥ 3, we define an Eisenstein series of level pr as follows:
Er (z) := αr∗−1 (Er−1 (z)) = Er−1 (z) = E2 (z).
The following proposition is clear from (a variant of) Lemma 3.2.
Proposition 5.2. For each r ≥ 2, Er is annihilated by I0 .
Note that Er has a pole only at the support of the divisor Cr .
Lemma 5.3. Let P(r ) be any cusp of level p on X0 ( pr ) with r ≥ 2. Then, the constant Fourier
coefficient of Er at P(r ) is
( p − 1)( p + 1)
.
24p
6
HWAJONG YOO
Proof. By direct computation, we get E2 =
( p+1) E1 − E2
,
p
where Ei are Eisenstein series in
[1, §3]. Thus by [1, Lemma 4.4], we get the result for r = 2.
By the same argument as in loc. cit., the constant Fourier coefficient of Er at P(r ) is
equal to that of Er−1 at P(r − 1) for all r ≥ 3. Therefore the result follows.
6. T HE PROOF OF THE MAIN THEOREM
From now on, we set T := T( pr ) and J := J0 ( pr ) for r ≥ 2. Let m be an ideal of T
containing ` and I0 .
Theorem 6.1. Let m0 = (`, U p , I0 ) and m1 := (`, U p − 1, I0 ).
(1) If m is maximal, then either m = m0 or m = m1 .
(2) If m = m0 (resp. m = m1 ) is maximal, then ` divides ab (resp. a).
(3) Conversely, if ` divides ab (resp. a), then m0 (resp. m1 ) is maximal.
(4) Assume that m is maximal. Then for each case, we have
J [m0 ] ' E p
and
J [m1 ] ' Z/`Z ⊕ µ` ,
where E p is the unique extension of µ` by Z/`Z (up to isomorphism), which is only
ramified at p and finite at `. In particular, dim J [m] = 2.
Proof. The first assertion follows from Lemma 3.1. If p = 2, then ab = 1 and hence the
fourth follows. If p ≥ 3, then it follows by Lemma 4.1.
Now we assume that m is maximal.
First, consider the case where m = m1 . Then, m cannot be new and hence there is a
corresponding maximal ideal of T( pr−1 ) to m. By abuse of notation, we denote it by
m. Then for any r ≥ 2, m is not new and the same argument can apply. Since αr∗−1 is
injective for each r, we get the following explicit isomorphisms:
J [m] = Jold [m] = αr∗−1 ( J0 ( pr−1 )[m]) ' J0 ( pr−1 )[m] ' · · · ' α1∗ ( J0 ( p)[m]) ' J0 ( p)[m].
By Mazur [7, §II, Corollary 16.3], ` divides a and J0 ( p)[m] ' Z/`Z ⊕ µ` .
Next, let m = m0 . Consider a mod ` cusp form δ of level pr whose q-expansion is
∑ (Tk
mod m) × qn .
n ≥1
• Suppose that ` = 3. Then by the argument of Mazur [7, p. 86], 24δ can be
regarded as a cusp form over Z/9Z. Since the q-expansion of 24δ and 24Er
coincide, 24δ = 24Er by the q-expansion principle. Since 24δ is a cusp form, the
constant Fourier coefficient of 24Er at all cusps must vanish modulo 9. Thus, 3
divides ab by Lemma 5.3.
ON EISENSTEIN IDEALS OF PRIME POWER LEVEL
7
• Suppose that ` = 5. Then by the same argument as above, the constant Fourier
coefficient of Er at all cusps must vanish modulo `. Thus, ` divides ab.
¨
By the argument of Mazur [7, §II, Proposition 14.1], all Jordan-Holder
factors of J [m]
are either Z/`Z or µ` . Since µ` * J [m] by Proposition 4.3, we get Z/`Z ⊆ J [m].
Let J [m]e´ t be the e´ tale part of the group scheme J [m]/F` . Then J [m]e´ t is 1-dimensional
by the same argument as in [7, §II, Corollary 14.8] because ` does not divide 2p. Moreover by the same argument as in [12, Proof of Theorem 4.2], we have the following.
Lemma 6.2. J [m] is an extension of µ` ⊕n by Z/`Z for some n ≥ 1, which is unramified
outside p and finite at `.
Let ι t be the embedding of µ` to the t-th component of µ` ⊕n . Then, ι∗t ( J [m]) is an
extension of µ` by Z/`Z, i.e., ι∗t ( J [m]) ∈ ExtZ[1/p] (µ` , Z/`Z) =: Ext. By BrumerKramer [2, Proposition 4.2.1], dim Ext = 1 and Ext = h E p i. By the same argument as
in the proof of [12, Theorem 4.2], we get J [m] ' E p . Moreover as above, we get the
following explicit isomorphisms:
J [m] ' αr∗−1 ( J0 ( pr−1 )[m]) ' J0 ( pr−1 )[m] ' · · · ' α1∗ ( J0 ( p2 )[m]) ' J0 ( p2 )[m] ' E p .
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8
HWAJONG YOO
[10] Kenneth Ribet and Hwajong Yoo, The structure of the kernel of Eisenstein primes on modular Jacobian
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[12] Hwajong Yoo, The index of an Eisenstein ideal and multiplicity one, submitted, available at http:
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//arxiv.org/pdf/1502.01571.pdf (2015).
E-mail address: [email protected]
U NIVERSIT E´ DU L UXEMBOURG , FACULT E´ DES S CIENCES , DE LA T ECHNOLOGIE ET DE LA C OMMUNI CATION ,
6, RUE R ICHARD C OUDENHOVE -K ALERGI , L-1359 L UXEMBOURG , L UXEMBOURG