aa r X i v : . [ m a t h . N T ] A p r ON RIBET’S ISOGENY FOR J (65) KRZYSZTOF KLOSIN AND MIHRAN PAPIKIAN
Abstract.
Let J be the Jacobian of the Shimura curve attached to theindefinite quaternion algebra over Q of discriminant 65. We study the isogenies J (65) → J defined over Q , whose existence was proved by Ribet. We provethat there is an isogeny whose kernel is supported on the Eisenstein maximalideals of the Hecke algebra acting on J (65), and moreover the odd part ofthe kernel is generated by a cuspidal divisor of order 7, as is predicted by aconjecture of Ogg. Introduction
Let N be a product of an even number of distinct primes. Let J ( N ) be theJacobian of the modular curve X ( N ). In [20], Ribet proved the existence of anisogeny defined over Q between the “new” part J ( N ) new of J ( N ) and the Jaco-bian J N of the Shimura curve X N attached to a maximal order in the indefinitequaternion algebra over Q of discriminant N . Although there are no morphisms X ( N ) → X N defined over Q , Ribet showed that the Q ℓ -adic Tate modules of J ( N ) new and J N are isomorphic as Gal( Q / Q )-modules, where ℓ is an arbitraryprime number; this is a consequence of a correspondence between automorphicforms on GL(2) and automorphic forms on the multiplicative group of a quaternionalgebra. The existence of the isogeny J ( N ) new → J N defined over Q then followsfrom a special case of Tate’s isogeny conjecture for abelian varieties over numberfields, also proved in [20] (the general case of Tate’s conjecture was proved a fewyears later by Faltings). Unfortunately, Ribet’s argument provides no informationabout the isogenies J ( N ) new → J N beyond their existence.In [16], Ogg made an explicit conjecture about the kernel of Ribet’s isogeny when N = pq is a product of two distinct primes and p = 2 , , , ,
13: the conjecturepredicts that there is an isogeny J ( N ) new → J N of minimal degree whose kernelis a specific group arising from the cuspidal divisor subgroup of J ( N ). Note that p = 2 , , , ,
13 are exactly the primes for which J ( pq ) has purely toric reductionat q . This fact is crucial for the calculations used by Ogg to come up with hisconjecture; the underlying idea is that the knowledge of the group of connectedcomponents of the N´eron models of J ( N ) new and J N at q yields restrictions on Mathematics Subject Classification.
Key words and phrases.
Modular curves, Ribet’s isogeny, Eisenstein ideal, cuspidal divisorgroup.The first author was supported by the Young Investigator Grant the isogenies between them. Ogg’s conjecture remains open except for the specialcases when J N has dimension ≤ J N ) = 1, equiv. N = 2 ·
7, 3 ·
5, 3 ·
7, 3 ·
11, 2 · J N is an ellipticcurve over Q which is uniquely determined by its component groups at p and q ,and J ( N ) new is the optimal elliptic curve of conductor N . Then one easily checksOgg’s conjecture using Cremona’s tables [5]. In general, the orders of componentgroups of J N can be computed using Brandt matrices [10], which is relatively easyto do with the help of a computer program such as Magma .When dim( J N ) = 2, equiv. N = 2 ·
13, 2 ·
19, 2 ·
29, Ogg’s conjecture is verifiedin [7]. In this case, the proof is based on the fact that X N is bielliptic and thelattices of J ( N ) new and J N can be computed through their elliptic quotients.When dim( J N ) = 3, equiv. N = 2 ·
31, 2 ·
41, 2 ·
47, 3 ·
13, 3 ·
17, 3 · ·
23, 5 ·
7, 5 ·
11, Ogg’s conjecture is verified in [6]. In this case, X N is alwayshyperelliptic. By utilizing this fact, Gonz´alez and Molina explicitly compute theequation for each X N . Then they obtain a basis of regular differentials for X N fromthese equations to produce a period matrix for J N . The period matrix of J ( N ) new can be computed using cusp forms with rational q -expansions. The problem thenreduces to comparing the period matrices of appropriate quotients of J ( N ) new withthe period matrix of J N .The goal of this paper is to study Ribet’s isogeny for N = 5 ·
13 = 65. In thiscase, dim( J N ) = 5 and X N is not hyperelliptic; cf. [14]. Our approach to the studyof Ribet isogenies is completely different from that in [7] and [6], and crucially relieson the Hecke equivariance of such isogenies. In this approach we need to know verylittle about X N or J N ; we only need to know the orders of component groups of J N ,which, as we mentioned, are easy to compute, and in fact were already computedin [16]. The difficulty shifts to the study of the structure of the Hecke algebra andits action on J ( N ).Let T ( N ) := Z [ T , T , . . . ] be the Z -algebra generated by the Hecke operators T n acting on be the space S ( N ) of weight 2 cups forms on Γ ( N ). This algebra isisomorphic to the subalgebra of End( J ( N )) generated by T n acting as correspon-dences on X ( N ). When N = 65, we have J ( N ) new = J ( N ), so there is a Ribetisogeny π : J ( N ) → J N . T ( N ) also naturally acts on J N and π is T ( N )-equivariant. This equivariance isimplicit in Ribet’s proof [20]; see also [9, Cor. 2.4].From now on we assume N = 65. To simplify the notation, we denote T := T ( N ), J := J ( N ), J ′ := J N , G Q := Gal( Q / Q ). Given a finite abelian group H , we denoteby H p its p -primary component ( p is a prime number), and by H odd its maximalsubgroup of odd order, so that H ∼ = H × H odd . Since the endomorphisms of J induced by Hecke operators are defined over Q , the actions of T and G Q on J commute with each other. Thus, ker( π ) is a T [ G Q ]-submodule of J . We show thatif the kernel of an isogeny from J to another abelian variety is a T [ G Q ]-module,then, up to endomorphisms of J , the kernel is supported on the Eisenstein maximalideals of T . We then classify all T [ G Q ]-submodules of J of odd order supported onthe Eisenstein maximal ideals. This leads to the following theorem, which is themain result of the paper: Theorem 1.1.
There is a Ribet isogeny π : J → J ′ such that ker( π ) odd ∼ = Z / Z isthe -primary component of the cuspidal divisor group of J . N RIBET’S ISOGENY FOR J (65) 3 Ogg’s conjecture in this case predicts that in fact ker( π ) = Z / Z . There is aunique Eisenstein maximal ideal m ✁ T of residue characteristic 2. In principle, itshould be possible to extend our analysis to finite T [ G Q ]-submodules of J supportedon m to show that ker( π ) = 0. But there are several technical difficulties whichat present we are not able to overcome: these stem from the fact that m is a primeof fusion, T m is not Gorenstein, and the groups of rational points of reductions of J usually have large 2-primary components.Our strategy can be applied also to cases when dim( J N ) = 3, which leads toresults similar to Theorem 1.1, at least when J ( N ) new = J ( N ) (equiv. N =3 · , · Remark . Given a prime ℓ , if H := ( J ( N ) new ( Q ) tor ) ℓ = 0 but ( J N ( Q ) tor ) ℓ = 0,then obviously H ⊂ ker( π ) for any Ribet isogeny π : J ( N ) new → J N . For anodd prime ℓ , in [24], Yoo gives sufficient conditions for the non-existence of rationalpoints of order ℓ on J N , when N = pq is a product of two distinct primes. This thencan be used to find non-trivial subgroups of the kernels of Ribet isogenies; see [24,Thm. 1.3]. In the case when N = 65, Yoo’s theorem implies that Z / Z ⊂ ker( π ).2. N´eron models
In this section we recall some terminology and facts from the theory of N´eronmodels. Let R be a complete discrete valuation ring, with fraction field K andresidue field k . Let A be an abelian variety over K . Denote by A its N´eron modelover R and denote by A k the connected component of the identity of the specialfiber A k of A . There is an exact sequence0 → A k → A k → Φ A → , where Φ A is a finite (abelian) group called the component group of A . We say that A has semi-abelian reduction if A k is an extension of an abelian variety A ′ k by anaffine algebraic torus T A over k (cf. [1, p. 181]):0 → T A → A k → A ′ k → . We say that A has good reduction , if A k = A ′ k (in this case, we also have A k = A k );we say that A has (purely) toric reduction if A k = T A . The character group (2.1) M A := Hom(( T A ) ¯ k , G m, ¯ k )is a free abelian group contravariantly associated to A .Let K ′ be a finite unramified extension of K , with ring of integers R ′ and residuefield k ′ . By the fundamental property of N´eron models, we have an isomorphismof groups A ( K ′ ) ∼ = A ( R ′ ), which defines a canonical reduction map(2.2) A ( K ′ ) → A k ( k ′ ) . Composing (2.2) with A k → Φ A , we get a homomorphism(2.3) A ( K ′ ) → Φ A . Proposition 2.1.
Let K ′ be a finite unramified extension of K . Let H ⊂ A ( K ′ ) bea finite subgroup. Assume that either H is coprime to the characteristic p of k ,or that K has characteristic and its absolute ramification index is < p − . Then (2.2) defines an injection H ֒ → A k ( k ′ ) .Proof. See [11, p. 502] and [1, Prop. 7.3/3]. (cid:3)
KRZYSZTOF KLOSIN AND MIHRAN PAPIKIAN
Let ϕ : A → B be an isogeny defined over K . By the N´eron mapping property, ϕ extends to a morphism ϕ : A → B of the N´eron models. On the special fiberswe get a homomorphism ϕ k : A k → B k , which induces an isogeny ϕ k : A k → B k ;[1, Cor. 7.3/7]. This implies that B has semi-abelian (resp. toric) reduction if A has semi-abelian (resp. toric) reduction. The isogeny ϕ k restricts to an isogeny ϕ t : T A → T B , which corresponds to an injective homomorphisms of charactergroups ϕ ∗ : M B → M A with finite cokernel. We also get a natural homomorphism ϕ Φ : Φ A → Φ B .Denote by ˆ A the dual abelian variety of A . Let ˆ ϕ : ˆ B → ˆ A be the isogenydual to ϕ . Assume A has semi-abelian reduction. In [8], Grothendieck defined anon-degenerate pairing u A : M A × M ˆ A → Z (called monodromy pairing ) with nicefunctorial properties, which induces an exact sequence(2.4) 0 → M ˆ A u A −−→ Hom( M A , Z ) → Φ A → . Using (2.4), one obtains a commutative diagram with exact rows (cf. [21, p. 8]):0 / / M ˆ A / / ˆ ϕ ∗ (cid:15) (cid:15) Hom( M A , Z ) Hom( ϕ ∗ , Z ) (cid:15) (cid:15) / / Φ Aϕ Φ (cid:15) (cid:15) / / / / M ˆ B / / Hom( M B , Z ) / / Φ B / / . From this diagram we get the exact sequence(2.5) 0 → ker( ϕ Φ ) → M ˆ B / ˆ ϕ ∗ ( M ˆ A ) → Ext Z ( M A /ϕ ∗ ( M B ) , Z ) → coker( ϕ Φ ) → . Since Ext Z ( M A /ϕ ∗ ( M B ) , Z ) ∼ = Hom( M A /ϕ ∗ ( M B ) , Q / Z ) =: ( M A /ϕ ∗ ( M B )) ∨ , we can rewrite (2.5) as(2.6) 0 → ker( ϕ Φ ) → M ˆ B / ˆ ϕ ∗ ( M ˆ A ) → ( M A /ϕ ∗ ( M B )) ∨ → coker( ϕ Φ ) → . Note that M A /ϕ ∗ ( M B ) ∼ = Hom(ker( ϕ t ) , G m,k ). On the other hand, ker( ϕ t ) canbe canonically identified with a subgroup scheme of H := ker( ϕ ); cf. [3, p. 762].Therefore, M A /ϕ ∗ ( M B ) divides H . Similarly, M ˆ B / ˆ ϕ ∗ ( M ˆ A ) divides ϕ ).Since ker( ˆ ϕ ) ∼ = Hom(ker( φ ) , G m,K ) (see [15, Thm.1, p. 143]), we conclude that M ˆ B / ˆ ϕ ∗ ( M ˆ A ) also divides H . Now one easily deduces from (2.6) the following: Lemma 2.2.
Assume A has semi-abelian reduction, and ϕ : A → B is an isogenydefined over K . If ℓ is a prime number which does not divide ϕ ) , then ϕ Φ induces an isomorphism (Φ A ) ℓ ∼ = (Φ B ) ℓ . Lemma 2.3.
Let K ′ be a finite unramified extension of K . Let ϕ : A → B be anisogeny defined over K such that H = ker( ϕ ) ⊂ A ( K ′ ) , i.e., H becomes a constantgroup-scheme over K ′ . Let H (resp. H ) be the kernel (resp. image) of thehomomorphism H → Φ A defined by (2.3) . Assume A has toric reduction. Assumethat either H is coprime to the characteristic p of k , or that K has characteristic and its absolute ramification index is < p − . Then there is an exact sequence → H → Φ A ϕ Φ −−→ Φ B → H → . N RIBET’S ISOGENY FOR J (65) 5 Proof.
Under these assumptions, we have
H ֒ → A k ( k ′ ) and H = ker( ϕ t ). Thisimplies ( M A /ϕ ∗ ( M B )) ∨ ∼ = H . Next, [3, Thm. 8.6] implies that M ˆ B / ˆ ϕ ∗ ( M ˆ A ) ∼ = H . Thus, we can rewrite (2.6) as0 → ker( ϕ Φ ) → H → H → coker( ϕ Φ ) → . Since ker( ϕ Φ ) = H , we conclude from this exact sequence that coker( ϕ Φ ) ∼ = H . (cid:3) Hecke Algebra
Since the Z -algebra T is free of finite rank as a Z -module, we can define thediscriminant disc( T ) of T with respect to the trace pairing; cf. [19, p. 66]. Analgorithm for computing the discriminants of Hecke algebras is implemented in Magma ; it gives disc( T ) = 2 ·
3. We then obtain T = Z T + Z T + Z T + Z T + Z T as a free Z -module by comparing the discriminants. We have T ⊗ Z Q ∼ = Q × Q ( √ × Q ( √ e T = Z × Z [ √ × Z [ √ T in T ⊗ Q . Viewing T as an order in e T , we have T = (1 , , T = ( − , − √ , √ T = ( − , √ , − √ T = ( − , , − T = (2 , − √ , − √ . One then observes that T = Z v + Z v + Z v + Z v + Z v , where v = (1 , , , v = (0 , , , v = (0 , , , v = (0 , √ , ,v = ( − , − √ , − √ , which implies(3.2) T ∼ = ( a, b + b √ , c + c √ (cid:12)(cid:12)(cid:12)(cid:12) a, b , b , c , c ∈ Z ,a ≡ b ≡ ( c + c ) mod 2 ,b ≡ c mod 2 . Given a maximal ideal m ✁ T , let T m = lim ←− n T / m n denote the completion of T at m . Proposition 3.1.
Every maximal ideal in T of odd residue characteristic is prin-cipal. In particular, T m is Gorenstein for any maximal ideal m ✁ T of odd residuecharacteristic; cf. [23, p. 329] .Proof. Since disc( T ) = [ e T : T ] · disc( e T ) = [ e T : T ] · · , we get [ e T : T ] = 2 . Let I e T , ′ be the set of ideals I ✁ e T such that e T /I is a finitering of odd order. Let I T , ′ be the set of ideals I ✁ T such that T /I is a finitering of odd order. The argument of the proof of Proposition 7.20 in [4] shows thatthe map I I ∩ T gives a bijection from I e T , ′ to I T , ′ , with the inverse given by KRZYSZTOF KLOSIN AND MIHRAN PAPIKIAN I I e T . Moreover, the proof of that proposition shows that for I ∈ I e T , ′ we have e T /I ∼ = T /I ∩ T , so that this bijection restricts to a bijection between the maximalideals of e T and T of odd residue characteristic.Since e T is a direct product of Euclidean domains, every ideal I ∈ I e T , ′ is principal.Write I = θ e T . If θ ∈ T , then I ∩ T = θ T is also principal, since ( θ T ) e T = θ e T .Therefore, to prove the proposition it is enough to show that for every maximalideal m ∈ I e T , ′ we can choose a generator which lies in T . Let p > m = θ e T . If we write m = m ′ × m ′′ × m ′′′ , where m ′ ✁ Z , m ′′ ✁ Z [ √ m ′′′ ✁ Z [ √ p , and theother two are equal to the corresponding ring. We consider three cases dependingon which of the three ideals is proper.Case 1: m ′ = p Z . Then θ = ( p, , ∈ T .Case 2: m ′′ is proper. If ( p ) is inert in Z [ √ θ = (1 , p, ∈ T .Now suppose p = ( α + β √ α − β √
2) splits, where α, β ∈ Z . Note that α must be odd. If β is even, then θ = (1 , α ± β √ , ∈ T . If β is odd, then θ = (1 , α ± β √ , √ ∈ T , as 2 + √ Z [ √ m ′′′ is proper. If ( p ) is inert in Z [ √ θ = (1 , , p ) ∈ T .If p = 3, then θ = (1 , √ , √ ∈ T , since 1 + √ Z [ √ p = ( α + β √ α − β √ α, β ∈ Z . Considering p = α − β modulo 2, we get 1 ≡ ( α + β ) mod 2, so that α and β have different parity. If α is odd and β is even, then θ = (1 , , α ± β √ ∈ T . If α is even and β is odd, then θ = (1 , √ , α ± β √ ∈ T . (cid:3) Remark . Let O = Z [ i ] be the Gaussian integers. Let O ′ = Z + 3 O = Z + 3 i Z be an order in O . We have [ O : O ′ ] = 3. The ideal m = (2 + i ) O is maximal and O / m ∼ = F . On the other hand, m ∩ O ′ = (5 , i ) O ′ is not principal, although(5 , i ) O = m . This indicates that Proposition 3.1 is not a special case of ageneral fact about orders. Definition 3.3.
The
Eisenstein ideal of T is the ideal E ✁ T generated by T ℓ − ( ℓ +1)for all primes ℓ ∤
65. A maximal ideal m ✁ T in the support of the Eisenstein idealis called an Eisenstein maximal ideal . Proposition 3.4.
We have T / E ∼ = Z / Z ∼ = Z / Z × Z / Z × Z / Z . Proof.
First, we explain how to compute the expansion of an arbitrary Hecke opera-tor T m ∈ T in terms of the Z -basis { T , T , T , T , T } of T . Up to Galois conjugacy,there are three normalized T -eigenforms in S (65). The three coordinates of T m inthe ring on the right hand-side of (3.2) are the eigenvalues with which T m acts onthese eigenforms. Once we have this representation of T m , thanks to (3.1), findingthe expansion of T m in terms of our basis amounts to solving a system of five linearequations in five variables. This strategy yields T = 2 T − T − T + 9 T − T ,T = 2 T + 2 T − T + 8 T − T ,T = − T + T + 12 T − T + 9 T . The Hecke operators T ℓ for primes ℓ ∤
65 are all congruent to integers modulo E . Since T = ( T − T ) + 3 T + 2 T + 2 T , we conclude that all Hecke operators N RIBET’S ISOGENY FOR J (65) 7 are congruent to integers. Hence the natural map Z → T / E is surjective. Wecannot have T / E = Z , for then there would exist a cusp form f ∈ S (65) suchthat T ℓ f = ( ℓ + 1) f , which would contradict the Ramanujan-Petersson bound.Therefore, T / E ∼ = Z /n Z for some integer n . Note that T ≡
29 (mod E ). Fromthe expansion of T , we obtain 168 = 2 · · ≡ E ); from the expansion of T , we obtain 252 = 2 · · ≡ E ); thus, n divides 4 · · E annihilates J ( Q ) tor ∼ = Z / Z × Z / Z × Z / Z × Z / Z ; see Proposition 4.2. Hence n is divisibleby the exponent of this group, which is 84. (cid:3) Lemma 3.5.
The Hecke operators T and T act on T / E ∼ = Z / Z × Z / Z × Z / Z as (1 , − , and (1 , , − , respectively.Proof. In the proof of Proposition 3.4 we computed that T ≡
29 (mod E ). Sim-ilarly, T = − T + T − T ≡
13 (mod E ). From this the claim of the lemmaimmediately follows since, for example, 29 ≡ ≡ − ≡ (cid:3) Remark . We note that T and T are actually equal to the negatives of theAtkin-Lehner involutions W and W acting on S (65). The conclusion ( T / E ) odd ∼ = Z / Z × Z / Z then can be deduced from Theorem 3.1.3 in [17].Proposition 3.4 implies that there are three Eisenstein maximal ideals in T : m := ( E ,
2) = ( E , , T − , T − , m := ( E ,
3) = ( E , , T + 1 , T − , m := ( E ,
7) = ( E , , T − , T + 1) . Proposition 3.7.
We have: (i)
The ideal m ✁ T is equal to the ideal (cid:16) (2 , , e T (cid:17) ∩ T = n ( a, b + b √ , c + c √ ∈ T (cid:12)(cid:12) a ∈ Z o , which is the unique maximal ideal of T of residue characteristic . (ii) m n is not principal for any n ≥ . (iii) T m is not Gorenstein.Proof. (i) The uniqueness of the maximal ideal of residue characteristic 2 impliesthat it must be the Eisenstein maximal ideal m . To prove the uniqueness, notethat each of the rings Z , Z [ √ Z [ √
3] has a unique maximal ideal of residuecharacteristic 2; these are generated by 2, √
2, and 1 + √
3, respectively. One easilychecks that m := ((2 , , e T ) ∩ T = ((1 , √ , e T ) ∩ T = ((1 , , √ e T ) ∩ T , and T / m ∼ = F .(ii) To prove this statement it is enough to observe that (1 , , ∈ e T is inEnd T ( m n ) but (1 , , T .(iii) We apply [23, Prop. 1.4 (iii)]: Let m denote the image of m in T / T .Then T m is Gorenstein if and only if dim F ( T / T )[ m ] = 1. Note that (2 , , , ,
0) have distinct non-zero images in T / T , since otherwise (2 , , ∈ T ,which would imply (1 , , ∈ T . On the other hand, for any θ ∈ m we have θ (2 , ,
0) = (4 a, ,
0) = 2(2 a, , ∈ T for some a ∈ Z . Therefore, m annihilates(2 , , m annihilates (0 , , F ( T / T )[ m ] ≥ (cid:3) KRZYSZTOF KLOSIN AND MIHRAN PAPIKIAN m m m Z [ √ Z [ √ Z Figure 1.
Spec( T )Spec( T ) can be sketched as in Figure 1. It has three irreducible componentsintersecting at m . The irreducible components containing the closed points m and m are determined by observing that T + 1 = (0 , ,
0) and T − − , , − T acts as − Z [ √ Z [ √ T m ∼ = Z and T m ∼ = Z [ √ Modular Jacobian
There are exactly four cusps, denoted [1], [ p ], [ q ] and [ pq ], on X ( pq ), where p and q are two distinct prime numbers. Let C ( pq ) be the subgroup of J ( pq ) generatedby all cuspidal divisors. Since all cusps are Q -rational, we have C ( pq ) ⊂ J ( pq )( Q ).Let Φ( p ) and Φ( q ) denote the component groups of J ( pq ) at p and q , and ℘ p , ℘ q : C ( pq ) → Φ( p ) , Φ( q ) be the homomorphisms induced by (2.3). Proposition 4.1.
Let p = 5 and q = 13 . Let c p and c q be the divisor classes of [1] − [ p ] and [1] − [ q ] in J ( pq ) . Denote C := C ( pq ) . (i) C is generated by c p and c q . The order of c p is ; the order of c q is ;the only relation between c p and c q in C is c p = 6 c q . This implies C ∼ = Z / Z × Z / Z × Z / Z × Z / Z . (ii) Φ( p ) ∼ = Z / Z and Φ( q ) ∼ = Z / Z . (iii) The order of ℘ p ( c p ) is , and ℘ p ( c q ) = 0 ; this implies that there is anexact sequence → h c q i → C ℘ p −→ Φ( p ) → Z / Z → . The order of ℘ q ( c q ) is , and ℘ q ( c p ) = 0 ; this implies that there is an exactsequence → h c p i → C ℘ q −→ Φ( q ) → . Proof. (i) follows from [2]. The groups Φ( p ) and Φ( q ) can be computed from thestructure of special fibres of X ( pq ) using a well-known method of Raynaud; see[16, p. 214] or the appendix in [13]. Finally, by considering the reductions of thecusps in the special fibre of the minimal regular model of X ( pq ) over Z p , one candetermine the homomorphism ℘ p and ℘ q ; cf. [18, p. 1161]. (cid:3) Proposition 4.2.
We have C = J ( Q ) tor .Proof. Obviously
C ⊆ J ( Q ) tor . On the other hand, J has good reduction at any oddprime p ∤
65, so by Proposition 2.1 we have an injective homomorphism J ( Q ) tor ֒ → J ( F p ), where J ( F p ) denotes the group of F p -rational points on the reduction of J at N RIBET’S ISOGENY FOR J (65) 9 p . The order of J ( F p ) can be computed using Magma . We have J ( F ) = 2 · · J ( F ) = 2 · · · ·
37. Since the greatest common divisor of these numbersis 2 · · C , the claim follows. (cid:3) The Hecke ring T is isomorphic to a subring of endomorphisms of J generatedby the Hecke operators T n acting as correspondences on X . In fact, in our case T is the full ring of endomorphisms of J (this can be proved as in [13, Prop. 9.5]).For a maximal ideal m ✁ T , we denote J [ m ] = \ α ∈ m ker( J α −→ J )Then J [ m ] ⊂ J [ p ], where p is the characteristic of T / m . By a theorem of Mazur[23, p. 341], T m is Gorenstein if and only if dim T / m J [ m ] = 2. Therefore, usingProposition 3.1, we conclude that dim T / m J [ m ] = 2 for any maximal ideal m of oddresidue characteristic.Let p = 3 , m p be the corresponding Eisenstein maximal ideal. The Eichler-Shimura congruence relation implies that E annihilates J ( Q ) tor = C . Hence Z /p Z ∼ = C p ⊂ J [ m p ]. We have(4.1) 0 −→ Z /p Z −→ J [ m p ] −→ µ p −→ , since G Q acts on ∧ J [ m p ] by the mod p cyclotomic character; cf. [22, p. 465]. By[12], the Shimura subgroup Σ (= kernel of the functorial homomorphims J (65) → J (65)) is(4.2) Σ ∼ = µ × µ , and the Eisenstein ideal E annihilates Σ. Therefore, (4.1) splits for p = 3: J [ m ] = C × Σ ∼ = Z / Z × µ . Lemma 4.3.
The sequence (4.1) does not split for p = 7 .Proof. If (4.1) splits then Z / Z × Z / Z ⊂ J ( Q ( µ )) tor . Since ℓ = 29 splits com-pletely in Q ( µ ), by Proposition 2.1 we must have 7 | J ( F ℓ ) = 2 · · · · . (cid:3) Remark . Let E be the elliptic curve defined by y + xy = x − x . It is easyto check that E has a rational 2-torsion point and E [2] as a Galois module is anon-split extension 0 −→ Z / Z −→ E [2] −→ Z / Z −→ . By Table 1 in [5], E is isomorphic to a subvariety of J . We claim that E [2] ⊂ J [ m ].To see this, consider a Hecke operator T p = ( a p , b p + √ c p , d p + √ e p ) for prime p ∤
65, given as in (3.2). T p acts on E by multiplication by a p . The fact that m is Eisenstein implies that a p − ( p + 1) is even; thus, T p − ( p + 1) annihilates E [2];thus m = (2 , E ) annihilates E [2]. On the other hand, clearly E [2]
6⊂ C [2], as C [2]is constant. Therefore, dim T / m J [ m ] ≥ dim F C [2] + 1 = 3. This gives a geometricproof of the fact that T m is not Gorenstein. Note that Proposition 4.2 implies thatΣ[2] ⊂ C [2], since µ ∼ = Z / Z is constant over Q . Proposition 4.5.
Let m ✁ T be an Eisenstein maximal ideal of odd residue charac-teristic p . Let H ⊂ J [ m s ] , s ≥ , be a T [ G Q ] -module. If J [ m ] H , then H ( J [ m ] . Proof.
We will assume that J [ m ] H and H J [ m ], and reach a contradiction.First, we make some simplifications. Since H [ m ] ⊂ J [ m ] is a T [ G Q ]-modulesatisfying the same assumptions, if we want to show that H does not exist, it isenough to prove the non-existence under the additional assumption that H ⊂ J [ m ]. Lemma 4.6.
We have H ∼ = T / m .Proof. We can consider H as a finite T m -module. Since T m is a DVR, we have H ∼ = T m / m s × · · · × T m / m s r ∼ = T / m s × · · · × T / m s r for some 1 ≤ s ≤ s ≤ · · · ≤ s r ≤
2. Since dim T / m J [ m ] = 2, and H [ m ] ∼ =( T / m ) r ( J [ m ], we must have r = 1, i.e., H ∼ = T / m s for s = 1 or s = 2. If s = 1,then H ⊂ J [ m ], contrary to our assumption, so s = 2. (cid:3) Note that T / m ∼ = ( Z /p Z if p = 7; F p [ x ] / ( x ) if p = 3 . Let K := Q ( H ). If K = Q , then p = H divides J ( Q ) tor . This contradictsProposition 4.2, so we will assume from now on that K = Q . Let η be a generator of m . Note that ηH = H [ η ] ⊂ J [ m ] is a proper non-trivial Galois invariant subgroup.On the other hand, the G Q -invariant subgroups of J [ m ] are Z /p Z and µ p , so either(4.3) 0 → Z /p Z → H η −→ Z /p Z → , or(4.4) 0 → µ p → H η −→ µ p → . Moreover, the second possibility does not occur for p = 7, since (4.1) does not split. Lemma 4.7.
Let K p denote the unique degree p extension of Q contained in Q ( µ p ) . (1) If p = 7 , then K = K p . (2) Assume p = 3 . In case of (4.3) , we have [ K : Q ] = p and K ⊂ K p Q ( µ ) .In case of (4.4) , we have Q ( µ p ) ⊆ K ⊂ Q ( µ p , µ ) .Proof. Since the actions of T and G Q on H commute, we haveGal( K/ Q ) ⊂ Aut T ( T / m ) ∼ = ( T / m ) × ∼ = Z / ( p − p Z . Hence K/ Q is an abelian extension. Since J has good reduction away from 5 and13, the extension K/ Q is unramified away from p, ,
13. By class field theory, K isa subfield of a cyclotomic extension Q ( µ p n , µ n , µ n ), for some n , n , n ≥ Q ( µ p n , µ n , µ n ) / Q ) ∼ = Gal( Q ( µ p n / Q ) × Gal( Q ( µ n / Q ) × Gal( Q ( µ n / Q ) ∼ = Z /p n − ( p − Z × Z / n − (5 − Z × Z / n − (13 − Z . Assume p = 7. Since in this case H is as in (4.3), G Q acts trivially on pH , soGal( K/ Q ) is in the subgroup of units ( Z /p Z ) × which satisfy ap ≡ p (mod p ),or equivalently, a ≡ p ). The units with this property form the cyclicsubgroup of order p in ( Z /p Z ) × . Hence K/ Q is an abelian extension of degree p . Since p does not divide (5 − n − or (13 − n − , the field K is fixedby Gal( Q ( µ n ) / Q ) × Gal( Q ( µ n ) / Q ). Therefore, K ⊂ Q ( µ p n ) is a subfield of N RIBET’S ISOGENY FOR J (65) 11 degree p over Q . There is a unique such field (as Gal( Q ( µ p n / Q ) is cyclic), and itis contained in Q ( µ p ).Assume p = 3 and H fits into an exact sequence (4.3). By the argument in theprevious paragraph, [ K : Q ] = p . Let F := Q ( µ ) and K ′ = F ( H ). We know that[ K ′ : F ] = 1 or p . Note thatGal( Q ( µ p n , µ n , µ n ) /F ) ∼ = Z / ( p − p n − × Z (5 − n − × Z / n − Z , so as in the case of p = 7, we get F ( H ) ⊂ K p F .Finally, assume p = 3 and H fits into an exact sequence (4.4). Then obviously Q ( µ p ) ⊂ K . Over L := Q ( µ p ), the group scheme H fits into an exact sequence(4.3), so, as in the earlier cases, L ( H ) /L is cyclic of order 1 or p . If H is notconstant over F L , then [
F L ( H ) : F L ] = p . On the other hand,Gal( Q ( µ p n , µ n , µ n ) /F L ) ∼ = Z /p n − × Z (5 − n − × Z / n − Z . As in the earlier cases, this implies that
F L ( H ) ⊂ K p F L = Q ( µ p , µ ). Overall,we see that K is always a subfield of Q ( µ p , µ ). (cid:3) Assume p = 7. By Lemma 4.7, we have K = K p . Let ℓ be a prime which splitscompletely in K p . Then H is constant over Q ℓ , so H ⊂ J ( Q ℓ ) tor . On the otherhand, under the canonical reduction map, we have an injection J ( Q ℓ ) tor ֒ → J ( F ℓ );see Proposition 2.1. Therefore, we must have p | J ( F ℓ ). It is easy to show thata prime ℓ splits completely in K p if and only if its order in ( Z /p Z ) × is coprimeto p . We can take 3 as a generator of ( Z /p Z ) × . The elements of orders coprimeto p are the powers of 3 ≡
31. These are { , , , , , } . Thus, the smallestprime that splits completely in K is 19, and J ( F ) = 2 · · · · . As 7 does not divide this number, we get a contradiction.Assume p = 3. By Lemma 4.7, we have Q ( H ) ⊂ Q ( µ , µ p ). Since µ p is constantover K ′ , we have Z /p Z × Z /p Z ∼ = J ( K ′ )[ m ] ⊂ J ( K ′ ) tor ⊂ J ( Q ℓ ). Since H is alsoconstant over K ′ , we also have Z /p Z × Z /p Z ∼ = H ⊂ J ( Q ℓ ). Since J [ m ] H , wesee that J ( Q ℓ ) contains a subgroup isomorphic to ( Z /p Z ) . As earlier, this impliesthat p | J ( F ℓ ). A prime ℓ splits completely in K ′ := Q ( µ , µ p ) if and onlyif ℓ ≡ ℓ ≡ ℓ = 937, and J ( F ) = 2 · · · · · · does not divide this number, weget a contradiction. This concludes the proof of Proposition 4.5. (cid:3) Let A be an abelian variety over Q and π : J → A an isogeny defined over Q . As-sume ker( π ) is invariant under the action of T , i.e., ker( π ) is a finite T [ G Q ]-module.We can decompose ker( π ) = ker( π ) × ker( π ) odd ; each of these subgroups is also a T [ G Q ]-module. Let the maximal ideal m ✁ T be in the support of H := ker( π ) odd .Since m has odd residue characteristic, m = η T is principal by Proposition 3.1.If ker( η ) = J [ m ] ⊂ H , then we can decompose π = π ′ ◦ η , where π ′ : J → A isanother isogeny whose kernel is a T [ G Q ]-module but with smaller odd componentthan π . We can apply the same argument to π ′ and continue this process untilwe obtain an isogeny whose kernel does not contain any J [ m ] with m having oddresidue characteristic. From now on we assume that π itself has this property.Since m has odd residue characteristic, the T [ G Q ]-module J [ m ] is 2-dimensionalover T / m . By [13, Prop. 14.2] and [22, Thm. 5.2], if m is not Eisenstein, then J [ m ] is irreducible. Since J [ m ] ∩ H = 0, we must have J [ m ] ⊂ H , which contradictsour assumption on π . Hence H is supported on the Eisenstein maximal ideals m and m . We decompose H = H × H into 3-primary and 7-primary components, which themselves are T [ G Q ]-modules. Now H p ⊂ J [ m sp ] for some s ≥ p = 3 , J [ m p ] H p . Applying Proposition 4.5, we conclude that H p ( J [ m p ]. Thus H = 0 or C , and H = 0 or Σ or C . Overall, H can be one of the followingsubgroups of J :(4.5) 0 , C , Σ , C , C × C , Σ × C . Theorem 4.8. If A = J ′ , then for π : J → J ′ chosen with the minimality conditiondiscussed above, we must have H = C .Proof. The reductions of J and J ′ at p = 5 or 13 are purely toric, cf. [16], [22]. LetΦ(5) ′ and Φ(13) ′ be the component groups of J ′ at 5 and 13. We have (see [16, p.214]): Φ(5) ′ ∼ = Z / Z , Φ(13) ′ ∼ = Z / Z . We decompose π : J → J ′ as J → J/H π ′ −→ J ′ , where ker( π ′ ) is isomorphicto the 2-primary part of ker( π ). Let Φ( p ) ′′ be the component group of J/H at p .By Lemma 2.2 we must have (Φ( p ) ′′ ) odd ∼ = (Φ( p ) ′ ) odd . On the other hand, sincewe know the image and kernel of ℘ p : C → Φ( p ), we can compute p ) ′′ ) odd for each possible H from the list (4.5) using Lemma 2.3. This simple calculationshows that the only possible H is C . (Note that the group scheme Σ becomesconstant over an unramified extension of Q p , but it is not important to knowwhether ℘ p : Σ → Φ( p ) is injective or trivial; neither of these possibilities givesthe correct Φ( p ) ′′ if Σ ⊂ H .) (cid:3) Remark . Let N = 5 ·
7. In this case, T = Z [ T ] ∼ = Z [ x ] / ( x − x + x − ∼ = { ( a, b + cα ) ∈ Z × Z [ α ] (cid:12)(cid:12) a, b, c ∈ Z , a ≡ b + c (mod 2) } , where α := − √ . Note that Z [ α ] is the ring of integers in Q ( √ Z [ α ] isa Euclidean domain with respect to the usual norm. We have C ∼ = Z / Z × Z / Z × Z / Z , Σ ∼ = µ × µ . There is a unique Eisenstein maximal ideal m ✁ T of odd residue characteristic.There is a unique Q -isogeny class of elliptic curves of level 35. The optimal curveis [5, p. 112] E : y + y = x + x + 9 x + 1 . We have E [3] ∼ = µ × Z / Z . Since T m is Gorenstein for any maximal ideal m ✁ T (as T is monogenic), J [ m ] is two dimensional over T / m , so J [ m ] = E [3] = C × Σ .Now it is easy to analyze all T [ G Q ]-submodules of J supported on m . An argumentsimilar to the argument of the proof of Theorem 4.8 then implies that there is aRibet isogeny π : J → J ′ with ker( π ) odd = 0. Ogg’s conjecture in this case predictsthat ker( π ) ∼ = Z / Z ⊂ C . Remark . Let N = 3 ·
13. In this case, T = Z [ T ] ∼ = Z [ x ] / ( x − x + 2 x − ∼ = { ( a, b + c √ ∈ Z × Z [ √ (cid:12)(cid:12) a, b, c ∈ Z , a ≡ b (mod 2) } , We have
C ∼ = Z / Z × Z / Z × Z / Z , Σ ∼ = µ . N RIBET’S ISOGENY FOR J (65) 13 There is a unique Eisenstein maximal ideal m ✁ T of odd residue characteristic. J [ m ] fits into the exact sequence (4.1), which is non-split in this case. One canclassify T [ G Q ]-submodules of J supported on m using an argument similar to theargument we used in Proposition 4.5. Finally, one deduces as in Theorem 4.8 thatthere is a Ribet isogeny π : J → J ′ with ker( π ) odd = C ∼ = Z / Z . Ogg’s conjecturein this case predicts that ker( π ) = C . Acknowledgements.
This work was carried out in part while the second authorwas visiting the Taida Institute for Mathematical Sciences in Taipei and the MaxPlanck Institute for Mathematics in Bonn in 2016. He thanks these institutes fortheir hospitality, excellent working conditions, and financial support. He is alsograteful to Fu-Tsun Wei for very useful discussions related to the topic of thispaper.
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