Preprint (June 11, 1999), Version 0.5

Component groups of optimal quotients of Jacobians

W.A. Stein

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Introduction

Let A be an abelian variety over a finite extension K of the p-adic numbers Q_{p}. Let O be the ring of integers of K, 𝔪 its maximal ideal and k = O/𝔪 the residue class field. The Néron model of A is a smooth commutative group scheme A over O such that A is its generic fiber and satisfying the property: the restriction map

Hom_{O}(S, A) ⟶ Hom_{K}(S/K, A)

is bijective for all schemes S over O. The special fiber A_{k} is a group scheme over k, which need not be connected. Denote by A_{k}^{0} the connected component containing the identity. There is an exact sequence

0 ⟶ A_{k}^{0} ⟶ A_{k} ⟶ Φ_{A} ⟶ 0

with Φ_{A} a finite étale group scheme over k, i.e., a finite abelian group equipped with an action of Gal(\overline{k}/k).

In this paper we study the group Φ_{A} with particular emphasis on quotients A of Jacobians of modular curves X_{0}(N). When A has semistable reduction Grothendieck described the component group in terms of a monodromy pairing on certain free abelian groups. When A = J is the Jacobian of X_{0}(N), this pairing can be explicitly computed, hence so can Φ_{J}, as has been done in many cases in [Maz77] and [Edi91]. Suppose now that A is a simple quotient of J and that the kernel of the map J → A is connected. There is a natural map Φ_{J} → Φ_{A}. In this paper we give a formula which can be used to compute the image and the order of the cokernel.

We now state our main result in more precise language. Suppose π : J → A is an optimal quotient, with J a semistable Jacobian and A purely toric. We express the component group of A in terms of the monodromy pairing associated to J. Let m_{A} = \sqrt{deg(θ_{A})} where θ_{A} : A′ → A is induced by the canonical principal polarization of J. Let X_{J} be the character group of the toric part of the special fiber of J. Let L be the saturation of the image of X_{A} in X_{J}. The monodromy pairing defines a map α : X_{J} → Hom(L, Z). Let Φ_{X} be the cokernel of α and m_{X} = [α(X_{J}) : α(L)] be the order of the finite group α(X_{J})/α(L). We prove that

\frac{|Φ_{A}|}{m_{A}} = \frac{|Φ_{X}|}{m_{X}}.

More precisely, Φ_{X} is the image of Φ_{J} → Φ_{A} and the cokernel of Φ_{J} → Φ_{A} has order m_{A}/m_{X}. If the optimal quotient J → A arises from a modular form on Γ_{0}(N), then the quantities m_{A}, m_{X} and Φ_{X} can be explicitly computed, hence so can |Φ_{A}|. Having done this, we present some tables and conjectures which they suggest.

Acknowledgement: I am deeply grateful for conversations with A. Agashe, R. Coleman, B. Edixhoven, D. Lorenzini, B. Mazur, L. Merel, K. Ribet, and S. Takahashi.

1 Optimal quotients of jacobians

Let J be a Jacobian equipped with its canonical principal polarization θ_{J}. An optimal quotient of J is an abelian variety A and a surjective map π : J → A whose kernel is an abelian subvariety B of J. Denote by J′ and A′ the abelian varieties dual to J and A, respectively. Dualizing π and composing with θ′_{J} = θ_{J}

we obtain a map A′ ⟶^{π′} J′ ⟶^{θ_{J}} J.

Proposition 1.1. A′ → J is injective.

Proof. Since θ_{J} is an isomorphism it suffices to prove that π′ is injective. Since the dual of π′ is (π′)′ = π and π is surjective, π′ must have finite kernel. Thus A′ → C = im(π′) is an isogeny. Let G denote the kernel, and dualize. By [Mil86, §11] we have have

\xymatrix{G \ar[r] & A′ \ar@{->>}[r] \ar[dr]_{π′} & C \ar[d] & \underrightarrow{dualize} & A & C \ar[l] & G′ \ar[l] \\  &  & J′ &  &  & J \ar[u]_{φ} \ar[ul]^{π}}

with G′ the Cartier dual of G. Since G′ is finite, ker(φ) ⊂ ker(π) is of finite index. Since ker(π) is an abelian variety it is divisible. Thus ker(φ) = ker(π) so G′ = 0 and hence G = 0.

We denote the map A′ → J by π′. The kernel of θ_{A} measures the intersection of A′ and B = ker(θ_{A}) inside of J as shown in the following diagram.

\xymatrix{A′ ∩ B \ar[r] \ar[d] & B \ar[d] \\ A′ \ar@{^{(}->}[r]^{π′} \ar[dr]_{θ_{A}} & J \ar[d]^{π} \\  & A}

Because θ_{A} is a polarization, | ker(θ_{A})| is a square [Mil86, Theorem 13.3]. The congruence modulus is the integer

m_{A} = \sqrt{| ker(θ_{A})|}.

2 The special fiber of the Néron model

Let K be a finite extension of Q_{p} with ring of integers O and residue class field k. Let A/K be an abelian variety and denote its Néron model by A. Let Φ_{A} be the group of connected components of the special fiber A_{k}. This group is a finite étale group scheme over k, i.e., a finite abelian group equipped with an action of Gal(\overline{k}/k). There is an exact sequence of group schemes

0 → A_{k}^{0} → A_{k} → Φ_{A} → 0.

The group A_{k}^{0} is an extension of an abelian variety B of dimension a by the product of a torus T of dimension t and of a unipotent group U of dimension u:

0 → U × T → A_{k}^{0} → B → 0.

The abelian variety A is said to have purely toric reduction if t = dim A, and is semistable if u = 0. The character group X_{A} = Hom(T , G_{m}) is a free abelian group of rank t contravariantly associated to A. If A is semistable there is a monodromy pairing X_{A} × X_{A′} → Z and an exact sequence

0 → X_{A′} → Hom(X_{A}, Z) → Φ_{A} → 0.

3 Rigid uniformization

In this section we review the rigid analytic uniformization of a semistable abelian variety over a finite extension K of the maximal unramified extension Q_{p}^{ur} of Q_{p}. We use this to prove that if A is purely toric, and ϕ : A′ → A is an isogeny, then

deg(ϕ) = | coker(X_{A} → X_{A′})|^{2}.

We also prove a few lemmas about the character groups X_{A}.

3.1 Raynaud-van der Put uniformization

Theorem 3.1 (Raynaud, van der Put). If A is a semistable Abelian variety, its universal covering is isomorphic to an extension G of an abelian variety B with good reduction by a torus T, the covering map from G to A is a homomorphism and its kernel is a twisted free Abelian group Γ of finite rank.

This may be summarized by the following diagram,

\xymatrix{ & Γ \ar[d] \\ T \ar[r] & G \ar[r] \ar[d] & B \\  & A}

which we call the uniformization cross of A. The group Γ can be identified with the character group X_{A} of the previous section. The uniformization cross of the dual abelian variety A′ is

\xymatrix{ & Γ′ \ar[d] \\ T′ \ar[r] & G′ \ar[r] \ar[d] & B′ \\  & A′}

where Γ′ = Hom(T, G_{m}) and T′ = Hom(Γ, G_{m}) and the morphisms Γ′ → G′ and T′ → G′ are the one-motive duals of the morphisms T → G and Γ → G, respectively. For more details see [Col99].

Example 3.2 (Tate curve). If E/Q_{p} is an elliptic curve with multiplicative reduction then the uniformization is E = G_{m}/q^{Z} where q = q(j) is obtained by inverting the expression for j as a function of q(z) = e^{2πiz}.

3.2 Some lemmas

Let π : J → A be an optimal quotient, with J semistable and A purely toric.

Lemma 3.3. The map Γ_{J} → Γ_{A} induced by π is surjective.

Proof. Because G_{J} is simply connected, π induces a map G_{J} → T_{A} and a map Γ_{J} → Γ_{A}. Because π is surjective and T (A) is a torus, the map G_{J} → T_{A} is surjective. The snake lemma applied to the following diagram gives a surjective map from B = ker(π) to M = coker(Γ_{J} → Γ_{A}).

\xymatrix{ & Γ_{J} \ar[r] \ar[d] & Γ_{A} \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\  & G_{J} \ar[r] \ar[d] & T_{A} \ar[d] \ar[r] & 0 \\ B \ar[r] & J \ar[r]^{π} & A}

Because π is optimal, B is connected so M must also be connected. Since M is discrete it follows that M = 0.

3.3 Purely toric abelian varieties

Assume that A is purely toric. Then B = 0, and the uniformization cross becomes

Γ

T

A

Let φ : A′ → A be a symmetric isogeny, i.e., φ′ : A′ → (A′)′ = A is equal to φ. Denote by φ_{t} : T′ → T and φ_{a} : Γ′ → Γ the induced maps.

Proposition 3.4. There is an exact sequence

0 → ker(φ_{t}) → ker(φ) → coker(φ_{a}) → 0,

and ker(φ_{t}) is dual to coker(φ_{a}).

Proof. Since φ is an isogeny we have the following diagram:

\xymatrix{0 \ar[r] \ar[d] & Γ′ \ar[r] \ar[d] & Γ \ar[r] \ar[d] & coker(φ_{a}) \ar[d] \\ ker(φ_{t}) \ar[r] \ar[d] & T′ \ar[d] \ar[r] & T \ar[r] \ar[d] & 0 \\ ker(φ) \ar[r] & A′ \ar[r]^{φ} & A}

The snake lemma then gives the claimed exact sequence. For the second assertion observe that the one-motive dual of the diagram

\xymatrix{ & Γ′ \ar[r] \ar[d] & Γ \ar[d] \ar[r] & coker(φ_{a}) \\ ker(φ_{t}) \ar[r] & T′ \ar[r]^{φ} & T}

is the diagram

\xymatrix{ & T & T′ \ar[l]_{φ′} & coker(φ_{a})′ \ar[l] \\ ker(φ_{t})′ & Γ \ar[l] \ar[u] & Γ′ \ar[l] \ar[u]}

Since φ is symmetric, φ′ = φ and so

ker(φ_{t}) = coker(φ_{a})′.

Lemma 3.5. | ker(φ)| = | coker(φ_{a})|^{2}

Proof. The order of a finite group scheme equals the order of its dual.

4 The main theorem

Let π : J → A be an optimal quotient, with J a semistable Jacobian and A purely toric. Let X_{A}, X_{A′}, and X_{J} denote the character groups of the toric parts of the special fibers.

4.1 Monodromy description of the component group

There is a pairing X_{A} × X_{A′} → Z called the monodromy pairing. We have an exact sequence

0 → X_{A′} → Hom(X_{A}, Z) → Φ_{A} → 0.

If J is a Jacobian then J is canonically self-dual so the monodromy pairing on J can be viewed as a pairing X_{J} × X_{J} → Z and there is an exact sequence

0 → X_{J} → Hom(X_{J}, Z) → Φ_{J} → 0.

Example 4.1 (Tate curve). Suppose E = G_{m}/q^{Z} is a Tate curve. The monodromy pairing on X_{E} = q^{Z} is

⟨q, q⟩ = ord_{p}(q) = − ord_{p}(j).

Thus Φ_{E} is cyclic of order − ord_{p}(j).

4.2 Proof of the main theorem

We now prove the main theorem. The key diagrams are

\xymatrix{A′ \ar@{^{(}->}[r]^{π′} \ar[dr]_{θ} & J \ar@{->>}[d]^{π} & X_{A} \ar@{^{(}->}[r]^{π^{∗}} \ar[dr]^{θ^{∗}} & X_{J} \ar@{->>}[d]^{π_{∗}} \\  & A &  & X_{A′} \ar[ul]^{θ_{∗}}}

The surjectivity of π_{∗} was proved in Lemma 3.3. The injectivity of π^{∗} follows because

θ_{∗}π_{∗}π^{∗} = θ_{∗}θ^{∗} = deg(θ) ≠ 0,

and multiplication by deg(θ) on a free abelian group is injective.

Let

α : X_{J} → Hom(π^{∗}X_{A}, Z)

be the map defined by the monodromy pairing restricted to X_{J} × π^{∗}X_{A}.

Lemma 4.2. ker(π_{∗}) = ker(α)

Proof. Suppose x ∈ ker(π_{∗}) and let y = π^{∗}z with z ∈ X_{A}. Then

⟨x, y⟩ = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩ = 0

so x ∈ ker(α). Next let x ∈ ker(α). Then for all z ∈ X_{A},

0 = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩

so π_{∗}x is in the kernel of the monodromy map

X_{A′} → Hom(X_{A}, Z).

Since X_{A′} and Hom(X_{A}, Z) are free of the same rank and the cokernel is torsion, the monodromy map is injective. Thus π_{∗}x = 0 and x ∈ ker(π_{∗}).

Lemma 4.3. There is an exact sequence

X_{J} → Hom(π^{∗}X_{A}, Z) → Φ_{A} → 0.

Proof. Lemma 4.2 gives the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & X_{J}/ ker(α) \ar[d]^{≅} \ar[r] & Hom(π^{∗}X_{A}, Z) \ar[r] \ar[d]^{≅} & coker(α) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & X_{A′} \ar[r] & Hom(X_{A}, Z) \ar[r] & Φ_{A} \ar[r] & 0}

By Lemma 4.2, the first vertical map is an isomorphism. The second is an isomorphism because it is induced by the isomorphism π^{∗} : X_{A} → π^{∗}X_{A}. It follows that coker(α) ≅ Φ_{A}, as claimed.

Let L be the saturation of π^{∗}X_{A} in X_{J}, i.e., [L : π^{∗}X_{A}] is finite and X_{J}/L is torsion free. Suppose L is of finite index in L. Define the congruence modulus of L

m_{L} = [α(X_{J}) : α(L)]

and the component group by

Φ_{L} = coker(X_{J} → Hom(L, Z)).

When L = L we often set m_{X} = m_{L} and Φ_{X} = Φ_{L} and think of m_{X} and Φ_{X} as the character group “congruence modulus and component group.”

Lemma 4.4. The rational number \frac{|Φ_{L}|}{m_{L}} does not depend on the choice of L.

Proof. If L′ is another choice let n = [L : L′] ∈ Q. Then since α is injective when restricted to L,

m_{L′} = [α(X_{J}) : α(L′)] = [α(X_{J}) : α(L)] · [α(L) : α(L′)] = m_{L} · n

and similarly |Φ_{L′}| = |Φ_{L}| · n.

Recall that we defined

m_{A} = \sqrt{deg(θ)}
Φ_{A} = coker(X_{A′} → Hom(X_{A}, Z))

Theorem 4.5. For any L of finite index in L the following relation holds:

\frac{|Φ_{A}|}{m_{A}} = \frac{|Φ_{L}|}{m_{L}}.

Proof. By Lemma 4.4 we may assume that L = π^{∗}X_{A}. With this choice of L, Lemma 4.3 says that Φ_{L} ≅ Φ_{A}. By Lemma 4.2, properties of the index, and Lemma 3.5 we have

m_{L} = [α(X_{J}) : α(L)]
= [π_{∗}(X_{J}) : π_{∗}(L)]
= [X_{A′} : π_{∗}(π^{∗}X_{A})]
= [X_{A′} : θ^{∗}X_{A}]
= coker(θ^{∗})
= \sqrt{deg(θ)} = m_{A}.

Proposition 4.6.

image(Φ_{J} → Φ_{A}) ≅ Φ_{L}.

Proof. Because π^{∗}X_{A} ⊂ L ⊂ X_{J}, by Lemma 4.3 we obtain a commutative diagram with exact rows

[figure: X_{J}; Hom(X_{J}, Z); Φ_{J}; 0; X_{J}; Hom(L, Z); Φ_{L}; 0; X_{J}; Hom(π^{∗}X_{A}, Z); Φ_{A}; 0]

The map Hom(L, Z) → Hom(π^{∗}X_{A}, Z) is an isomorphism so Φ_{L} → Φ_{A} is injective, hence

image(Φ_{J} → Φ_{A}) ≅ image(Φ_{J} → Φ_{L}).

The cokernel of Hom(X_{J}, Z) → Hom(L, Z) surjects onto the cokernel of Φ_{J} → Φ_{L}. Using the exact sequence

0 → L → X_{J} → X_{J}/L → 0,

we find that

coker(Hom(X_{J}, Z) → Hom(L, Z)) ⊂ Ext^{1}(X_{J}/L, Z) = 0,

where Ext^{1} vanishes because L is saturated so that X_{J}/L is torsion free. Thus the cokernel of Φ_{J} → Φ_{L} is 0, from which the proposition follows.

The following corollary follows from Theorem 4.5 and Proposition 4.6.

Corollary 4.7.

| coker(X_{J} → X_{A})| = \frac{m_{A}}{m_{L}} .

As a consequence, m_{L}|m_{A}.

5 Optimal quotients of J_{0}(N)

Let X_{0}(N)/Q be the modular curve associated to the congruence subgroup Γ_{0}(N) ⊂ SL_{2}(Z) of matrices which are upper triangular modulo N. Let p be a prime divisor of N which is coprime to M = N/p. The Jacobian J = J_{0}(N) of X_{0}(N) has semistable reduction at p. The Hecke algebra

T = Z[… T_{n} …] ⊂ End(J)

is a commutative ring of endomorphisms of J of Z-rank = dim J. The character group X_{J} is equipped with a functorial action of T. The Hecke algebra T also act on the cusp forms

S = S_{2}(Γ_{0}(N), C).

A newform f is an eigenform normalized so that the coefficient of q in the expansion of f at the cusp ∞ is 1, and such that f does not occur at any level N′ | N with N′ ≠ N. If f is a newform, let I_{f} be the ideal in T of elements which annihilate f. Then O_{f} = T/I_{f} is an order in the ring of integers of the totally real number field K_{f} obtained by adjoining the Fourier coefficients of f to Q. The quotient

A_{f} = J_{0}(N)/I_{f}J_{0}(N)

is a purely toric optimal quotient of dimension [K_{f} : Q].

Let H = H_{1}(X_{0}(N), Z) be the integral homology of the complex algebraic curve X_{0}(N). Integration defines a T-equivariant nondegenerate pairing S × H → C which we view as a map α : H → Hom_{C}(S, C).

Theorem 5.1. We have the following commutative diagram of T-modules:

\xymatrix{H[I_{f}] \ar@{^{(}->}[r] \ar@{^{(}->}[d] & H \ar@{->>}[r] \ar@{^{(}->}[d] & α(H) \ar[d] \ar@{^{(}->}[d] \\ Hom_{C}(S, C)[I_{f}] \ar@{^{(}->}[r] \ar@{->>}[d] & Hom_{C}(S, C) \ar@{->>}[r] \ar@{->>}[d] & Hom_{C}(S[I_{f}], C) \ar@{->>}[d] \\ A′_{f}(C) \ar@{^{(}->}[r] \ar[rr]_{θ_{A}} & J(C) \ar@{->>}[r] & A_{f}(C)}

Proof. This can be deduced from [Shi73].

Corollary 5.2. m_{A}^{2} = [α(H) : α(H[I_{f}])].

Proof. Recall that m_{A} is defined to be \sqrt{deg(θ_{A})}. The kernel of an isogeny of complex tori is isomorphic to the cokernel of the induced map on lattices. The corollary now follows from the diagram of Theorem 5.1 which indicates that the index [α(H) : α(H[I_{f}])] is the cokernel of the map H[I_{f}] → α(H).

Let Frob_{p} : X_{J} → X_{J} denote the map induced by Frobenius. One has Frob_{p} = −W_{p}, where W_{p} is the map induced by the Atkin-Lehner involution on J_{0}(p). Let f be a newform, A = A_{f} the corresponding optimal quotient, and w_{p} the sign of the eigenvalue of W_{p} on f.

Proposition 5.3.

Φ_{A}(F_{p}) = \begin{cases}Φ_{A}(\overline{F}_{p}) & if w_{p} = −1, \\ Φ_{A}(\overline{F}_{p})[2] & if w_{p} = 1. \end{cases}

Proof. If w_{p} = −1, then Frob_{p} = 1 and the Gal(\overline{F}_{p}/F_{p})-action of Φ_{A}(\overline{F}_{p}) is trivial. Thus in this case, Φ(F_{p}) = Φ(\overline{F}_{p}). Next suppose w_{p} = 1. We have an exact sequence

0 → X_{A′} → Hom(X_{A}, Z) → Φ_{A} → 0.

Since W_{p} acts as +1 on f it acts as +1 on A, on X_{A}, on Hom(X_{A}, Z), and on Φ_{A}. Thus Frob_{p} = −W_{p} acts as −1 on Φ_{A}. The 2-torsion in a finite abelian group equals the fixed points under −1.

5.1 Computation

Suitable generalizations of the algorithms described in [Cre97] can be used to enumerate the optimal quotients A_{f} and to compute m_{A}. These will be described in the author’s Berkeley Ph.D. thesis [Ste99a]. The method of graphs [Mes86] and [Koh98] can be used to compute X = X_{J_{0}(N)} with its T-action and the monodromy pairing. We can then compute

L = ⋂_{t∈I_{f}} ker(t|_{X}),

m_{X} := m_{L}, and Φ_{X} := Φ_{L}. By Theorem 4.5 we can now compute

|Φ_{A}| = |Φ_{X}| · \frac{m_{A}}{m_{X}} .

We have computed Φ_{A} in a number of cases. In the next subsection we discuss two conjectures suggested by our numerical computations.

5.2 Conjectures

Our numerical computations suggest the following conjectures. Suppose that N = pM with (p, M) = 1. Let

H_{new} = ker (H_{1}(X_{0}(N), Z) ⟶ (H_{1}(X_{0}(M), Z) ⊕ H_{1}(X_{0}(M), Z)),

where the map is induced by the two natural degeneracy maps X_{0}(N) → X_{0}(M). The Hecke algebra T acts on H_{new}, and on the submodule H_{new}[I_{f}] of elements annihilated by I_{f}. Integration defines a map α : H_{new} → Hom(S[I_{f}], C). Define the homology congruence modulus m_{H} by

m_{H}^{2} = [α(H_{new}) : α(H_{new}[I_{f}])].

We expect that there is a very close relationship between m_{X} and m_{H}.

Conjecture 5.4. Up to powers of 2,

m_{X} = m_{H}.

When N = p is prime we make the following conjecture.

Conjecture 5.5. Let p be a prime and let f_{1}, …, f_{n} be a set of representatives of the Galois conjugacy classes for newforms in S_{2}(Γ_{0}(p)). Let A_{1}, …, A_{n} be the corresponding optimal quotients. Then the natural maps

Φ_{J_{0}(p)} ⟶ ∏_{i=1}^{n} Φ_{A_{i}}

J_{0}(p)(Q) ⟶ ∏_{i=1}^{n} A_{i}(Q)

are isomorphisms.

We offer the tables in the next section as evidence that an assertion such as the above two conjectures may be true.

6 Tables

We computed several component groups of optimal quotients A_{f} of J_{0}(N) associated to newforms f. We denote such an optimal quotient by

N isogeny-class dimension

The dimension frequently determines the factor, so it is included in the notation.

6.1 Table 1: Some large component groups predicted by the Birch and Swinnerton-Dyer conjecture

Using the algorithm described in [Ste99b] we computed the special value L(A, 1)/Ω (up to a Manin constant) for every optimal quotient A = A_{f} of level ≤ 1500. We found exactly five for which the numerator of L(A, 1)/Ω is nonzero and divisible by a prime number > 10^{9}. These are given below.

A  N  L(A, 1)/Ω · Manin constant
1154E20 2 · 577  2^{?} · 85495047371/17^{2}
1238G19 2 · 619  2^{?} · 7553329019/5 · 31
1322E21 2 · 661  2^{?} · 57851840099/331
1382D20 2 · 691  2^{?} · 37 · 1864449649/173
1478J20 2 · 739  2^{?} · 7 · 29 · 1183045463/5 · 37

The Birch and Swinnerton-Dyer conjecture predicts that these large prime divisors must divide either |Φ_{A}| or the Shafarevich-Tate group of A. We computed Φ_{A} and found that this was the case.

A  p w |Φ_{X}|  m_{X}  |Φ_{A}|
1154E20 2 − 17^{2}  2^{24}  2^{?} · 17^{2} · 85495047371
577 + 1  2^{26} · 85495047371  2^{?}
1238G19 2 − 5 · 31  2^{26}  2^{?} · 5 · 31 · 7553329019
619 + 1  2^{28} · 7553329019  2^{?}
1322E21 2 − 331  2^{28}  2^{?} · 331 · 57851840099
661 + 1  2^{32} · 57851840099  2^{?}
1382D20 2 − 173  2^{29}  2^{?} · 37 · 173 · 1864449649
691 + 1  2^{31} · 37 · 1864449649  2^{?}
1478J20 2 − 5 · 37  2^{31}  2^{?} · 5 · 7 · 29 · 37 · 1183045463
739 + 1 2^{33} · 7 · 29 · 1183045463  2^{?}

6.2 Table 3: Some quotients of J_{0}(N)

In this table we give the invariants defined above for the optimal quotients of levels 65, 66, 68, and 69.

A  p w |Φ_{X}| m_{X} m_{H} m_{A} |Φ_{A}|
65A1 5 + 1  2  ?  2  1
13 + 1  2  ?  1
65B2 5 + 3  2^{2}  ?  2^{2}  3
13 − 3  2^{2}  ?  3
65C2 5 − 7  2^{2}  ?  2^{2}  7
13 + 1  2^{2}  ?  1
66A1 2 + 1  2  ?  2^{2}  2
3 − 3  2^{2}  ?  3
11 + 1  2^{2}  ?  1
66B1 2 − 2  2  ?  2^{2}  2^{2}
3 + 1  2^{2}  ?  1
11 + 1  2^{2}  ?  1
66C1 2 − 1  2  ? 2^{2} · 5 2 · 5
3 − 1  2^{2}  ?  5
11 − 1 2^{2} · 5 ?  1
68A2 17 + 2  2 · 3 ? 2 · 3  2
69A1 3 − 2  2  ?  2  2
23 + 1  2  ?  1
69B2 3 + 2  2  ? 2 · 11 2 · 11
23 − 2 2 · 11 ?  2

6.3 Table 3: Some quotients of J_{0}(p)

Using the method of graphs and modular symbols we computed the quantities m_{A}, m_{L} and Φ_{L} for each abelian variety A = A_{f} associated to a newform of prime level p ≤ 757. The results were as follows:

1. In all cases m_{A} = m_{L}, so the map Φ_{J} → Φ_{A} is surjective.

2. Φ_{A} = 1 whenever the sign of the Atkin-Lehner involution w_{p} on A is 1.

3. ∏ |Φ_{A}| = |Φ_{J}| Table 1 lists those A of level ≤ 631 for which w_{p} = −1, along with the order of the component group.

Table 1: Some quotients of J_{0}(p)

A  |Φ_{A}|
11A1  5
17A1  2^{2}
19A1  3
23A2  11
29A2  7
31A2  5
37B1  3
41A3  2 · 5
43B2  7
47A4  23
53B3  13
59A5  29
61B3  5
67A1  1
67C2  11
71A3  5
71B3  7
73A1  2
73C2  3
79B5  13
83B6  41
89B1  2
89C5  11
97B4  2^{3}
101B7  5^{2}
103B6  17
107B7  53
109A1  1
109C4  3^{2}
113A1  2
113B2  2
113D3  7
127B7 3 · 7
131B10 5 · 13
137B7 2 · 17
139A1  1
139C7  23
149B9  37
151B3  1
151C6  5^{2}

A  |Φ_{A}|
157B7  13
163C7  3^{3}
167B12 83
173B10 43
179A1  1
179C11 89
181B9  3 · 5
191B14 5 · 19
193C8  2^{4}
197C10 7^{2}
199A2  1
199C10 3 · 11
211A2  5
211D9  7
223C12 37
227B2  1
227C2  1
227E10 113
229C11 19
233A1  2
233C11 29
239B17 7 · 17
241B12 2^{2} · 5
251B17  5^{3}
257B14  2^{6}
263B17 131
269C16 67
271B16 3^{2} · 5
277B3  1
277D9  23
281B16 2 · 5 · 7
283B14 47
293B16 73
307A1  1
307B1  1
307C1  1
307D1  1
307E2  3
307F9  17
311B22 5 · 31

A  |Φ_{A}|
313A2  1
313C12 2 · 13
317B15 79
331D16 5 · 11
337B15 2^{2} · 7
347D19 173
349B17 29
353A1  2
353B3  2
353D14 2 · 11
359D24 179
367B19 61
373C17 31
379B18 3^{2} · 7
383C24 191
389A1  1
389E20 97
397B2  1
397C5  11
397D10  3
401B21 2^{2} · 5^{2}
409B20 2 · 17
419B26 11 · 19
421B19 5 · 7
431B1  1
431D3  1
431F24 5 · 43
433A1  1
433B3  1
433D16 2^{2} · 3^{2}
439C25 73
443C1  1
443E22 13 · 17
449B23 2^{4} · 7
457C20 2 · 19
461D26 5 · 23
463B22 7 · 11
467C26 233
479B32 239
487A2  1

A  |Φ_{A}|
487B2  3
487C3  1
487D16  3^{3}
491C29 5 · 7^{2}
499C23  83
503B1  1
503C1  1
503D3  1
503F26  251
509B28  127
521B29 2 · 5 · 13
523C26 3 · 29
541B24 3^{2} · 5
547C25 7 · 13
557B1  1
557D26 139
563A1  1
563E31  281
569B31 2 · 71
571A1  1
571B1  1
571C2  1
571D2  1
571F4  1
571I18  5 · 19
577A2  3
577B2  1
577C3  1
577D18  2^{4}
587C31  293
593B1  2
593C2  1
593E27 2 · 37
599C37 13 · 23
601B29 2 · 5^{2}
607D31 101
613C27 3 · 17
617B28 2 · 7 · 11
619B30  103
631B32 3 · 5 · 7

References

[Col99] R. Coleman, The monodromy pairing, Asian Math. Journal (1999).

[Cre97] J.E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[Edi91] B. Edixhoven, L’action de l’algèbre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est “Eisenstein”, Astérisque (1991), no. 196-197, 7–8, 159–170 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).

[Koh98] D. Kohel, Hecke module structure of quaternions.

[Maz77] B. Mazur, Modular curves and the eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).

[Mes86] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata) (1986), 217–242.

[Mil86] J.S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150.

[Shi73] G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), no. 3, 523–544.

[Ste99a] W. Stein, Optimal quotients of J_{0}(N), U.C. Berkeley Ph.D. thesis, in preparation (1999).

[Ste99b] W. Stein, Some abelian varieties with visible shafarevich-tate groups, Preprint (1999).