Some Abelian Varieties with Visible Shafarevich-Tate Groups

William Stein^{∗}

May 28, 1999

Abstract We give examples of abelian variety quotients of the modular jacobian J_{0}(N) with nontrivial visible Shafarevich-Tate groups. In computing these examples we developed formulas for invariants of higher dimensional modular abelian varieties.

Introduction

In [4] Cremona and Mazur studied visibility of Shafarevich-Tate groups of elliptic curves E ⊂ J_{0}(N). This paper extends some of these computations to higher dimensional A ⊂ J_{0}(N). For each N ≤ 1001 and N = 1028, 1061, we compute the analytic rank 0 new optimal A in J_{0}(N) having nontrivial odd visible Shafarevich-Tate group, visible in the new part of J_{0}(N). A total of 19 such A were found having � of order the squares of: 3, 5, 7, 3^{2}, 11, 13, 151. Our computations say little about invisible elements of � , however see [1]. The algorithms developed for computing invariants of modular abelian varieties are also of some interest.

Acknowledgment: It is a pleasure to thank Barry Mazur for his brilliant lectures on visibility, Ken Ribet and Robert Coleman for explaining monodromy, and John Cremona for explaining modular symbols. The author would also like to thank Richard Taylor and Loic Merel for comments, and Amod Agashe for many useful discussions and proving Theorem 2.7.

Notation

Let S_{2}(N) = S_{2}(Γ_{0}(N), C) be the space of cusp forms of weight 2 for the subgroup Γ_{0}(N) ⊂ SL_{2}(Z). Thus S_{2}(N) can be identified with the differentials on the modular curve X_{0}(N). Let J_{0}(N) denote the jacobian of X_{0}(N). Denote by H_{1}(X_{0}(N), Z) the integral homology of X_{0}(N). The Hecke algebra T acts in a compatible manner on S_{2}(N), H_{1}(X_{0}(N), Z) and J_{0}(N). The letters f and g are reserved for newforms, i.e., normalized eigenforms for T in the complement of the old subspace of S_{2}(N). Let f_{1}, …, f_{d} denote the Galois conjugates of f, and I_{f} = Ann(f) the annihilator of f in T.

∗Amod Agashe will be a joint author of this paper as soon as he has had a chance to agree with what it says.

1 Optimal abelian varieties and newforms

An abelian variety quotient of J_{0}(N) is called optimal if the kernel is connected. We associate to f an optimal quotient A_{f} of J_{0}(N):

0 → I_{f}J_{0}(N) → J_{0}(N) → A_{f} → 0.

Let H = H_{1}(X_{0}(N), Z), S_{2} = S_{2}(N), and recall that integration defines a nondegenerate pairing S_{2} × H → C, hence a map H → Hom_{C}(S_{2}, C). Composing with restriction to S_{2}[I_{f}] defines a map Φ_{f} : H → Hom(S_{2}[I_{f}], C).

Theorem 1.1. A_{f} is an abelian variety of dimension d with canonical L-series

L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s).

The complex uniformization of the tori A_{f}(C) and A_{f}^{∨}(C) is described by the following diagram

0  0  0
↓  ↓  ↓
H[I_{f}]  ⟶  H  ⟶  Φ_{f}(H)
↓  ↓  ↓
Hom(S_{2}, C)[I_{f}] ⟶ Hom(S_{2}, C) ⟶ Hom(S_{2}[I_{f}], C)
↓  ↓  ↓
A^{∨}(C)  ⟶ J_{0}(N)(C) ⟶  A(C)
↓  ↓  ↓
0  0  0

in which the vertical columns are exact but the rows are not.

Proof. [16] and section 1.7 of [5].

1.1 The Birch and Swinnerton-Dyer Conjecture

Let A/Z be the Neron model of A = A_{f}. The Tamagawa number c_{p} is the number of F_{p}-rational components of the special fiber A_{F_{p}}. A basis h_{1}, …, h_{d} for the Neron differentials defines a measure μ on A(R) and we let Ω_{A} = μ(A(R)). Let w : H^{1}(Q, A) → ∏_{v} H^{1}(Q_{v}, A) and set � = Ker(w). If L(A, 1) ≠ 0 it is known [9] that A(Q) and � (A) are both finite. One then has the following fundamental and still open

Conjecture 1.2 (Birch, Swinnerton-Dyer, Tate).

L(A, 1) = Ω_{A} · \frac{|� | · ∏_{p|N} c_{p}}{|A(Q)| · |A^{∨}(Q)|}.

1.2 The Manin Constant

Let S_{f}(C) ⊂ S_{2}(N, C) denote the subspace of cusp forms spanned by the conjugates of f. There are two lattices in S_{f}(C). One is the lattice S_{f}(Z) of cusp forms with integer Fourier expansion at infinity. The other S_{f}(A/Z) is got by pulling back the Neron differentials defined above. The Manin constant is

c_{f} = [S_{f}(Z) : S_{f}(A/Z)].

We are aware of no examples of newforms f for which c_{f} ≠ 1. It is reasonable to expect that one can extend methods known for elliptic curves (e.g., [11]) to show that c_{f} is at least coprime to 2N.

Later in this paper we give a formula for L(A_{f}, 1)/Ω_{f} where Ω_{f} is computed using the lattice S_{f}(Z). Thus our BSD special value is off by the Manin constant. For the remainder of this paper we officially assume

Conjecture 1.3. c_{f} = 1

1.3 Connecting Mordell-Weil and Shafarevich-Tate

Let f, g ∈ S_{2}(N) be nonconjugate newforms and A_{f}, A_{g} the corresponding optimal quotients of J_{0}(N). Let 𝔪 ⊂ T be a maximal ideal such that A_{f}^{∨}[𝔪] = A_{g}^{∨}[𝔪] ⊂ J_{0}(N). Let p be the residue characteristic of 𝔪. Assume that p ∤ 2N · ∏_{p|N} c_{p}c′_{p} where c′_{p} are the Tamagawa numbers of A_{g}.

Under hypothesis such as these we expect there to be a commutative diagram

0 → A_{f}(Q)/𝔪A_{f}(Q) → X → � (A_{f})[𝔪] → 0
||
0 → A_{g}(Q)/𝔪A_{g}(Q) → X → � (A_{g})[𝔪] → 0

with exact rows. Here X is an abelian group, H^{1}(Spec Z, A_{f}[𝔪]). A precise statement will not be given here as the purpose of this paper is merely to present a few algorithms and computational results.

2 Algorithms

2.1 Modular Symbols

Modular symbols give a presentation of the homology of the modular curve X_{0}(N). Here we briefly review modular symbols for Γ_{0}(N). More information on how to actually compute with them can be found in [3], [7], and [12].

Define the space of modular symbols \underline{M}(N, Z) to be the free abelian group generated by symbols {α, β} such that α, β ∈ P^{1}(Q) = Q ∪ {∞} subject to the relations

0 = {α, β} + {β, γ} + {γ, α},
{α, β} = {g(α), g(β)}, all g ∈ Γ_{0}(N).

The space of boundary symbols \underline{B}(N, Z) is the free abelian group generated by symbols {α}, α ∈ P^{1}(Q), modulo the relations

{α} = {g(α)}, all g ∈ Γ_{0}(N).

The cuspidal symbols \underline{S}(N, Z) are the kernel of the boundary map ∂(\{α, β\}) = {β} − {α}:

0 → \underline{S}(N, Z) → \underline{M}(N, Z) ⟶^{∂} \underline{B}(N, Z).

The Hecke algebra act on \underline{M}(N, Z) and there is an involution ∗\{α, β\} = \{−α, −β\}. Integration defines a pairing

S_{2}(N) × \underline{M}(N, Z) → C.

The Manin-Drinfeld theorem asserts that the image of \underline{M}(N, Z) in Hom(S_{2}(N), C) is a lattice. There is a natural isomorphism between \underline{S}(N, Z) and H_{1}(X_{0}(N), Z).

2.2 The Method of Graphs

We briefly review the method of graphs, see [13] and [15] for more details. Let M be a positive integer, p a prime not dividing M, and put N = pM. Let D be the finitely generated free abelian group on the superingular points of X_{0}(M)(\overline{F}_{p}), i.e., the enhanced elliptic curves E = (E, C) where E is a supersingular elliptic curve defined over \overline{F}_{p} and C is a cyclic subgroup of order M, and enhanced curves are identified if they are isomorphic in the evident way. Let w_{E} = \frac{| Aut(E)|}{2} where Aut(E) is the group of \overline{F}_{p}-automorphisms of E. We have w_{E} ≤ 12 and if p ≥ 5 then w_{E} ≤ 3. The monodromy pairing on D is

⟨E, E′⟩ = \begin{cases}w_{E} & if E = E′ \\ 0 & if E ≠ E′ \end{cases}

The Hecke operators act on D in a way compatible with this pairing. Define

X_{N,p} = {∑ a_{E}E : ∑ a_{E} = 0}

It is known that X_{N,p} ⊗ C is isomorphic as a Hecke module to the subspace of S_{2}(N, C) generated by newforms and oldforms of level pd for d|M.

David Kohel [8] has implimented an algorithm which computes the action of the Hecke operators on X_{N,p} using the arithmetic of quaternion algebras.

2.3 Enumerating quotients of J_{0}(N)

It is necessary to list all newforms of a given level N. This can be done by decomposing the new subspace of the modular symbols \underline{S}(N, Q) using the Hecke operators. The characteristic polynomial of T_{2} is computed, and then T_{2} is used to break up the space. The process is applied recursively with T_{3}, T_{5}, … until it terminates. After computing the decomposition we order the newforms as suggested by Cremona: First by dimension. Within each dimension, in binary by the signs of the Atkin-Lehner involutions, e.g., + + +, + + −, + − +, + − −, − + +, etc. When two forms have the same involutions, order by | Tr(a_{p})| with ties broken by taking the positive trace first. For historical reasons this does not always agree with the ordering in Cremona’s tables (page 5 of [3]). There is only one case in our table in which the two ordering schemes disagree, our 446B is Cremona’s 446D.

2.4 The Modular Polarization

A polarization of an abelian variety A is an isogeny between A and its dual arising from a very amply invertible sheaf (see [14]). J_{0}(N) is a Jacobian so it possesses a canonical polarization arising from the θ-divisor and this induces the modular polarization θ_{f} : A_{f}^{∨} → A_{f}.

[figure: A_{f}^{∨}; J_{0}(N); A_{f}; θ_{f}]

By ([14], Theorem 13.3) deg(θ_{f}) is a perfect square so we may define the modular degree δ_{f} = \sqrt{deg(θ_{f})}. The kernel of θ_{f} is the intersection of A_{f}^{∨} with I_{f}J_{0}(N) so it measure intersections between A_{f}^{∨} and other factors of J_{0}(N).

Proposition 2.1. With notation as in Theorem 1.1,

Ker(θ_{f}) ≅ Coker(H[I_{f}] → Φ_{f}(H))

Proof. Delete the middle column of the diagram in Theorem 1.1 and apply the snake lemma.

Using Lemma 2.5, to be proved later, we see that the modular degree can be computed as follows: Let φ_{1}, …, φ_{2d} be a basis for Hom(H, Z)[I_{f}] and a_{1}, …, a_{2d} a basis for H[I_{f}]. Then δ_{f} is the square root of the absolute value of the determinant of the matrix (φ_{i}(a_{j})).

2.5 Torsion

We can obtain both upper and lower bounds on A_{f}(Q)_{tor} and A_{f}^{∨}(Q)_{tor}. For the examples considered in our table these bounds were sufficient to determine the odd parts of these groups. Let χ_{p}(X) ∈ Z[X] denote the characteristic polynomial of T_{p} acting on A_{f}. It is a polynomial having integer coefficients and degree equal to dim A_{f}.

Proposition 2.2. Both |A_{f}(Q)_{tor}| and |A_{f}^{∨}(Q)_{tor}| divide

gcd{χ_{p}(p + 1) : (p, 2N) = 1, p prime}.

Proof. Use the Eichler-Shimura relation and injectivity of rational torsion under reduction modulo an odd prime p (since 1 = e < p − 1).

The difference of cusps α, β ∈ X_{0}(N) define a point (α) − (β) ∈ J_{0}(N)(C).

Proposition 2.3. The order of the image of (α) − (β) in A_{f}(C) equals the order of the image of the modular symbol {α, β} in

\frac{Φ_{f}(\underline{M}(N, Z))}{Φ_{f}(\underline{S}(N, Z))}.

Proof. By the classical Abel-Jacobi theorem ([10], ch IV, Theorem 2.2), the modular symbol {α, β} maps, via the period map, to the point (α) − (β) ∈ J_{0}(N)(C). Now use Theorem 1.1.

In particular, the point (0) − (∞) ∈ J_{0}(N)(Q) generates a cyclic subgroup of A_{f}(Q) and this gives a lower bound on A_{f}(Q)_{tor}.

2.6 Component Groups

Let p be a prime exactly dividing N and Φ_{A,p} the component group of A = A_{f}. Thus we have the exact sequence

0 → A_{F_{p}}^{0} → A_{F_{p}} → Φ_{A,p} → 0

with A the Neron model of A and A_{F_{p}}^{0} connected. Let X[I_{f}] be the submodule of X = X_{N,p} cut out by the annihilator of f. The monodromy pairing defines a map X → Hom(X[I_{f}], Z). Let δ_{f} be the modular degree and w_{p} the sign of the Atkin-Lehner involution W_{p} on f. The following will be proved in [18].

Theorem 2.4.

|Φ_{A,p}(\overline{F}_{p})| = δ_{f} · \frac{| Coker(X → Hom(X[I_{f}], Z))|^{2}}{Disc(X[I_{f}] × X[I_{f}] → Z)} ,

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

2.7 Rational part of the special value

Let \underline{M}(Q) = \underline{M}(N, Q) and extend Φ_{f} to a map \underline{M}(Q) → C. Then Φ_{f} has a rational structure in the following sense.

Lemma 2.5. Let φ_{1}, …, φ_{n} be a Q-basis for Hom(\underline{M}(Q), Q)[I_{f}] and set

Ψ = φ_{1} × ⋯ × φ_{n} : \underline{M}(Q) → Q^{n}.

Then n = 2d and Ker(Ψ) = Ker(Φ_{f}).

Proof. This result is due to Shimura ([17]), but we sketch a proof. To compute dim Hom(\underline{M}(Q), Q)[I_{f}] we may first tensor with C. Let \overline{S}_{2} denote the weight 2 anti-holomorphic cusp forms and E_{2} the weight 2 Eisenstein series for Γ_{0}(N). Then \underline{M}(C) is isomorphic as a T-module to S_{2} ⊕ \overline{S}_{2} ⊕ E_{2} (prop. 9 of [12] and the Eichler-Shimura embedding). Because of the Peterson inner product, the dual Hom(\underline{M}(C), C) is also isomorphic as a T-module to S_{2} ⊕ \overline{S}_{2} ⊕ E_{2}. Since f is new, by the Atkin-Lehner multiplicity one theory,

(S_{2} ⊕ \overline{S}_{2} ⊕ E_{2})[I_{f}] = S_{2}[I_{f}] ⊕ \overline{S}_{2}[I_{f}]

has complex dimension 2d, which gives the first assertion.

Next note that Ker(Φ_{f}) ⊗ C ⊂ Ker(Ψ) ⊗ C because each map x ↦ ⟨f_{i}, x⟩ lies in Hom(\underline{M}(Q), C)[I_{f}] and Ker(Ψ) ⊗ C is the intersection of the kernels of all maps in Hom(\underline{M}(Q), C)[I_{f}]. By Theorem 1.1 the image of Φ_{f} is a lattice, so dim_{Q} Ker(Φ_{f}) = dim_{Q} \underline{M}(Q) − 2d. Since Ψ is the intersection of the kernels of n = 2d independent linear functionals φ_{1}, …, φ_{n}, Ker(Ψ) also has dimension dim \underline{M}(Q)−2d. Since the dimensions are the same and there is an inclusion, we have an equality Ker(Φ_{f})⊗C = Ker(Ψ)⊗C which forces Ker(Φ_{f}) = Ker(Ψ).

Let V be a finite dimensional vector space over R. A lattice L ⊂ V is a free abelian group of rank = dim V such that RL = V. If L, M ⊂ V are lattices, the lattice index [L : M] is the absolute value of the determinant of an automorphism of V taking L isomorphically onto M. Extend the definition to the case when M has rank strictly smaller than dim V by defining [L : M] = 0.

Lemma 2.6. Suppose τ_{i} : V → W_{i}, i = 1, 2 are surjective linear maps such that Ker(τ_{1}) = Ker(τ_{2}). Then

[τ_{1}(L) : τ_{1}(M)] = [τ_{2}(L) : τ_{2}(M)].

Proof. Surjectivety and equality of kernels insures that there is a unique isomorphism ι : W_{1} → W_{2} such that ιτ_{1} = τ_{2}. Let σ be an automorphism of W_{1} such that σ(τ_{1}(L)) = τ_{1}(M). Then

ισι^{−1}(τ_{2}(L)) = ιστ_{1}(L) = ιτ_{1}(M) = τ_{2}(M).

Since conjugation doesn’t change the determinant,

[τ_{2}(L) : τ_{2}(M)] = | det(ισι^{−1})| = | det(σ)| = [τ_{1}(L) : τ_{1}(M)].

Let S_{2}(N, Z) be the space of cusp forms whose q-expansion at infinity hass integer coefficients. Let Ω_{f}^{0} be the measure of the identity component of A_{f}(R) with respect to an integral basis for S_{f}(Z) = S_{2}(N, Z)[I_{f}]. Let e = {0, ∞} ∈ \underline{M}(N, Z) denote the winding element.

Theorem 2.7. Let Ψ be as in Lemma 2.5. Then

±\frac{L(A_{f}, 1)}{Ω_{f}^{0}} = [Ψ(\underline{S}(N, Z)^{+}) : Ψ(Te)]

Proof. Let Φ = Φ_{f} be the period map defined by a basis f_{1}, …, f_{d} of conjugate newforms. The image of Φ, which we identify with C^{d}, is an algebra with unit element 1 = (1, …, 1) equipped with an action of the Hecke operators: T_{p} acts as (a_{p}^{(1)}, …, a_{p}^{(d)}) where the components are the Galois conjugates of a_{p}. Let Z^{d} ⊂ R^{d} ⊂ C^{d} be the usual submodules. Let Vol(\underline{S}^{+}) be the volume of the image of \underline{S}^{+} = \underline{S}(N, Z)^{+} under Φ. Observe that Vol(\underline{S}^{+}) = [Z^{d} : Φ(\underline{S}^{+})] and |L(A_{f}, 1)| = [Z^{d} : Φ(e)Z^{d}]. Let W ⊂ C^{d} be the Z-module spanned by the columns of a basis for S_{f}(Z). Because Ω_{f}^{0} is computed with respect to a basis for S_{f}(Z),

Vol(\underline{S}^{+}) = [W : T1] · Ω_{f}^{0}.

Because S_{2}(N, Z) is saturated, [Z^{d} : W] = 1 so [Z^{d} : T1] = [W : T1]. The following calculation involves lattices in R^{d}:

[Φ(\underline{S}^{+}) : Φ(Te)] = [Φ(\underline{S}^{+}) : Z^{d}] · [Z^{d} : Φ(Te)]
= \frac{1}{[Z^{d} : Φ(\underline{S}^{+})]} · [Z^{d} : Φ(Te)]
= \frac{1}{Vol(\underline{S}^{+})} · [Z^{d} : Φ(e)Z^{d}] · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(\underline{S}^{+})} · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(\underline{S}^{+})} · [Φ(e)Z^{d} : Φ(e)T1]
= \frac{|L(A_{f}, 1)|}{Ω_{f}^{0} · [W : T1]} · [Z^{d} : T1]
= \frac{|L(A_{f}, 1)|}{Ω_{f}^{0}}.

The theorem now follows from lemmas 2.5, 2.6, and the fact that f has real Fourier coefficients so L(A_{f}, 1) ∈ R hence |L(A_{f}, 1)| = ±L(A_{f}, 1).

Corollary 2.8. Let n_{f} be the order of the image in A_{f}(Q) of the point (0) − (∞) ∈ J_{0}(N)(Q). Then

\frac{L(A_{f}, 1)}{Ω_{f}^{0}} ∈ \frac{1}{n_{f}} Z.

Proof. Let x denote the image of (0) − (∞) ∈ A_{f}(Q) and set I = Ann(x) ⊂ T. Since f is a newform the Hecke operators T_{p} for p|N act as 0 or ±1 on A_{f}(Q) (end of section 6 of [6]). If p ∤ N a standard calculation (section 2.8 of [3]) combined with the Abel-Jacobi theorem shows that T_{p}(x) = (p + 1)x. Let C = Zx denote the (finite, by Manin-Drinfeld) cyclic subgroup of A_{f}(Q) generated by x, so n_{f} is the order of C. There is an injection T/I ↪ C sending T_{p} to T_{p}(x). By the theorem, we have

±L(A_{f}, 1)/Ω_{f}^{0} = [Ψ(\underline{S}^{+}) : Ψ(Te)]
= [Ψ(\underline{S}^{+}) : Ψ(Ie)] · [Ψ(Ie) : Ψ(Te)]
= [Ψ(\underline{S}^{+}) : IΨ(e)] · [IΨ(e) : TΨ(e)]
= \frac{[Ψ(\underline{S}^{+}) : IΨ(e)]}{[TΨ(e) : IΨ(e)]} ∈ \frac{1}{n_{f}} Z.

The final inclusion follows from two observations. By Abel-Jacobi, I is exactly those elements of T which send Ψ(e) into Ψ(\underline{S}^{+}), so [Ψ(\underline{S}^{+}) : IΨ(e)] ∈ Z. Second, there is a surjective map

T/I → \frac{TΨ(e)}{IΨ(e)}

sending t to tΨ(e), so [TΨ(e) : IΨ(e)] divides n_{f} = |C| = |T/I|.

2.8 Intersections

Let f, g be nonconjugate newforms and H = H_{1}(X_{0}(N), Z).

Proposition 2.9. (A_{f}^{∨} ∩ A_{g}^{∨})[p] ≠ 0 iff the mod p rank of H[I_{f}] + H[I_{g}] is strictly less than rank H[I_{f}] + rank H[I_{g}].

Proof. By (1.1) Λ_{f} = H[I_{f}] (resp., Λ_{g} = H[I_{g}]) is the submodule of H which defines A_{f} (resp., A_{g}). By reduction mod p we mean the map H → H ⊗ F_{p}. Suppose

rank(Λ_{f} + Λ_{g}) mod p < rank Λ_{f} + rank Λ_{g}.

Since Λ_{f} (resp., Λ_{g}) is a kernel, it is saturated, so rank Λ_{f} mod p = rank Λ_{f} (resp., for Λ_{g}). We conclude that the mod p linear dependence must involve vectors from both Λ_{f} and Λ_{g}; there is v ∈ Λ_{f} and w ∈ Λ_{g} so that v, w ≢ 0 mod p but v + w ≡ 0 mod p. Thus \frac{v+w}{p} ∈ H is integral, i.e., in J_{0}(N)(C) we have \frac{1}{p}v − (−\frac{1}{p}w) = 0. But \frac{1}{p}v ∉ Λ_{f} and \frac{1}{p}w ∉ Λ_{g} (otherwise v and w would be 0 mod p), so \frac{1}{p}v and −\frac{1}{p}w are both nontrivial p-torsion in A_{f}^{∨}, A_{g}^{∨}, resp. Conclusion: 0 ≠ \frac{1}{p}v = −\frac{1}{p}w ∈ (A_{f}^{∨} ∩ A_{g}^{∨})[p].

Conversely, suppose 0 ≠ x ∈ (A_{f}^{∨} ∩ A_{g}^{∨})[p]. Choose lifts modulo H to x_{f} ∈ \frac{1}{p}Λ_{f} and x_{g} ∈ \frac{1}{p}Λ_{g}. Then px_{f} ∈ Λ_{f} (resp., px_{g} ∈ Λ_{g}), but px_{f} ∉ pH (resp., px_{g} ∉ pH) because x ≠ 0. Since x_{f} −x_{g} ∈ H, px_{f} −px_{g} = p(x_{f} −x_{g}) ≡ 0 mod p. This is a nontrivial linear relation between Λ_{f} and Λ_{g}.

Corollary 2.10. If p > 2 and the sign of some Atkin-Lehner involution for f is different than that for g then (A_{f}^{∨} ∩ A_{g}^{∨})[p] = 0.

Proof. Suppose w_{q}(f) ≠ w_{q}(g) and let G = (A_{f}^{∨} ∩ A_{g}^{∨})[p]. Observe that W_{q} acts as w_{q}(f) mod p on A_{f}^{∨}[p] and as w_{q}(g) mod p on A_{g}^{∨}[p]. Hence W_{q} acts as both w_{q}(f) mod p and w_{q}(g) mod p on G. Since p > 2, this is not possible when G ≠ 0.

3 Results

This section contains tables computed using the above algorithms as implimented in the author’s program HECKE (a C++ program using LiDIA and NTL), David Kohel’s Magma software, and PARI. Each factor A_{f} of J_{0}(N) is denoted as follows:

N isogeny-class dimension

The dimension frequently determines the factor, so it is included in the notation. We consider only the odd part of � so we only computed the odd parts of the arithmetic invariants of A_{f}. Thus at this point we make the

WARNING: ONLY ODD PARTS OF INVARIANTS ARE GIVEN!

Tables 1-3: New Visible � Let n_{f} be the largest odd square dividing the numerator of L(A_{f}, 1)/Ω_{f}. Table 1 lists those A_{f} such that, for p|n_{f} there exists a new factor B_{g} of J_{0}(N), of positive analytic rank, and such that (A_{f}^{∨} ∩ B_{g}^{∨})[p] ≠ 0. This is necessary (and usually sufficient) for the p-torsion in the new visible part of � to be nonzero. In many cases it could be seen that there were no other appropriate new factors by looking at the signs of the Atkin-Lehner involutions. Up to level 1001 our search was systematic. The two examples after level 1001 were not found by systematic search (i.e., there may be a gap). In those cases for which 4|N, we put c_{2} = a, as we don’t know how to compute c_{2} exactly when the reduction is additive. Table 2 contains further arithmetic information about each explanatory factor.

The explanatory factors of level ≤ 1028 are exactly the set of rank 2 elliptic curves of level ≤ 1028. By [2], the explanatory factor at level 1061 is the first surface of rank 4 (and prime level). Table 4: Component groups Table 4 gives the quantities involved in the formula for Tamagawa numbers, for each of the A_{f} from table 1. Table 5: Odd square numerator In order to find the A_{f}, we first enumerated those A_{f} for which the numerator of L(A_{f}, 1)/Ω_{f} is divisible by an odd square n_{f}. For N < 1000, these are given in table 5. Any odd visible � coprime to primes dividing torsion and c_{p} must show up as a divisor of the numerator, and given BSD, it must show up as a square divisor because the Mordell-Weil rank of the explanatory factor is even. It would be interesting to compute the conjectural order of � for each abelian variety in this table, but not in table 1, and show (when possible) that the visible � is old.

A_{f}  n_{f}  w_{q}  c_{p}  T T L(1)/Ω_{f}  δ_{A}  B_{g}
389E20  5^{2}  −  97  97  5^{2}  5  389A1
433D16  7^{2}  −  3^{2}  3^{2}  7^{2}  3 · 7 · 37  433A1
446F8  11^{2}  +−  1, 3  3  11^{2}  11 · 359353  446B1
563E31  13^{2}  −  281 281  13^{2}  13  563A1
571D2  3^{2}  −  1  1  3^{2}  3^{2} · 127  571B1
655D13  3^{4}  +−  1, 1  1  3^{4}  3^{2} · 19 · 515741 655A1
664F8  5^{2}  −+  a, 1  1  5^{2}  5  664A1
681B1  3^{2}  +−  1, 1  1  3^{2}  3 · 5^{3}  681C1
707G15  13^{2}  +−  1, 1  1  13^{2}  13 · 800077  707A1
709C30  11^{2}  −  59  59  11^{2}  11  709A1
718F7  7^{2}  +−  1, 1  1  7^{2}  7 · 151 · 35573  718B1
794G14  11^{2}  +−  3, 1  3  11^{2}  3 · 7 · 11 · 47 · 35447 794A1
817E15  7^{2}  +−  1, 5  5  7^{2}  7 · 79  817A1
916G9  11^{2}  −+  a, 1  1  11^{2}  3^{9} · 11 · 17 · 239 916C1
944O6  7^{2}  +−  a, 1  1  7^{2}  7  944E1
997H42  3^{4}  −  83  83  3^{4}  3^{2}  997B1, C1
1001L7  7^{2}  + − + 1, 1, 1 1  7^{2}  7 · 19 · 47 · 2273 1001C1
1028E14 3^{2} · 11^{2} −+  a, 1  3  3^{4} · 11^{2}  3^{13} · 11  1028A1
1061D46 151^{2}  −  5 · 53 5 · 53  151^{2}  61 · 151 · 179  1061B1

Table 1: New visible �

B_{g} rank w_{q} c_{p} T δ_{B} Comments

389A1

2  −  1  1 5  first curve of rank 2
433A1  2  −  1  1 7

446B1

2  +−  1, 1 1 11  this is 446D in [3]
563A1  2  −  1  1 13
571B1  2  −  1  1 3
655A1  2  +−  1, 1 1 3^{2}
664A1  2  −+  1, 1 1 5
681C1  2  +−  1, 1 1 3
707A1  2  +−  1, 1 1 13
709A1  2  −  1  1 11
718B1  2  +−  1, 1 1 7
794A1  2  +−  1, 1 1 11
817A1  2  +−  1, 1 1 7
916C1  2  −+  3, 1 1 3 · 11
944E1  2  +−  1, 1 1 7
997B1  2  −  1  1 3
997C1  2  −  1  1 3
1001C1 2 + − + 1, 3, 1 1 3^{2} · 7

1028A1

2  −+  3, 1 1 3 · 11 intersects 1028E mod 11

1061B2

4  −  1  1 151 first surface of rank 4 [2]

Table 2: Explanatory factors

\begin{matrix}446 = 2 · 223 & 655 = 5 · 131 & 664 = 2^{3} · 83 & 681 = 3 · 227 \\ 707 = 7 · 101 & 718 = 2 · 359 & 794 = 2 · 397 & 817 = 19 · 43 \\ 916 = 2^{2} · 229 & 944 = 2^{4} · 59 & 1001 = 7 · 11 · 13 & 1028 = 2^{2} · 257 \end{matrix}

Table 3: Factorizations

A_{f}  p w_{p} | Coker |  | Disc(X[I_{f}])|  |Φ(\overline{F}_{p})|
389E20 389 −  97  5 · 97  97
433D16 433 −  3^{2}  3^{3} · 7 · 37  3^{2}
446F8  223 −  3  3 · 11 · 359353  3
2 +  3  3 · 11  3 · 359353
563E31 563 −  281  13 · 281  281
571D2  571 −  1  3^{2} · 127  1
655D13 131 −  1  3^{2} · 19 · 515741  1
5 +  1  3^{2}  19 · 515741
664F8  83 +  1  5  1
681B1  227 −  1  3 · 5^{3}  1
3 +  1  3 · 5^{2}  5
707G15 101 −  1  13 · 800077  1
7 +  1  13  800077
709C30 709 −  59  11 · 59  59
718F7  359 −  1  7 · 151 · 35573  1
2 +  1  7  151 · 35573
794G14 397 −  3  3^{2} · 7 · 11 · 47 · 35447  3
2 +  3  3 · 11  3^{2} · 7 · 47 · 35447
817E15  43 −  5  5 · 7 · 79  5
19 +  1  7  79
916G9  229 +  1  3^{9} · 11 · 17 · 239  1
944O6  59 −  1  7  1
997H42 997 −  83  3^{2} · 83  83
1001L7  13 +  1  7 · 19 · 47 · 2273  1
11 −  1  7 · 19 · 47 · 2273  1
7 +  1  7 · 19 · 47  2273
1028E14 257 +  1  3^{13} · 11  1
1061D46 1061 − 5 · 53 5 · 53 · 61 · 151 · 179  5 · 53

Table 4: Component groups

305D7 : 3  309D8 : 5  335E11 : 3^{2}  389E20 : 5 394A2 : 5  399G5 : 3^{4}
433D16 : 7 435G2 : 3  436C4 : 3  446E7 : 3  446F8 : 11 455D4 : 3
473F9 : 3  500C4 : 3  502E6 : 11  506I4 : 5  524D4 : 3  530G4 : 7
538E7 : 3  551H18 : 3 553D13 : 3  555E2 : 3  556C7 : 3  563E31 : 13
564C3 : 3  571D2 : 3  579G13 : 3 · 5 597E14 : 19 602G3 : 3  604C6 : 3
615F6 : 5  615G8 : 7  620D3 : 3  620E4 : 3  626F12 : 5 629G15 : 3
642D2 : 3  644C5 : 3  644D5 : 3  655D13 : 3^{2} 660F2 : 3  662E10 : 43
664F8 : 5  668B5 : 3  678I2 : 3  681B1 : 3  681I10 : 3  682I6 : 11
707G15 : 13 709C30 : 11 718F7 : 7  721F14 : 3^{2} 724C8 : 3  756G2 : 3
764A8 : 3  765M4 : 3  766B4 : 3  772C9 : 3  790H6 : 3  794G12 : 11
794H14 : 5^{2} 796C8 : 3  817E15 : 7  820C4 : 3  825E2 : 3  844C10 : 3^{2}
855M4 : 3  860D4 : 3  868E5 : 3  876E5 : 3  878C2 : 3  884D6 : 3
885L9 : 3^{2}  894H2 : 3  902I5 : 3  913G17 : 3 916G9 : 11 918O2 : 5
918P2 : 3  925K7 : 3  932B13 : 3^{2}  933E14 : 19 934I12 : 7  944O6 : 7
946K7 : 3  949B2 : 3  951D19 : 3  959D24 : 3 964C12 : 3^{2} 966J1 : 3
970I5 : 3  980F1 : 3  980J2 : 3  986J7 : 5  989E22 : 5 993B3 : 3^{2}
996E4 : 3  997H42 : 3^{2} 998A2 : 3  998H9 : 3  999J10 : 3

Table 5: Odd square numerator

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