Visibility of Shafarevich-Tate groups of modular abelian varieties

Amod Agashe and William A. Stein

March 28, 2000

Abstract We give examples of abelian varieties, which are quotients of the modular Jacobian J_{0}(N), with nontrivial visible Shafarevich-Tate groups. Our examples are not conditional on the truth the Birch and Swinnerton-Dyer conjecture; instead, they provide evidence for this conjecture, and may perhaps lead to new connections between the conjecture and the wellunderstood theory of congruences among modular forms.

Introduction

Now that the Shimura-Taniyama conjecture has been proved [5], the main outstanding problem in the field is the Birch and Swinnerton-Dyer conjecture (BSD conjecture) which ties together the arithmetic invariants of an elliptic curve. At present, there is no general class of elliptic curves for which the full BSD conjecture is known, though a slightly weakened form is known for a broad class of complex multiplication elliptic curves of analytic rank 0 (see [21]).

The notion of visibility of Shafarevich-Tate groups provides a framework in which to discuss the Birch and Swinnerton-Dyer conjecture for certain abelian varieties over Q within the context of weight-two modular forms. Let N be a positive integer, and consider an abelian subvariety A of J_{0}(N), where J_{0}(N) is the Jacobian of the modular curve X_{0}(N) (see, e.g., [9] for a survey of the basic facts about J_{0}(N)). The given inclusion A ↪ J_{0}(N) induces a map H^{1}(Q, A) → H^{1}(Q, J_{0}(N)), which restricts to a map � (A) → � (J_{0}(N)). Following [8] and [1], we call an element c ∈ � (A) visible in J_{0}(N) if it lies in the kernel of this map.

Though the notion of visibility is relatively young, B. Mazur has obtained some partial results. In [8] he asked: “if E ⊂ J_{0}(N) is a elliptic curve, how much of � (E/Q) is visible in an abelian surface inside J_{0}(N)?” In the range of their numerical experiment, the answer is frequently affirmative; however, they warn that the conductors considered (N ≤ 5500) are quite small, and the situation is likely to change completely when the conductor is large. In [15], Mazur proved that if E is an elliptic curve over a number field K, then every element of order 3 in � (E/K) is visible in some abelian surface.

We consider abelian varieties A of dimension possibly greater than one. The numbers become large at much smaller conductor, so the sort of behavior that remains hidden when carrying out numerical experiments with elliptic curves of small conductor is perhaps no longer out of our reach. Certain moduli surfaces are of general type, so one does not expect visibility, in abelian surfaces, of large order Shafarevich-Tate group of elliptic curves; it is thus essential to consider visibility in higher dimensional abelian varieties.

Acknowledgment. It is a pleasure to thank Barry Mazur for his brilliant lectures and conversations about visibility, K. Ribet for explaining congruences, and L. Merel and R. Coleman for helping us to understanding semistable reduction. The authors would also like to thank B. Conrad, B. Poonen, and R. Taylor for their helpful comments.

1 A criterion for the existence of visible elements of the Shafarevich-Tate group

The following theorem makes it possible to provably compute examples in which the visible part of � (A) is nontrivial.

Theorem 1.1. Let A and B be abelian subvarieties of an abelian variety J such that A ∩ B is finite and A(Q) is finite. Assume that B has purely toric reduction at each prime at which J has bad reduction. Suppose p is an odd prime at which J has good reduction and that p does not divide the order of any of the (geometric) component groups of A and B, and of the torsion subgroups of (J/B)(Q) and B(Q). Suppose further that B[p] ⊂ A ∩ B. Then

B(Q)/pB(Q) ↪ � (A)^{∘}.

Proof. Let C = J/A. The long exact sequence of Galois cohomology associated to the short exact sequence

0 → A → J → C → 0

begins

0 → A(Q) → J(Q) → C(Q) ⟶^{δ} H^{1}(Q, A) → · · · .

Because B[p] ⊂ A, the map B → C, obtained by composing the inclusion B ↪ J with J → C, factors through multiplication-by-p, giving the following commutative diagram:

\xymatrix{ & B \ar[d] \ar[r]^{p} & B \ar[d] \\ A \ar[r] & J \ar[r] & C.}

Because B(Q)[p] = 0 and B(Q) ∩ A(Q) = 0, we deduce the following exact diagram:

\xymatrix{ & 0 \ar[d] & K_{1} \ar[d] & K_{2} \ar[d] \\ 0 \ar[r] & B(Q) \ar[r]^{p} \ar[d] & B(Q) \ar[dr]^{π} \ar[r] \ar[d] & B(Q)/pB(Q) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & J(Q)/A(Q) \ar[r] \ar[d] & C(Q) \ar[r] & δ(C(Q)) \ar[r] & 0 \\  & K_{3},}

where K_{1} and K_{2} are the indicated kernels and K_{3} is the cokernel. We have the snake lemma exact sequence

0 → K_{1} → K_{2} → K_{3}.

Because B(Q)[p] = 0 and K_{2} is a p-torsion group, we have K_{1} = 0; the quotient J(Q)/B(Q) has no p-torsion because it is a subgroup of (J/B)(Q); also, A(Q) is a finite group of order coprime to p, so K_{3} = J(Q)/(A(Q) + B(Q)) has no p-torsion. Thus K_{2} = 0.

The above argument shows that B(Q)/pB(Q) is a subgroup of H^{1}(Q, A); however, the latter group contains infinitely many elements of order p, whereas � (A)[p] is a finite group, so we must work harder in order to deduce that B(Q)/pB(Q) actually lies in � (A)[p]. Let x ∈ B(Q); we must show that π(x) ∈ � (A)[p]. It suffices to show that res_{v}(π(x)) = 0 for all places v of Q.

At the archimedian place v = ∞, the restriction res_{v}(π(x)) is killed by 2 and the odd prime p, hence res_{v}(π(x)) = 0.

Suppose that v is a place at which J has bad reduction. By hypothesis, B has purely toric reduction, so over Q_{v}^{ur} there is an isomorphism B ≅ G_{m}^{d}/Γ of Gal(\overline{Q}_{v}/Q_{v}^{ur})-modules, for some “lattice” Γ. For example, when dim B = 1, this is the Tate curve representation of B. Let n be the order of the component group of B at v; thus n equals the order of the cokernel of the valuation map Γ → Z^{d}. Choose a representative P = (x_{1}, …, x_{d}) ∈ G_{m}^{d} for the point x. Then nP can be adjusted by elements of Γ so that each of its components x_{i} ∈ G_{m} has valuation 0; it follows that there is a point Q ∈ G_{m}^{d}(Q_{v}^{ur}) such that pQ = nP. Thus the cohomology class res_{v}(π(nx)) is unramified at v. By [20, Prop. 3.8],

H^{1}(Q_{v}^{ur}/Q_{v}, A(Q_{v}^{ur})) = H^{1}(Q_{v}^{ur}/Q_{v}, Φ_{A,v}(\overline{F}_{p})),

where Φ_{A,v} is the component group of A at v. Since the component group Φ_{A,v}(\overline{F}_{p}) has order n, it follows that

res_{v}(π(nx)) = n res_{v}(π(x)) = 0.

Since the order p of res_{v}(π(x)) is coprime to n, we conclude that res_{v}(π(x)) = 0.

Next suppose that J has good reduction at v and that v is odd. Let A, J, C, be the Néron models of A, J, and C, respectively (for more on Néron models, see Chapter ??). Let A, J, C, also denote the sheaves on the étale-site over Spec(Z_{v}) determined by the group schemes A, J, and C, respectively. Since v is odd, 1 = e < v − 1, so we may apply [4, Thm. 7.5.4] to conclude that the sequence of group schemes

0 → A → J → C → 0

is exact; in particular, it is exact as a sequence of sheaves on the étale site (see the proof of [4, Thm. 7.5.4]). Thus it is exact on the stalks, so by [18, 2.9(d)] the sequence

0 → A(Z_{v}^{ur}) → B(Z_{v}^{ur}) → C(Z_{v}^{ur}) → 0

is exact; by the Néron mapping property the sequence

0 → A(Q_{v}^{ur}) → B(Q_{v}^{ur}) → C(Q_{v}^{ur}) → 0

is also exact. Thus res_{v}(π(x)) in unramified. By [20, Prop. I.3.8],

H^{1}(Q_{v}^{ur}/Q_{v}, A) ≅ H^{1}(Q_{v}^{ur}/Q_{v}, Φ_{A,v}),

where Φ_{A,v} is the component group of A at v. Since A has good reduction, Φ_{A,v} = 0, hence res_{v}(π(x)) = 0.

If J has bad reduction at v = 2, then we already dealt with 2 above. Consider the case when J has good reduction at 2. Because the absolute ramification index e of Z_{2} is 1, which is not less than v − 1 = 1, we can not apply [4, Thm. 7.5.4]. However, we can modify everything by an isogeny of degree a power of 2 and apply a different theorem, as follows. The 2-primary subgroup Ψ of A ∩ B is rational over Q. The abelian varieties J̃ = J/Ψ, Ã = A/Ψ, and B̃ = B/Ψ also satisfy the hypothesis of the theorem we are proving. By [4, Prop. 7.5.3(a)], the corresponding sequence of Néron models

0 → Ã → J̃ → C̃ → 0

is exact, so the sequence

0 → Ã(Q_{v}^{ur}) → J̃(Q_{v}^{ur}) → C̃(Q_{v}^{ur}) → 0

is exact. Thus the image of res_{v}(π(x)) in H^{1}(Q_{v}, Ã) is unramified. It equals 0, again by [20, Prop. 3.8], since the component group of Ã at v has order a power of 2, whereas π(x) has odd prime order p. Thus res_{v}(π(x)) = 0, since the kernel of H^{1}(Q_{v}, A) → H^{1}(Q_{v}, Ã) is a finite group of 2-power order.

2 The BSD conjectural formula for the order of the Shafarevich-Tate group

The Jacobian J_{0}(N) is equipped with a standard action of the Hecke algebra

T := Z[… T_{n} …].

We call an abelian variety A a new modular abelian variety if it is an abelian subvariety of some J_{0}(N) that is invariant under the action of the corresponding Hecke algebra T, and the annihilator of A in T is a new prime ideal, in the sense that it is the annihilator of some newform f ∈ S_{2}(Γ_{0}(N), C). The integer N is the level of A. Given a new prime ideal ℘ of some Hecke algebra T ⊂ End(J_{0}(N)) there is an associated new modular abelian variety

A_{℘} = J_{0}(N)[℘] = ∩{ker(t) : t ∈ ℘}.

These are the abelian varieties that populate the land in which we will seek visible elements of Shafarevich-Tate groups.

The Birch and Swinnerton-Dyer conjecture, as generalized by Tate in [24], furnishes a conjectural formula for the order of the Shafarevich-Tate group of any new optimal quotient A. For general A it is an open problem to compute this conjectural order, so we consider only new modular abelian varieties A such that L-function L(A, s) does not vanish at s = 1. For these A we have devised an algorithm which we use to compute the BSD conjectural order of � (A), usually up to a power of 2 and primes whose square divides the level of A.

Let A be a new modular abelian variety such that L(A, 1) ≠ 0. The BSD conjecture asserts that

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

Here the Shafarevich-Tate group

� (A) := ker (H^{1}(Q, A) → ∏_{v} H^{1}(Q_{v}, A))

is a measure of the failure of the local-to-global principle; the Tamagawa numbers c_{p} are the orders of the component groups of A; the real number Ω_{A} is the measure of A(R) with respect to a basis of differentials having everywhere nonzero good reduction; and A^{∨} is the abelian variety dual to A.

3 Computing with modular abelian varieties

Our intent in this section is to describe an algorithm that in many cases can be used to compute the order of � (A) as predicted by the BSD conjecture, at least up to a power of 2 and primes whose square divides the level N of A. At the end, we will also see how to verify the visibility criterion.

In broad outline, our algorithm appears as follows. First select a positive integer N. Using modular symbols, enumerate the finitely many new prime nonmaximal ideals ℘ of T ⊂ J_{0}(N). For each new modular abelian variety A = A_{℘}, use modular symbols again to compute the rational number L(A, 1)/Ω_{A}, up to a bounded “Manin constant.” If this rational number is 0, then we are finished; otherwise, we attempt to compute or bound the numbers c_{p}, #A(Q) and #A^{∨}(Q), then solve for the conjectural order of � (A). In general we remain incapable of computing any of the quantities c_{p}, #A(Q), #A^{∨}(Q), but fortunately in many cases we can compute all of them. When p||N, a formula involving the monodromy pairing on the character group of the toric part of the mod p reduction of the Néron model of J_{0}(N) yields c_{p}, up to a power of 2. Modular symbols and the Eichler-Shimura relation combine to yield both upper and lower bounds on #A(Q) and #A^{∨}(Q).

3.1 The Manin constant

Fix a new modular abelian subvariety A of J_{0}(N), and let ℘ be annihilator of A in T. In this subsection we establish a theoretical bound on the “Manin constant” of A.

Let A be the Néron model of A over Z. The Manin constant c_{A} ∈ Z of A is the index

c_{A} := [S_{2}(Γ_{0}(N); Z)[℘] : H^{0}(A, Ω_{A/Z})].

That c_{A} ∈ Z is a theorem of Edixhoven (see, e.g., [2]). In the special case dim A = 1, the Manin constant is the classical Manin constant of A, and in [14] Mazur proved that c_{A} is a unit in Z[\frac{1}{2m}]. We generalize his proof to obtain the analogous result in dimension greater than 1.

Theorem 3.1. Let A be the new optimal quotient of J_{0}(N) corresponding to a newform f. Then the Manin constant c_{A} is a unit in Z[\frac{1}{2m}], where m is the largest square dividing N.

Proof. Let π denote the map J_{0}(N) → A; let A denote the Néron model of A over R := Z[\frac{1}{2m}], and J the Néron model of J_{0}(N) over R. Let X be the smooth locus a minimal proper regular model for X_{0}(N) over R. Consider the diagram

H^{0}(A, Ω_{A}) ⟶^{π^{∗}} H^{0}(J , Ω_{J}) ≅ H^{0}(X , Ω_{X}) ⟶^{q-exp} R[[q]]  (1)

The map π^{∗} must be an inclusion because H^{0}(A, Ω_{A}) is torsion-free and π^{∗} is an inclusion after tensoring with C. To show that the Manin constant is a unit in R, it suffices to check that the image of H^{0}(A, Ω_{A}) in R[[q]] is saturated, in the sense that the cokernel is torsion free; indeed, the image of S_{2}(Γ_{0}(N); R)[I] is saturated and S_{2}(Γ_{0}(N); R)[I] ⊗ Q = H^{0}(A, Ω_{A}) ⊗ Q.

For the image of H^{0}(A, Ω_{A}) in R[[q]] to be saturated means that the quotient D is torsion free. Let ℓ be a prime not dividing 2m; tensoring

0 → H^{0}(A, Ω_{A}) ⟶^{q-exp} R[[q]] → D → 0

with F_{ℓ} we obtain

0 → D[ℓ] → H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] → D ⊗ F_{ℓ} → 0.

Here we have used that Tor^{1}(D, F_{ℓ}) is the ℓ-torsion in D, and that Tor^{1}(−, F_{ℓ}) vanishes on the torsion free group R[[q]]. To show that D[ℓ] = 0, it suffices to prove that the map Ψ : H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] is injective.

Since ℓ ≠ 2 and A is an optimal quotient, [14, Cor 1.1] gives an exact sequence

0 → H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) → H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) → 0

where B = ker(J → A). In particular, H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) is torsion free, so

H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ} → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) ⊗ F_{ℓ} ≅ H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}})

is injective. We also remark that

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} ≅ H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ},

because Z_{ℓ} is torsion-free, hence flat over R. Thus the map

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}})

is injective.

If ℓ ∤ N, then injectivity of Ψ now follows from the q-expansion principle, which asserts that the q-expansion map H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}}) → F_{ℓ}[[q]] is injective.

Suppose ℓ does divide N, and let ω ∈ ker(Ψ). Since ℓ | N and ℓ ∤ 2m, we have that ℓ || N; thus X /F_{ℓ} breaks up into a union of two irreducible components, and the q-expansion principle implies only that ω vanishes on the irreducible component containing the cusp ∞. However, since A is new and corresponds to a singe eigenform, ω is an eigenvector for the involution W_{N} (since f and all of its conjugates are). Since W_{N} permutes the two components, ω must be 0 on all X /F_{ℓ}. Therefore ω = 0, and hence Ψ is injective.

3.2 Modular Symbols

We require a finite presentation of H_{1}(X_{0}(N), Z); modular symbols provide one that we briefly review (see [7, 11, 16]). Consider the space M(N, Z) of modular symbols. This is the free abelian group generated by symbols {α, β}, with α, β ∈ P^{1}(Q) = Q ∪ {∞}, modulo the relations {α, β} + {β, γ} + {γ, α} = 0, and {α, β} = {g(α), g(β)}, for all g ∈ Γ_{0}(N). Manin observed that this presentation is finite by proving that M(N, Z) is generated by symbols of the form {g(0), g(∞)}, where g is a right coset representative for Γ_{0}(N) in SL_{2}(Z). The space of boundary symbols B(N, Z) is the free abelian group generated by symbols {α}, with α ∈ P^{1}(Q), modulo the relation {α} = {g(α)}, for all g ∈ Γ_{0}(N). The cuspidal symbols S(N, Z) are the kernel of the boundary map {α, β} ↦ {β}−{α}, and we have S(N, Z) ≅ H_{1}(X_{0}(N), Z). The Hecke algebra acts on M(N, Z), and there is an involution ∗{α, β} = {−α, −β}. Integration defines a perfect pairing S_{2}(N) × (S(N, Z)^{+} ⊗ C) → C. By the Manin-Drinfeld theorem, the image of M(N, Z) in Hom(S_{2}(N), C) is a lattice.

3.3 The Method of Graphs

The method of graphs will be used in some of our component group computations. We briefly review it here; for more details, see [17]. Suppose N = pM with M a positive integer and p a prime not dividing M. Let D_{N,p} be the finitely generated free abelian group on the superingular points of X_{0}(M)(\overline{F}_{p}). We view the supersingular points as pairs E = (E, C) where E is a supersingular elliptic curve over \overline{F}_{p} and C is a cyclic subgroup of order M. 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_{N,p} is

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

There is a natural monodromy-compatible action of T_{0}(N) on D_{N,p}. If X_{N,p} := {∑ a_{E}E : ∑ a_{E} = 0}, then X_{N,p} ⊗ C is isomorphic, as a T_{0}(N)-module, to the subspace of S_{2}(N, C) generated by newforms and oldforms of level pd for d|M.

David Kohel [12] has implemented an algorithm which computes the action of the Hecke operators on X_{N,p} using the arithmetic of quaternion algebras. We have found this program extremely useful when X_{0}(M) has positive genus.

3.4 Enumerating quotients of J_{0}(N)

We list all newforms of a given level N by decomposing the new subspace of H_{1}(X_{0}(N), Z)^{+} using the Hecke operators. The characteristic polynomial of T_{2} is computed, and then T_{2} is used to break up the space. This process is applied recursively with T_{3}, T_{5}, … until it terminates, either because we have exceeded the bound in [3], or we have found a Hecke operator T_{n} whose characteristic polynomial is irreducible. 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 (see [7, pg. 5]). There is only one case, in our table, in which the two ordering schemes disagree: our 446B is Cremona’s 446D.

3.5 The modular degree

A polarization of an abelian variety A is an isogeny between A and its dual arising from a very amply invertible sheaf (see [19]). 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}.

\xymatrix{A_{f}^{∨} \ar[r]^{π^{∨}} \ar[dr]^{θ_{f}} & J_{0}(N) \ar@{->>}[d]^{π} \\  & A_{f}}

By ([19], 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 3.2. With notation as in Theorem ??,

ker(θ_{f}) ≅ coker(H[I_{f}] → Φ_{f}(H))

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

Using Lemma 3.6, 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})).

3.6 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 3.3. 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 3.4. 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}(M(N, Z))}{Φ_{f}(S(N, Z))}.

Proof. By the classical Abel-Jacobi theorem ([13], 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 ??.

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}.

3.7 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 [23].

Theorem 3.5.

|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}

3.8 Rational part of the special value

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

Lemma 3.6. Let φ_{1}, …, φ_{n} be a Q-basis for Hom(M(Q), Q)[I_{f}] and set

Ψ = φ_{1} × ⋯ × φ_{n} : M(Q) → Q^{n}.

Then n = 2d and ker(Ψ) = ker(Φ_{f}).

Proof. This result is due to Shimura ([22]), but we sketch a proof. To compute dim Hom(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 M(C) is isomorphic as a T-module to S_{2} ⊕ \overline{S}_{2} ⊕ E_{2} (prop. 9 of [16] and the Eichler-Shimura embedding). Because of the Peterson inner product, the dual Hom(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 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(M(Q), C)[I_{f}] and ker(Ψ) ⊗ C is the intersection of the kernels of all maps in Hom(M(Q), C)[I_{f}]. By Theorem ?? the image of Φ_{f} is a lattice, so dim_{Q} ker(Φ_{f}) = dim_{Q} 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 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 3.7. 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, i∞} ∈ M(N, Z) denote the winding element.

Theorem 3.8. Let Ψ be as in Lemma 3.6. Then

±\frac{L(A_{f}, 1)}{Ω_{f}^{0}} = [Ψ(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 codomain 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(S^{+}) be the volume of a fundamental domain for the real lattice Φ(S^{+}) = Φ(S(N, Z)^{+}). Observe that Vol(S^{+}) = [Z^{d} : Φ(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(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(S^{+}) : Φ(Te)] = [Φ(S^{+}) : Z^{d}] · [Z^{d} : Φ(Te)]
= \frac{1}{[Z^{d} : Φ(S^{+})]} · [Z^{d} : Φ(Te)]
= \frac{1}{Vol(S^{+})} · [Z^{d} : Φ(e)Z^{d}] · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(S^{+})} · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(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 3.6, 3.7, 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 3.9. 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 [9]). If p ∤ N a standard calculation (section 2.8 of [7]) 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(S^{+}) : Ψ(Te)]
= [Ψ(S^{+}) : Ψ(Ie)] · [Ψ(Ie) : Ψ(Te)]
= [Ψ(S^{+}) : IΨ(e)] · [IΨ(e) : TΨ(e)]
= \frac{[Ψ(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(S^{+}), so [Ψ(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|.

3.9 Intersections

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

Proposition 3.10. (A_{f}^{∨} ∩ A_{g}^{∨})[p] ≠ 0 if and only if 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 (??) 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 3.11. 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.

4 Tables

This section contains tables computed using the above algorithms as implemented 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 [6], 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. Large Sha Upon seeing an early draft of this paper, Armand Brumer provided us with the first quotients of J_{0}(p) having positive rank and dimension 3 and 4. Computing further I’ve found the following spectacular table. The listed orders of � really is the numerator of the L-function as a long computation showed.

A_{f}  divisior of |� | B_{g}
389E20  5  389A1
1061D46  151  1061B2
1693B72  1301  1693A3
2593B109  67 · 2213  2593A4

For 1693 the odd part of the special value is 1301^{2}/141. For 2593 it is 67^{2} · 2213^{2}/3^{3}.

Table 1: New visible Shafarevich-Tate

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  1061B2
2593B109 67^{2} · 2213^{2}  −  ?  ?  ?  ?  2593A4

Table 2: Explanatory factors

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 [7]
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 prime level surface of pos. even rank [6]

Table 3: Factorizations

\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 4: Component groups

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 5: Odd square numerator

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 6: Odd parts of Shafarevich-Tate groups at prime level

A  dim � _{an}  mod deg  B  dim mod deg
389E  20  5^{2}  5  389A  1  5
433D 16  7^{2}  3 · 7 · 37  433A  1  7
563E  31  13^{2}  13  563A  1  13
571D  2  3^{2}  3^{2} · 127  571B  1  3
709C 30  11^{2}  11  709A  1  11
997H 42  3^{4}  3^{2}  997B  1  3
1061D 46  151^{2}  61 · 151 · 179  1061B 2  151
1091C 62  7^{2}  1  NONE
1171D 53  11^{2}  3^{4} · 11  1171A 1  11
1283C 62  5^{2}  5 · 41 · 59  NONE
1429B 64  5^{2}  1  NONE
1481C 71  13^{2}  5^{2} · 2833  NONE
1483D 67 3^{2} · 5^{2}  3 · 5  1483A 1  3 · 5
1531D 73  3^{2}  3  1531A 1  3
1559B 90  11^{2}  1  NONE
1567D 69 7^{2} · 41^{2}  7 · 41  1567B 3  7 · 41
1613D 75  5^{2}  5 · 19  1613A 1  5
1621C 70  17^{2}  17  1621A 1  17
1627C 73  3^{4}  3^{2}  1627A 1  3^{2}
1693C 72 1301^{2}  1301  1693A 3  1301
1811D 98  31^{2}  1  NONE
1847B 98  3^{6}  1  NONE
1871C 98  19^{2}  14699  NONE
1877B 86  7^{2}  1  NONE
1907D 90  7^{2}  3 · 5 · 7 · 11  1907A 1  7
1913B 1  3^{2}  3 · 103  1913A 1  3 · 5^{2}
1913E 84 5^{4} · 61^{2}  5^{2} · 61 · 103  1913A 1  3 · 5^{2}
1933C 83 3^{2} · 7^{2}  3 · 7  1933A 1  3 · 7
1997C 93  17^{2}  1  NONE
2027C 94  29^{2}  29  2027A 1  29
2029C 90 5^{2} · 269^{2}  5 · 269  2029A 2  5 · 269
2039F 99 3^{4} · 5^{2} 19 · 29 · 7759 · 3214201 NONE
2063C 106  13^{2}  61 · 139  NONE
2089J 91  11^{2}  3 · 5 · 11 · 19 · 73 · 139 2089B 1  11
2099B 106  3^{2}  1  NONE
2111B 112 211^{2}  1  NONE
2113B 91  7^{2}  1  NONE
2161C 98  23^{2}  1  NONE
2333C 101 83341^{2}  83341  2333A 4  83341
2339C 114  3^{8}  6791  NONE
2411B 123  11^{2}  1  NONE

References

[1] A. Agashé, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369–374.

[2] A. Agashé and W. Stein, On the generalized manin constant for quotients of J_{0}(N), in preparation.

[3] A. Agashe and W. A. Stein, Generating the Hecke algebra, preprint (1999).

[4] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990.

[5] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q, or Wild 3-adic exercises, http://www.math.harvard.edu/HTML/Individuals/Richard Taylor.html (2000).

[6] A. Brumer, The rank of J_{0}(N), Astérisque (1995), no. 228, 3, 41–68, Columbia University Number Theory Seminar (New York, 1992).

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

[8] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, to appear in Experiment. Math.

[9] F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem, Providence, RI, 1995, pp. 39–133.

[10] E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, submitted (1999).

[11] G. Frey and M. Müller, Arithmetic of modular curves and applications, Algorithmic algebra and number theory (Heidelberg, 1997), Springer, Berlin, 1999, pp. 11–48.

[12] D. R. Kohel, Hecke module structure of quaternions, preprint (1998).

[13] S. Lang, Introduction to modular forms, Springer-Verlag, Berlin, 1995, With appendixes by D. Zagier and Walter Feit, Corrected reprint of the 1976 original.

[14] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.

[15] , Visualizing elements of order three in the Shafarevich-Tate group, preprint (1999).

[16] L. Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations (Berlin), Springer, 1994, pp. 59–94.

[17] 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.

[18] J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980.

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

[20] , Arithmetic duality theorems, Academic Press Inc., Boston, Mass., 1986.

[21] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68.

[22] G. Shimura, On the periods of modular forms, Math. Ann. 229 (1977), 211–221.

[23] W. A. Stein, Component groups of optimal quotients of Jacobians, preprint (1999).

[24] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440.