CONSTRUCTING ELEMENTS IN SHAFAREVICH-TATE GROUPS OF MODULAR MOTIVES (NOT FOR DISTRIBUTION!)

NEIL DUMMIGAN, WILLIAM STEIN, AND MARK WATKINS

Abstract.

1. Introduction

Mention [CM].

2. Motives and Galois representations

Let f = ∑ a_{n}q^{n} be a newform of weight k ≥ 2 for Γ_{0}(N), with coefficients in an algebraic number field E. A theorem of Deligne [De1] implies the existence, for each (finite) prime λ of E, of a two-dimensional vector space V_{λ} over E_{λ}, and a continuous representation

ρ_{λ} : Gal(\overline{ℚ}/ℚ) → Aut(V_{λ}),

such that (1) ρ_{λ} is unramified at p for all primes p not dividing lN (where λ | l);

(2) if Frob_{p} is an arithmetic Frobenius element at such a p then the characteristic polynomial of Frob_{p}^{−1} acting on V_{λ} is x^{2} − a_{p}x + p^{k−1}.

Following Scholl [Sc], V_{λ} may be constructed as the λ-adic realisation of a Grothendieck motive M_{f}. There are also Betti and de Rham realisations V_{B} and V_{dR}, both 2-dimensional E-vector spaces. For details of the construction see [Sc]. The de Rham realisation has a Hodge filtration

V_{dR} = F^{0} ⊃ F^{1} = ⋯ = F^{k−1} ⊃ F^{k} = {0}.

The Betti realisation V_{B} comes from singular cohomology, while V_{λ} comes from étale l-adic cohomology. There are natural isomorphisms V_{B} ⊗ E_{λ} ≃ V_{λ}. Using a basis for singular cohomology with ℤ-coefficients, we get Gal(\overline{ℚ}/ℚ)-stable O_{λ}-modules T_{λ} inside each V_{λ}. Define A_{λ} = V_{λ}/T_{λ}. There are two kinds of twist we shall have to consider. There is the Tate twist V_{λ}(j) (for an integer j), which amounts to multiplying the action of Frob_{p} by p^{j}. For D the discriminant of a quadratic field, there is the quadratic twist V_{λ}(χ_{D}), which is the tensor product of V_{λ} with a one-dimensional space on which Gal(\overline{ℚ}/ℚ) acts via the quadratic character χ_{D}.

Following [BK] (Section 3), for p ≠ l (including p = ∞) let

H_{f}^{1}(ℚ_{p}, V_{λ}(χ_{D}, j)) = ker(H^{1}(D_{p}, V_{λ}(χ_{D}, j)) → H^{1}(I_{p}, V_{λ}(χ_{D}, j))).

Date: February 24, 2002.

1991 Mathematics Subject Classification. 11F33, 11F67, 11G40.

Key words and phrases. modular form, L-function, visibility, Bloch-Kato conjecture, Shafarevich-Tate group.

The subscript f stands for “finite part”. D_{p} is a decomposition subgroup at a prime above p, I_{p} is the inertia subgroup, and the cohomology is for continuous cocycles and coboundaries. For p = l let

H_{f}^{1}(ℚ_{l}, V_{λ}(χ_{D}, j)) = ker(H^{1}(D_{l}, V_{λ}(χ_{D}, j)) → H^{1}(D_{l}, V_{λ}(χ_{D}, j) ⊗ B_{cris}))

(see Section 1 of [BK] for definitions of Fontaine’s rings B_{cris} and B_{dR}). Let H_{f}^{1}(ℚ, V_{λ}(χ_{D}, j)) be the subspace of elements of H^{1}(ℚ, V_{λ}(χ_{D}, j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, V_{λ}(χ_{D}, j)) for all primes p.

There is a natural exact sequence

0 ⟶ T_{λ}(χ_{D}, j) ⟶ V_{λ}(χ_{D}, j) ⟶^{π} A_{λ}(χ_{D}, j) ⟶ 0.

Let H_{f}^{1}(ℚ_{p}, A_{λ}(χ_{D}, j)) = π_{∗}H_{f}^{1}(ℚ_{p}, V_{λ}(χ_{D}, j)). Define the λ-Selmer group H_{f}^{1}(ℚ, A_{λ}(χ_{D}, j)) to be the subgroup of elements of H^{1}(ℚ, A_{λ}(χ_{D}, j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, A_{λ}(χ_{D}, j)) for all primes p. Note that the condition at p = ∞ is superfluous unless l = 2. Define the Shafarevich-Tate group

� (χ_{D}, j) = ⊕_{λ}H_{f}^{1}(ℚ, A_{λ}(χ_{D}, j))/π_{∗}H_{f}^{1}(ℚ, V_{λ}(χ_{D}, j)).

The length of its λ-component may be taken for the exponent of λ in an ideal of O_{E}, which we call #� (χ_{D}, j). We shall only concern ourselves with the case j = k/2, and write � (χ_{D}) for � (χ_{D}, j).

Define the set of global points

Γ_{ℚ}(χ_{D}) = ⊕_{λ}H^{0}(ℚ, A_{λ}(χ_{D}, k/2)).

This is analogous to the group of rational torsion points on an elliptic curve. The length of its λ-component may be taken for the exponent of λ in an ideal of O_{E}, which we call #Γ_{ℚ}(χ_{D}).

3. The Bloch-Kato conjecture

Suppose that L(f, k/2) ≠ 0. The Bloch-Kato conjecture for the motive M_{f}(k/2) predicts that

\frac{L(f, k/2)}{vol_{∞}} = \frac{(∏_{p} c_{p}(k/2)) #�}{(#Γ_{ℚ})^{2}} .

The local factors vol_{∞} and c_{p}(k/2) depend on the choice of a basis for V_{dR}, but their product does not. For l ≠ p, ord_{λ}(c_{p}(χ_{D}, j)) is defined to be

lengthH_{f}^{1}(ℚ_{p}, T_{λ}(χ_{D}, j))_{tors.} − ord_{λ}((1 − χ_{D}(p)a_{p}p^{−j} + p^{k−1−2j})).

We omit the definition of ord_{λ}(c_{p}(χ_{D}, j)) for λ | p, which requires one to assume Fontaine’s de Rham conjecture ([Fo], Appendix A6). The above formula is to be interpreted as an equality of fractional ideals of E. (Strictly speaking, the conjecture in [BK] is only given for E = ℚ.)

Similarly, if L(f, χ_{D}, k/2) ≠ 0 then the Bloch-Kato conjecture for M_{f}(χ_{D}, k/2) predicts that

\frac{L(f, χ_{D}, k/2)}{vol_{∞}(χ_{D})} = \frac{(∏_{p} c_{p}(k/2, χ_{D})) #� (χ_{D})}{(#Γ_{ℚ}(χ_{D}))^{2}}.

CONSTRUCTING ELEMENTS IN SHAFAREVICH-TATE GROUPS OF MODULAR MOTIVES(NOT FOR DISTRIBUTION!) 3

Note that if L(f, k/2) ≠ 0 then it is necessary that χ_{D}(−N) = 1 if the sign in the functional equation is to allow L(f, χ_{D}, k/2) ≠ 0. A careful treatment of real periods such as vol_{∞}(χ_{D}) is given in [De2]. The calculation of the periods of Artin motives in terms of generalized Gauss sums, in Section 6 of [De2], yields the following lemma.

Lemma 3.1. If χ_{D}(−1) = 1 (i.e. if D > 0) then vol_{∞}(χ_{D}) = vol_{∞}/\sqrt{D}.

Lemma 3.2. Let p ∤ N be a prime, j an integer and D a quadratic discriminant. Then the fractional ideal c_{p}(χ_{D}, j) is supported at most on divisors of 2 and divisors of p, while c_{p}(j) is supported at most on divisors of p.

Proof. ^{1} As on p. 30 of [Fl1], for odd l ≠ p, ord_{λ}(c_{p}(χ_{D}, j)) is the length of the 1 finite O_{λ}-module H^{0}(ℚ_{p}, H^{1}(I_{p}, T_{λ}(χ_{D}, j))_{tors}), where I_{p} is an inertia group at p. If p does not divide D then T_{λ}(χ_{D}, j) is a trivial I_{p}-module, so H^{1}(I_{p}, T_{λ}(χ_{D}, j)) is torsion-free. In general, since l ≠ p,

H^{1}(I_{p}, T_{λ}(χ_{D}, j)) ≃ T_{λ}(χ_{D}, j)/(1 − τ)T_{λ}(χ_{D}, j),

where τ is a topological generator of the tame quotient of I_{p}. If p | D then thanks to the quadratic twist ramified at p, τ acts as multiplication by −1 on T_{λ}(χ_{D}, j), hence H^{1}(I_{p}, T_{λ}(χ_{D}, j)) is trivial (remember l ≠ 2). □

The assertion about c_{p}(j) may be proved similarly to the case p ∤ D above.

Lemma 3.3. Let D be a quadratic discriminant, and p a prime such that χ_{D}(p) =

1. Then c_{p}(χ_{D}, j) = c_{p}(j).

Proof. This is simply a consequence of the fact that the definition of ord_{λ}(c_{p}(j)) depends only on T_{λ} as a Gal(\overline{ℚ}_{p}/ℚ_{p})-module. □

Lemma 3.4. ord_{λ}(#Γ_{ℚ}(χ_{D}, j)) = 0 unless the coefficients of f = ∑ a_{n}q^{n} satisfy the congruence a_{p} ≡ χ_{D}(p)(p^{j} + p^{k−1−j}) (mod λ) for all primes p ∤ lN.

This follows from the interpretation of a_{p} as a trace of Frobenius. See [SwD].

Putting together the above lemmas we arrive at the following:

Proposition 3.5. Let q ∤ N be an odd prime. Let D > 0 be a quadratic discriminant such that χ_{D}(q) = 1 and χ_{D}(p) = 1 for all primes p | N. Let 𝔮 | q be a prime of E such that neither

a_{p} ≡ χ_{D}(p)(p^{k/2} + p^{(k/2)−1}) (mod 𝔮)

nor

a_{p} ≡ p^{k/2} + p^{(k/2)−1} (mod 𝔮)

holds for all p ∤ qN. If

ord_{𝔮}(L(f, k/2)/\sqrt{D}L(f, χ_{D}, k/2)) > 0

(with numerator and denominator both non-zero) then the Bloch-Kato conjecture predicts that ord_{𝔮}(#� ) > 0.

Proposition 3.6. Suppose that p ∤ DNϕ(N), p > k and that f is not congruent to another cusp form for Γ_{0}(N) modulo any divisor of p. Then, for (k/2) ≤ j ≤ k −1, c_{p}(j) is trivial and c_{p}(χ_{D}, j) is at worst supported on divisors of 2.

1I switched to using proof environments instead of doing it by hand. – was

This may be proved in a similar manner to Theorem 7.6 of [Du1]. The conditions in the statement of the proposition ensure, among other things, that the denominator of the projector used to cut out the motive M_{f} is coprime to p. This is essential if the proofs of the lemmas in Section 7 of [Du1] are to carry across.

Bearing in mind Flach’s generalisation of the Cassels-Tate pairing [Fl2], we see the following corollary.

Corollary 3.7. The Bloch-Kato conjecture predicts that, for D as in Proposition 3.5, the fractional ideal (L(f, k/2)/\sqrt{D}L(f, χ_{D}, k/2)) is a square, up to divisors of 2 and of primes not satisfying the conditions of Proposition 3.6.

4. Constructing elements of the Shafarevich-Tate group

Let f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} be newforms of equal weight k ≥ 2 for Γ_{0}(N). Let E be a number field large enough to contain all the coefficients a_{n} and b_{n}. Suppose that 𝔮 | q is a prime of E such that f ≡ g (mod 𝔮), i.e. a_{n} ≡ b_{n} (mod 𝔮) for all n. For f we have defined V_{λ}, T_{λ} and A_{λ}. Let V′_{λ}, T′_{λ} and A′_{λ} be the corresponding objects for g. Let A[λ] denote the λ-torsion in A_{λ}. Since a_{p} is the trace of Frob_{p}^{−1} on V_{λ}, it follows from the Cebotarev Density Theorem that A[𝔮] and A′[𝔮], if irreducible, are isomorphic as Gal(\overline{ℚ}/ℚ)-modules.

Suppose that L(g, k/2) = 0. If the sign in the functional equation is positive, this implies that the order of vanishing at s = k/2 is at least 2. According to the Beilinson-Bloch conjecture [B], the order of vanishing of L(g, s) at s = k/2 is the rank of the group CH_{0}^{k/2}(M_{g})(ℚ) of ℚ-rational, null-homologous, codimension k/2 algebraic cycles on the motive M_{g}, modulo rational equivalence.

Via the 𝔮-adic Abel-Jacobi map, CH_{0}^{k/2}(M_{g})(ℚ) maps to H^{1}(ℚ, V′_{𝔮}(k/2)), and Nekovár shows in [Ne1] that its image is contained in the subspace H_{f}^{1}(ℚ, V′_{𝔮}(k/2)). See also 0.13 of [Ne2]. If, as expected, the 𝔮-adic Abel-Jacobi map is injective, we get (assuming also the Beilinson-Bloch conjecture) a subspace of H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) of dimension equal to the order of vanishing of L(g, s) at s = k/2.

If L(f, k/2) ≠ 0 then it is expected that H_{f}^{1}(ℚ, V_{𝔮}(k/2)) = 0, so that � = 2 H_{f}^{1}(ℚ, A_{𝔮}(k/2)).^{2} (For j ≥ k/2, the rank of H_{f}^{1}(ℚ, V_{𝔮}(j)) is expected to be the

order of vanishing of L(f, s) at s = k − j.)

Theorem 4.1. Let q ∤ N be a prime satisfying q > k. Suppose that H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) has dimension r > 0. Suppose that A[𝔮] and A′[𝔮] are irreducible representations of Gal(\overline{ℚ}/ℚ) and that, for all primes p | N, p ≠ ±1 (mod q). Suppose also that

3 neither f nor g is congruent modulo 𝔮 to any newform^{3} of weight k, trivial character 4 and level dividing N/p, with p any^{4} prime that exactly divides N. Then the 𝔮-torsion

subgroup of H_{f}^{1}(ℚ, A_{𝔮}(k/2)) has 𝔽_{𝔮}-rank at least r − 1.

5 Proof. ^{5} Take a non-zero class d ∈ H_{f}^{1}(ℚ, V′_{𝔮}(k/2)). By continuity and rescaling we may assume that it lies in H_{f}^{1}(ℚ, T′_{𝔮}(k/2)) but not in 𝔮H_{f}^{1}(ℚ, T′_{𝔮}(k/2)). Then by reduction we get a non-zero class c ∈ H^{1}(ℚ, A′[𝔮](k/2)) ≃ H^{1}(ℚ, A[𝔮](k/2)). By irreducibility, H^{0}(ℚ, A[𝔮](k/2)) is trivial, so H^{1}(ℚ, A[𝔮](k/2)) injects into H^{1}(ℚ, A_{𝔮}(k/2)), and we get a non-zero class γ ∈ H^{1}(ℚ, A_{𝔮}(k/2)).

2Does one really get all of � as you claim, or just the 𝔮-torsion or 𝔮-part??

3I added this weight k assertion. – was

4Change “a” to “any”. – was

5This proof is really long. Could it benefit by being broken up into more maneagable chunks (lemmas, etc.)? If nothing else, a summary paragraph at the beginning would be useful. – was

First we will show that res_{p}(γ) ∈ H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)), for all (finite) primes p ∤ qN. Since in this case A′_{𝔮}(k/2) is unramified at p, H^{0}(I_{p}, A′_{𝔮}(k/2)) = A′_{𝔮}(k/2), which is 𝔮-divisible. Therefore H^{1}(I_{p}, A′[𝔮](k/2)) injects into H^{1}(I_{p}, A′_{𝔮}(k/2)), and it follows from d ∈ H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) that the image of γ in H^{1}(I_{p}, A_{𝔮}(k/2)) is zero. By line 3 of p. 125 of [Fl2], H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)) is equal to (not just contained in) the kernel of the map from H^{1}(ℚ_{p}, A_{𝔮}(k/2)) to H^{1}(I_{p}, A_{𝔮}(k/2)), so we have shown that res_{p}(γ) ∈ H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)).

Next we will show that res_{p}(γ) ∈ H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)) for p | N. Our first task is to show that H^{0}(I_{p}, A′_{𝔮}(k/2)) is 𝔮-divisible. It suffices to show that

dim H^{0}(I_{p}, A′[𝔮](k/2)) = dim H^{0}(I_{p}, V′_{𝔮}(k/2)).

If the dimensions differ then, given that g is not congruent modulo 𝔮 to a newform of level strictly dividing N, and since p ≠ ±1 (mod q), Propositions 3.1 and 2.3 of [L] tell us that A′[𝔮](k/2) is unramified at p and that ord_{p}(N) = 1. (Here we are using Carayol’s result that N is the prime-to-q part of the conductor of V_{𝔮} [Ca1].) But then Theorem 1 of [JL] (which uses the condition q > k) implies the existence of a newform of weight k, trivial character and level dividing N/p, congruent to g modulo 𝔮. This contradicts our hypotheses, and we have established that H^{0}(I_{p}, A′_{𝔮}(k/2)) is 𝔮-divisible.

It follows that the image of c ∈ H^{1}(ℚ, A[𝔮](k/2)) in H^{1}(I_{p}, A[𝔮](k/2)) is zero. By inflation-restriction, res_{p}(c) then comes from H^{1}(D_{p}/I_{p}, H^{0}(I_{p}, A[𝔮](k/2))). The order of this group is the same as the order of the group H^{0}(ℚ_{p}, A[𝔮](k/2)), which we claim is trivial. By the work of Carayol [Ca1], p | N implies that V_{𝔮}(k/2) is ramified at p, so dim H^{0}(I_{p}, V_{𝔮}(k/2)) = 0 or 1. As above, we see that dim H^{0}(I_{p}, V_{𝔮}(k/2)) = dim H^{0}(I_{p}, A[𝔮](k/2)), so we need only consider the case where this common dimension is 1. The (motivic) Euler factor at p for M_{f} is (1 − αp^{−s})^{−1}, where Frob_{p}^{−1} acts as multiplication by α on the one-dimensional space H^{0}(I_{p}, V_{𝔮}(k/2)). It follows from Theoréme A of [Ca1] that this is the same as the Euler factor at p of L(f, s). By the work of Atkin and Lehner [AL], it then follows that p^{2} ∤ N and α = −w_{p}p^{(k/2)−1}, where w_{p} = ±1 is such that W_{p}f = w_{p}f. Twisting by k/2, Frob_{p}^{−1} acts on H^{0}(I_{p}, V_{𝔮}(k/2)) (hence also on H^{0}(I_{p}, A[𝔮](k/2)) as ±p^{−1}. Since p ≠ ±1 (mod q), we see that H^{0}(ℚ_{p}, A[𝔮](k/2)) is trivial. Hence res_{p}(c) = 0 so res_{p}(γ) = 0 and certainly lies in H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)).

It remains to deal with the local condition at p = q. Since q ∤ N is a prime of good reduction for the motive M_{f}, V_{𝔮} is a crystalline representation of Gal(\overline{ℚ}_{q}/ℚ_{q}), meaning D_{cris}(V_{𝔮}) and V_{𝔮} have the same dimension, where D_{cris}(V_{𝔮}) := H^{0}(ℚ_{q}, V_{𝔮}⊗_{ℚ_{q}} B_{cris}). (This is a consequence of Theorem 5.6 of [Fa].) It follows from Theorem 4.1(ii) of [BK] that

D_{dR}(V_{𝔮})/(F^{k/2}D_{dR}(V_{𝔮})) ≃ H_{f}^{1}(ℚ_{q}, V_{𝔮}(k/2)),

via their exponential map.

Figure 1. Newforms Satisfying Theorem 4.1

f  deg(f)  g  deg(g) q′s
127k4A  1  127k4C 17  43
159k4B  1  159k4E 16  5, 23
365k4A  1  365k4E 18  29
369k4B  1  369k4I  9  13
453k4A  1  453k4E 23  17
465k4B  1  465k4I  7  11
477k4B  1  477k4L  12  73
567k4B  1  567k4H  8  23
581k4A  1  581k4E 34  19
95k6A  1  95k6D  9  31, 59
122k6A  1  122k6D  6  73

Since p is a prime of good reduction, the de Rham conjecture is a consequence of the crystalline conjecture, which follows from Theorem 5.6 of [Fa]. Hence

(V_{dR} ⊗_{E} E_{𝔮})/(F^{k/2}V_{dR} ⊗_{E} E_{𝔮}) ≃ H_{f}^{1}(ℚ_{q}, V_{𝔮}(k/2)).

Observe that the dimension of the left-hand-side is 1. Meanwhile, by 3.8 of [BK], H_{f}^{1}(ℚ_{q}, V_{𝔮}(k/2)) is its own annihilator in H^{1}(ℚ_{q}, V_{𝔮}(k/2)) with respect to the Tate duality pairing. Hence H_{f}^{1}(ℚ_{q}, V_{𝔮}(k/2)) has codimension one in H^{1}(ℚ_{q}, V_{𝔮}(k/2)). It follows that our r-dimensional 𝔽_{𝔮}-subspace of the 𝔮-torsion in H^{1}(ℚ, A_{𝔮}(k/2)) has at least an (r − 1)-dimensional subspace landing in H_{f}^{1}(ℚ_{𝔮}, A_{𝔮}(k/2)).

To justify this argument carefully, we should check that the natural map from H^{1}(ℚ_{q}, V_{𝔮}(k/2)) to H^{1}(ℚ_{𝔮}, A_{𝔮}(k/2)) is surjective. Its cokernel injects into H^{2}(ℚ_{𝔮}, T_{𝔮}(k/2)), which is dual to H^{0}(ℚ_{𝔮}, A_{𝔮}(k/2)) (via local Tate duality). It suffices to check that H^{0}(ℚ_{𝔮}, A[𝔮](k/2)) is trivial. This follows from the description of A[𝔮](k/2) as a Gal(\overline{ℚ}_{q}/ℚ_{q})-module provided by theorems of Deligne and Fontaine (Theorems 2.5 and 2.6 of [Ed]). □

Theorem 2.7 of [AS] is concerned with verifying local conditions in the case k = 2, where f and g are associated with abelian varieties A and B. (Their theorem also applies to abelian varieties over number fields.) Our restriction outlawing congruences modulo 𝔮 with cusp forms of lower level is analogous to theirs forbidding q from dividing Tamagawa factors c_{A,l} and c_{B,l}. (In the case where A is an elliptic curve with ord_{l}(j(A)) < 0, consideration of a Tate parametrisation shows that if q | c_{A,l}, i.e. if q | ord_{l}(j(A)), then it is possible that A[q] is unramified at l.)

5. Eleven examples of Theorem 4.1

Table 1 on page 6 lists eleven pairs of newforms f and g along with at least one prime q such that there is a prime 𝔮 | q such that by Theorem 4.1 the Beilinson-Bloch conjecture implies that

dim_{𝔽_{𝔮}} #H_{f}^{1}(ℚ, A_{𝔮}(k/2))[𝔮] > 0.

Recall that H_{f}^{1}(ℚ, A_{𝔮}(k/2))[𝔮] is a piece of the finite cohomology associated to the motive attached to g. As discussed just before Theorem 4.1, since L(g, \frac{k}{2} ) ≠ 0, we expect that H_{f}^{1}(ℚ, A_{𝔮}(k/2))[𝔮] equals the 𝔮-torsion of the Shafarevich-Tate group of the motive attached to g. (Note also that the subscript of f means “finite” and has nothing to do with the newform that we denote by “f”.)

5.1. Notation. Table 1 is laid out as follows. The first column contains a label whose structure is

[Level]k[Weight][GaloisOrbit]

This label determines a newform f = ∑ a_{n}q^{n}, up to Galois conjugacy. For example, 127k4C denotes a newform in the third Galois orbit of newforms in S_{4}(Γ_{0}(127)). The Galois orbits are ordered first by the degree of ℚ(…, a_{n}, …), then by |Tr(a_{p}(f))| for p not dividing the level, with positive trace being first in the event that the two absolute values are equal, and the first Galois orbit is denoted A, the second B, and so on. For the purposes of the present paper, the ordering is not important; if you repeat the experiment, you will quickly find which form must be f and which must be g. The second column contains the degree of the field ℚ(…, a_{n}, …). The third and forth columns contain g and its degree, respectively. The fifth columns contains at least one prime q such that there is a prime 𝔮 | q with f ≡ g (mod 𝔮), and so that the hypothesis of Theorem 4.1 are satisfied for f, g, and 𝔮.

5.2. The first example in detail. We describe the first line of Table 1 in more detail. See the next section for further details on how the computations were performed.

Using modular symbols, we find that there is a newform

f = q − q^{2} − 8q^{3} − 7q^{4} − 15q^{5} + 8q^{6} − 25q^{7} + ⋯ ∈ S_{4}(Γ_{0}(127))

with L(f, 2) = 0. Because W_{127}(f) = f, the functional equation has sign +1, so L′(f, 2) = 0 as well. We also find a newform g ∈ S_{4}(Γ_{0}(127)) whose Fourier coefficients generate a number field K of degree 17, and by computing the image of the modular symbol XY {0, ∞} under the period mapping, we find that L(g, 2) ≠ 0. The newforms f and g are congruent modulo a prime 𝔮 of K of residue characteristic 43. The mod 𝔮 reductions of f and g are both equal to

\overline{f} = q + 42q^{2} + 35q^{3} + 36q^{4} + 28q^{5} + 8q^{6} + 18q^{7} + ⋯ ∈ 𝔽_{43}[[q]].

There is no form in the Eisenstein subspaces of M_{4}(Γ_{0}(127)) whose Fourier coefficients of index n, with (n, 127) = 1, are congruent modulo 43 to those of \overline{f}, so ρ_{f,𝔮} ≈ ρ_{g,𝔮} is irreducible. Since 127 is prime and S_{4}(SL_{2}(ℤ)) = 0, \overline{f} does not arise from a level 1 form of weight 4. Thus, if we assume the Beilinson-Bloch conjecture, Theorem 4.1 implies that the Shafarevich-Tate group attached to g has order divisible by 43.

5.3. Some remarks on how the computation was performed. We give a brief summary of how the computation was performed. The algorithms that we used were implemented by the second author, and most are a standard part of the MAGMA V2.8 (see [BCP]).

Let f, g, and q be some data from a line of Table 1 and let N denote the level of f. We verified the existence of a congruence modulo q, that L(f, \frac{k}{2} ) = L′(f, \frac{k}{2} ) = 0 and L(g, \frac{k}{2} ) ≠ 0, and that ρ_{f,𝔮} = ρ_{g,𝔮} is irreducible and does not arise from any S_{k}(Γ_{0}(N/p)), as follows:

To prove there is a congruence, we showed that the corresponding integral spaces of modular symbols satisfy an appropriate congruence, which forces the existence of a congruence on the level of Fourier expansions. We showed that ρ_{f,𝔮} is irreducible by computing a set that contains all possible residue characteristics of congruences between f and any Eisenstein series of level dividing N, where by congruence, we mean a congruence for all Fourier coefficients of index n with (n, N) = 1. Similarly, we checked that f is not congruent to any form h of level N/p for any p that exactly divides N by listing a basis of such h and finding the possibly congruences, where again we disregard the Fourier coefficients of index not coprime to N.

To verify that L(f, \frac{k}{2} ) = 0, we computed the image of the modular symbol e = X^{\frac{k}{2} −1}Y^{k−2−( \frac{k}{2} −1)}\{0, ∞\} under a map with the same kernel as the period mapping, and found that the image was 0. The period mapping sends the modular symbol e to a nonzero multiple of L(f, \frac{k}{2} ), so that e maps to 0 implies that L(f, \frac{k}{2} ) = 0. In a similar way, we verified that L(g, \frac{k}{2} ) ≠ 0. Next, we checked that W_{N}(f) = f which, because of the functional equation, implies that L′(f, \frac{k}{2} ) = 0.

6. Some Remarks on Visibility

It would be nice to add a section on visibility. I.e., systematically list examples where the rational part of the central value has a big square factor, which probably should be nontrivial � , and say whether or not the theory of this paper predicts that it is visible. Answer the question: “Does it look like all Sha be visible for motives?” probably “no”.

References

[AS] A. Agashe, W. Stein, Visibility of Shafarevich-Tate groups of abelian varieties: evidence for the Birch and Swinnerton-Dyer conjecture, in preparation.

[AL] A. O. L. Atkin, J. Lehner, Hecke operators on Γ_{0}(m), Math. Ann. 185 (1970), 135–160.

[B] S. Bloch, Algebraic cycles and values of L-functions, J. reine angew. Math. 350 (1984), 94–108.

[BCP] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993).

[BK] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift Volume I, 333–400, Progress in Mathematics, 86, Birkhäuser, Boston, 1990.

[Ca1] H. Carayol, Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4)19 (1986), 409–468.

[Ca2] H. Carayol, Sur les représentations Galoisiennes modulo ℓ attachées aux formes modulaires, Duke Math. J. 59 (1989), 785–801.

[CM] J. E. Cremona, B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), 13–28.

[De1] P. Deligne, Formes modulaires et représentations ℓ-adiques. Sém. Bourbaki, éxp. 355, Lect. Notes Math. 179, 139–172, Springer, 1969.

[De2] P. Deligne, Valeurs de Fonctions L et Périodes d’Intégrales, AMS Proc. Symp. Pure Math., Vol. 33 (1979), part 2, 313–346.

[Du1] N. Dummigan, Period ratios of modular forms, Math. Ann. 318 (2000), 621–636.

[Du2] N. Dummigan, Congruences of modular forms and Selmer groups, Math. Res. Lett., to appear.

[Ed] S. J. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), 563–594.

[Fa] G. Faltings, Crystalline cohomology and p-adic Galois representations, in Algebraic analysis, geometry and number theory (J. Igusa, ed.), 25–80, Johns Hopkins University Press, Baltimore, 1989.

[Fl2] M. Flach, A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990), 113–127.

[Fl1] M. Flach, On the degree of modular parametrisations, Séminaire de Théorie des Nombres, Paris 1991-92 (S. David, ed.), 23–36, Progress in mathematics, 116, Birkhäuser, Basel Boston Berlin, 1993.

[Fo] J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local, construction d’un anneau de Barsotti-Tate, Ann. Math. 115 (1982), 529–577.

[JL] B. W. Jordan, R. Livné, Conjecture “epsilon” for weight k > 2, Bull. Amer. Math. Soc. 21 (1989), 51–56.

[L] R. Livné, On the conductors of mod ℓ Galois representations coming from modular forms, J. Number Theory 31 (1989), 133–141.

[Ne1] J. Nekovár, Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992), 99–125.

[Ne2] J. Nekovár, On the p-adic heights of Heegner cycles, Math. Ann. 302 (1995), 609–686.

[Sc] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419–430.

[SwD] H. P. F. Swinnerton-Dyer, On ℓ-adic representations and congruences for coefficients of modular forms,Modular functions of one variable III, Lect. Notes Math. 350, Springer, 1973.

University of Sheffield, Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, U.K.

Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138, U.S.A.

Penn State Mathematics Department, University Park, State College, PA 16802, U.S.A.

   E-mail address: n.p.dummigan@shef.ac.uk
   E-mail address: was@math.harvard.edu
   E-mail address: watkins@math.psu.edu