CONSTRUCTING ELEMENTS IN SHAFAREVICH-TATE GROUPS OF MODULAR MOTIVE

NEIL DUMMIGAN, WILLIAM STEIN, AND MARK WATKINS

Not for distribution

1. Introduction

Let E be an elliptic curve defined over ℚ and let L(E, s) be the associated Lfunction. The conjecture of Birch and Swinnerton-Dyer predicts that the order of vanishing of L(E, s) at s = 1 is the rank of the group E(ℚ) of rational points, and also gives an interpretation of the leading term in the Taylor expansion in terms of various quantities, including the order of the Shafarevich-Tate group of E.

Cremona and Mazur [2000] look, among all strong Weil elliptic curves over ℚ of conductor N ≤ 5500, at those with non-trivial Shafarevich-Tate group (according to the Birch and Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate group has predicted elements of prime order m. In most cases they find another elliptic curve, often of the same conductor, whose m-torsion is Galois-isomorphic to that of the first one, and which has positive rank. The rational points on the second elliptic curve produce classes in the common H^{1}(ℚ, E[m]). They show [CM2] that these lie in the Shafarevich-Tate group of the first curve, so rational points on one curve explain elements of the Shafarevich-Tate group of the other curve.

The Bloch-Kato conjecture [BK] is the generalisation to arbitrary motives of the leading term part of the Birch and Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture [B] generalises the part about the order of vanishing at the central point, identifying it with the rank of a certain Chow group.

The present work may be considered as a partial generalisation of the work of Cremona and Mazur, from elliptic curves over ℚ (which are associated to modular forms of weight 2) to the motives attached to modular forms of higher weight. (See [AS] for a different generalisation, to modular abelian varieties of higher dimension.) It may also be regarded as doing, for congruences between modular forms of equal weight, what [Du2] did for congruences between modular forms of different weights.

We consider the situation where two newforms f and g, both of weight k > 2 and level N, are congruent modulo a maximal ideal 𝔮 of odd residue characteristic, and L(g, k/2) = 0 but L(f, k/2) ≠ 0. It turns out that this forces L(g, s) to vanish to order at least 2 at s = k/2. We are able to find sixteen examples (all with k = 4 and k = 6), and in each 𝔮 divides the numerator of the algebraic number L(f, k/2)/ vol_{∞}, where vol_{∞} is a certain canonical period. In fact, we show how this divisibility may be deduced from the vanishing of L(g, k/2) using recent work of Vatsal [V]. The point is the congruence betweenf and g leads to a congruence between suitable “algebraic parts” of the special values L(f, k/2) and L(g, k/2). If one vanishes then the other is divisible by 𝔮. Under certain hypotheses, the Bloch-Kato conjecture then implies that the Shafarevich-Tate group attached to f has non-zero 𝔮-torsion. Under certain hypotheses and assumptions, the most substantial of which is the Beilinson-Bloch conjecture relating the vanishing of L(g, k/2) to the existence of algebraic cycles, we are able to construct the predicted elements of � , using the Galois-theoretic interpretation of the congruences to transfer elements from a Selmer group for g to a Selmer group for f. In proving the local conditions at primes dividing the level, and also in examining the local Tamagawa factors at these primes, we make use of a higher weight level-lowering result due to Jordan and Livné [JL].

Date: May 29th, 2002.

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

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

One might say that algebraic cycles for one motive explain elements of � for the other. A main point of [CM1] was to observe the frequency with which those elements of � predicted to exist for one elliptic curve may be explained by finding a congruence with another elliptic curve containing points of infinite order. One shortcoming of our work, compared to the elliptic curve case, is that, due to difficulties with local factors in the Bloch-Kato conjecture, we are unable to compute the exact order of � predicted by the Bloch-Kato conjecture. We have to start with modular forms between which there exists a congruence. However, Vatsal’s work allows us to explain how the vanishing of one L-function leads, via the congruence, to the divisibility by 𝔮 of (an algebraic part of) another, independent of observations of computational data. The computational data does however show that there exist examples to which our results apply. Moreover, it displays factors of 𝔮^{2}, whose existence we do not prove theoretically, but which are predicted by Bloch-Kato.

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, which is necessarily totally real. 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}. We may choose a Gal(\overline{ℚ}/ℚ)-stable O_{λ}-module T_{λ} inside each V_{λ}. Define A_{λ} = V_{λ}/T_{λ}. Let A[λ] denote the λ-torsion in A_{λ}. There is the Tate twist V_{λ}(j) (for any integer j), which amounts to multiplying the action of Frob_{p} by p^{j}.

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

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

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}(j)) = ker(H^{1}(D_{l}, V_{λ}(j)) → H^{1}(D_{l}, V_{λ}(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}(j)) be the subspace of elements of H^{1}(ℚ, V_{λ}(j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, V_{λ}(j)) for all primes p.

There is a natural exact sequence

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

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

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

Define an ideal #� (j) of O_{E}, in which the exponent of any prime ideal λ is the length of the λ-component of � (j). We shall only concern ourselves with the case j = k/2, and write � for � (j).

Define the group of global torsion points

Γ_{ℚ} = ⊕_{λ}H^{0}(ℚ, A_{λ}(k/2)).

This is analogous to the group of rational torsion points on an elliptic curve. Define an ideal #Γ_{ℚ} of O_{E}, in which the exponent of any prime ideal λ is the length of the λ-component of Γ_{ℚ}.

3. Canonical periods

We assume from now on for convenience that N ≥ 3. We need to choose convenient O_{E}-lattices T_{B} and T_{dR} in the Betti and deRham realisations V_{B} and V_{dR} of M_{f}. We do this in a way such that T_{B}⊗_{O_{E}} O_{E}[1/Nk!] and T_{dR}⊗_{O_{E}} O_{E}[1/Nk!] agree with the O_{E}[1/Nk!]-lattices 𝔐_{f,B} and 𝔐_{f,dR} defined in [DFG1]. (See especially Sections 2.2 and 5.4 of [DFG1].)

For any finite prime λ of O_{E} define the O_{λ} module T_{λ} inside V_{λ} to be the image of T_{B} ⊗ O_{λ} under the natural isomorphism V_{B} ⊗ E_{λ} ≃ V_{λ}. Then for λ ∤ Nk!, the O_{λ} module T_{λ} is Gal(\overline{ℚ}/ℚ)-stable, since it comes from ℓ-adic cohomology with O_{λ} coefficients. We may assume that T_{λ} is Gal(\overline{ℚ}/ℚ)-stable for all finite λ, by adjusting T_{B} locally at primes λ | Nk! if necessary.

Let M(N) be the modular curve over ℤ[1/N] parametrising generalised elliptic curves with full level-N structure. Let 𝔈 be the universal generalised elliptic curve over M(N). Let 𝔈^{k−2} be the (k − 2)-fold fibre product of 𝔈 over M(N). Realising M(N)(ℂ) as the disjoint union of ϕ(N) copies of the quotient Γ(N)\ℌ^{∗}, and letting τ be a variable on ℌ, the fibre 𝔈_{τ} is isomorphic to the elliptic curve with period lattice generated by 1 and τ. Let z_{i} ∈ ℂ/⟨1, τ⟩ be a variable on the i^{th} copy of 𝔈_{τ} in the fibre product. Then 2πif(τ) dτ ∧dz_{1} ∧…∧dz_{k−2} is a well-defined differential form on (a desingularisation of) 𝔈^{k−2} and naturally represents a generating element of F^{k−1}T_{dR}. (At least, we can make our choices locally at primes dividing Nk! so that this is the case.) We shall call this element e(f).

Under the deRham isomorphism between V_{dR} ⊗ ℂ and V_{B} ⊗ ℂ, e(f) maps to some element ω_{f}. There is a natural action of complex conjugation on V_{B}, breaking it up into one-dimensional E-vector spaces V_{B}^{+} and V_{B}^{−}. Let ω_{f}^{+} and ω_{f}^{−} be the projections of ω_{f} to V_{B}^{+} ⊗ ℂ and V_{B}^{−} ⊗ ℂ, respectively. Let T_{B}^{±} be the intersections of V_{B}^{±} with T_{B}. These are rank one O_{E}-modules, but not necessarily free, since the class number of O_{E} may be greater than one. Choose non-zero elements δ_{f}^{±} of T_{B}^{±} and let 𝔞^{±} be the ideals [T_{B}^{±} : O_{E}δ_{f}^{±}]. Define complex numbers Ω_{f}^{±} by ω_{f}^{±} = Ω_{f}^{±}δ_{f}^{±}.

4. The Bloch-Kato conjecture

Let L(f, s) be the L-function attached to f. For ℜ(s) > \frac{k+1}{2} it is defined by the Dirichlet series with Euler product ∑_{n=1}^{∞} a_{n}n^{−s} = ∏_{p}(P_{p}(p^{−s}))^{−1}, but there is an analytic continuation given by an integral, as described in the next section. Suppose that L(f, k/2) ≠ 0. The Bloch-Kato conjecture for the motive M_{f}(k/2) predicts the following equality of fractional ideals of E:

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

(Strictly speaking, the conjecture in [BK] is only given for E = ℚ.) Here, ± represents the parity of (k/2) − 1, and vol_{∞} is equal to (2πi)^{k/2} multiplied by the determinant of the isomorphism V_{B}^{±} ⊗ℂ ≃ (V_{dR}/F^{k/2})⊗ℂ, calculated with respect to the lattices O_{E}δ_{f}^{±} and the image of T_{dR}. For l ≠ p, ord_{λ}(c_{p}(j)) is defined to be

length H_{f}^{1}(ℚ_{p}, T_{λ}(j))_{tors} − ord_{λ}(P_{p}(p^{−j}))
= length (H^{0}(ℚ_{p}, A_{λ}(j))/H^{0} (ℚ_{p}, V_{λ}(j)^{I_{p}}/T_{λ}(j)^{I_{p}})) .

We omit the definition of ord_{λ}(c_{p}(j)) for λ | p, which requires one to assume Fontaine’s de Rham conjecture ([Fo1], Appendix A6), and depends on the choices of T_{dR} and T_{B}, locally at λ. (We shall mainly be concerned with the q-part of the Bloch-Kato conjecture, where q is a prime of good reduction. For such primes, the de Rham conjecture follows from Theorem 5.6 of [Fa].) Lemma 4.1. vol_{∞} = c(2πi)^{k/2}Ω_{±}, with c ∈ E and ord_{λ}(c) = 0 for λ ∤ 𝔞^{±}Nk!.

Proof. vol_{∞} is also equal to the determinant of the period map from F^{k/2}V_{dR} ⊗ ℂ to V_{B}^{±} ⊗ ℂ, with respect to lattices dual to those we used above in the definition of vol_{∞} (c.f. the last paragraph of 1.7 of [De2]). We are using here the natural pairings. Recall that the index of O_{E}δ_{f}^{±} in T_{B}^{±} is the ideal 𝔞^{±}. Then the proof is completed by noting that, locally away from primes dividing Nk!, the index of T_{dR} in its dual is equal to the index of T_{B} in its dual, both being equal to the ideal denoted η in [DFG2]. □

Lemma 4.2. Let p ∤ N be a prime and j an integer. Then the fractional ideal c_{p}(j) is supported at most on divisors of p.

Proof. As on p. 30 of [Fl2], for odd l ≠ p, ord_{λ}(c_{p}(j)) is the length of the finite O_{λ}-module H^{0}(ℚ_{p}, H^{1}(I_{p}, T_{λ}(j))_{tors}), where I_{p} is an inertia group at p. But T_{λ}(j) is a trivial I_{p}-module, so H^{1}(I_{p}, T_{λ}(j)) is torsion free. □

Lemma 4.3. Let q ∤ N be a prime satisfying q > k. Suppose that A[𝔮] is an irreducible representation of Gal(\overline{ℚ}/ℚ), where 𝔮 | q. Let p | N be a prime, and if p^{2} | N suppose that p ≢ −1 (mod q). Suppose also that f is not congruent modulo 𝔮 to any newform of weight k, trivial character and level dividing N/p. Then for j any integer, ord_{𝔮}(c_{p}(j)) = 0.

Proof. It suffices to show that

dim_{O_{E}/𝔮} H^{0}(I_{p}, A[𝔮](j)) = dim_{E_{𝔮}} H^{0}(I_{p}, V_{𝔮}(j)),

since this ensures that H^{0}(I_{p}, A_{𝔮}(j)) = V_{𝔮}^{I_{p}}/T_{𝔮}^{I_{p}}, hence that H^{0}(ℚ_{p}, A_{𝔮}(j)) = H^{0}(ℚ_{p}, V_{𝔮}^{I_{p}}/T_{𝔮}^{I_{p}}). If the dimensions differ then, given that f is not congruent modulo 𝔮 to a newform of level dividing N/p, Proposition 2.2 of [L] shows that we are in the situation covered by one of the three cases in Proposition 2.3 of [L]. Since p ≢ −1 (mod q) if p^{2} | N, Case 3 is excluded, so A[𝔮](j) is unramified at p and 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. □

Remark 4.4. For an example of what can be done when f is congruent to a form of lower level, see the first example in Section 7.4 below. Lemma 4.5. If 𝔮 | q is a prime of E such that q ∤ Nk!, then ord_{𝔮}(c_{q}) = 0.

Proof. It follows from Lemma 5.7 of [DFG1] (whose proof relies on an application, at the end of Section 2.2, of the results of [Fa]) that T_{𝔮} is the O_{λ}[Gal(\overline{ℚ}_{q}/ℚ_{q})]- module associated to the filtered module T_{dR} ⊗ O_{𝔮} by the functor they call 𝕍. (This property is part of the definition of an S-integral premotivic structure given in Section 1.2 of [DFG1].) Given this, the lemma follows from Theorem 4.1(iii) of [BK]. (That 𝕍 is the same as the functor used in Theorem 4.1 of [BK] follows from the first paragraph of 2(h) of [Fa].) □

Lemma 4.6. If A[λ] is an irreducible representation of Gal(\overline{ℚ}/ℚ), then ord_{λ}(#Γ_{ℚ}(j)) = 0.

This follows trivially from the definition.

Putting together the above lemmas we arrive at the following: Proposition 4.7. Let q ∤ N be a prime satisfying q > k and suppose that A[𝔮] is an irreducible representation of Gal(\overline{ℚ}/ℚ), where 𝔮 | q. Assume the same hypotheses as in Lemma 4.3, for all p | N. Choose T_{dR} and T_{B} which locally at 𝔮 are as in the previous section. If

ord_{𝔮}(L(f, k/2)𝔞^{±}/ vol_{∞}) > 0

(with numerator non-zero) then the Bloch-Kato conjecture predicts that

ord_{𝔮}(#� ) > 0.

5. Congruences of special values

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. Suppose that q ∤ Nϕ(N)k! It is easy to see that we may choose the δ_{f}^{±} ∈ T_{B}^{±} in such a way that ord_{𝔮}(𝔞^{±}) = 0, i.e. δ_{f}^{±} generates T_{B}^{±} locally at 𝔮. Let us suppose that such a choice has been made.

We shall now make two further assumptions:

(1) L(f, k/2) ≠ 0;

(2) L(g, k/2) = 0.

Proposition 5.1. With assumptions as above, ord_{𝔮}(L(f, k/2)/ vol_{∞}) > 0.

Proof. This is based on some of the ideas used in Section 1 of [V]. Note the apparent typo in Theorem 1.13 of [V], which presumably should refer to “Condition 2”. Since ord_{𝔮}(𝔞^{±}) = 0, we just need to show that ord_{𝔮}(L(f, k/2)/((2πi)^{k/2}Ω_{±})) > 0, where ±1 = (−1)^{(k/2)−1}. It is well-known, and easy to prove, that

∫_{0}^{∞} f(iy)y^{s−1}dy = (2π)^{−s}Γ(s)L(f, s).

Hence, if for 0 ≤ j ≤ k − 2 we define the j^{th} period

r_{j}(f) = ∫_{0}^{i∞} f(z)z^{j}dz,

where the integral is taken along the positive imaginary axis, then

r_{j}(f) = j!(−2πi)^{−(j+1)}L_{f}(j + 1).

Thus we are reduced to showing that ord_{𝔮}(r_{(k/2)−1}(f)/Ω_{±}) > 0.

Let D_{0} be the group of divisors of degree zero supported on ℙ^{1}(ℚ). For a ℤalgebra R and integer r ≥ 0, let P_{r}(R) be the additive group of homogeneous polynomials of degree r in R[X, Y]. Both these groups have a natural action of Γ_{1}(N). Let S_{Γ_{1}(N)}(k, R) := Hom_{Γ_{1}(N)}(D_{0}, P_{k−2}(R)) be the R-module of weight k modular symbols for Γ_{1}(N).

Via the isomorphism (8) in Section 1.5 of [V], ω_{f}^{±} corresponds to an element Φ_{f}^{±} ∈ S_{Γ_{1}(N)}(k, ℂ), and δ_{f}^{±} corresponds to an element Δ_{f}^{±} ∈ S_{Γ_{1}(N)}(k, O_{E}[1/Nϕ(N)]). (See also Section 4.2 of [St].) Inverting Nϕ(N) takes into account the fact that we are now dealing with X_{1}(N) rather that M(N). Up to some small factorials which do not matter locally at 𝔮,

Φ_{f}^{±}([∞] − [0]) = ∑_{j=0,j≡(k/2)−1 (mod 2)}^{k−2} r_{f}(j)X^{j}Y^{k−2−j}.

Since ω_{f}^{±} = Ω_{f}^{±}δ_{f}^{±}, we see that

Δ_{f}^{±}([∞] − [0]) = ∑_{j=0,j≡(k/2)−1 (mod 2)}^{k−2}(r_{f}(j)/Ω_{f}^{±})X^{j}Y^{k−2−j}.

The coefficient of X^{(k/2)−1}Y^{(k/2)−1} is what we would like to show is divisible by 𝔮. Similarly

Φ_{g}^{±}([∞] − [0]) = ∑_{j=0,j≡(k/2)−1 (mod 2)}^{k−2} r_{g}(j)X^{j}Y^{k−2−j}.

The coefficient of X^{(k/2)−1}Y^{(k/2)−1} in this is 0, since L(g, k/2) = 0. Therefore it would suffice to show that, for some μ ∈ O_{E}, the element Δ_{f}^{±} − μΔ_{g}^{±} is divisible by 𝔮 in S_{Γ_{1}(N)}(k, O_{E}[1/Nϕ(N)]). It suffices to show that, for some μ ∈ O_{E}, the element δ_{f}^{±} − μδ_{g}^{±} is divisible by 𝔮, considered as an element of 𝔮-adic cohomology of X_{1}(N) with non-constant coefficients. This would be the case if δ_{f}^{±} and δ_{g}^{±} generate the same one-dimensional subspace upon reduction (mod 𝔮). But this is a consequence of Theorem 2.1(1) of [FJ]. □

Remark 5.2. By Proposition 4.7 (assuming, for all p | N the same hypotheses as in Lemma 4.3, together with q ∤ ϕ(N)), the Bloch-Kato conjecture now predicts that ord_{𝔮}(#� ) > 0. The next section provides a conditional construction of the required elements of � .

Remark 5.3. The signs in the functional equations of L(f, s) and L(g, s) have to be equal, since they are determined by the action of the involution W_{N} on the common subspace generated by the reductions (mod 𝔮) of δ_{f}^{±} and δ_{g}^{±}. Specifically, the sign is (−1)^{k/2}w_{N}, where w_{N} is the common eigenvalue of W_{N} acting on f and g.

This is analogous to the remark at the end of Section 3 of [CM1], which shows that if L(f, k/2) ≠ 0 but L(g, k/2) = 0 then L(g, s) must vanish to order at least two, as in all the examples below. It is worth pointing out that there are no examples of g of level one, and positive sign in the functional equation, such that L(g, k/2) = 0, unless Maeda’s conjecture (that all the normalised cuspidal eigenforms of weight k and level one are Galois conjugate) is false. See [CF].

6. Constructing elements of the Shafarevich-Tate group

Let f and g be as in the first paragraph of the previous section. 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. 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 (as it must be if L(f, k/2) ≠ 0, see Remark 5.3), this implies that the order of vanishing of L(g, s) 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 rational equivalence classes of null-homologous, codimension k/2 algebraic cycles on the motive M_{g}. (This generalises the part of the Birch–Swinnerton-Dyer conjecture which says that for an elliptic curve E/ℚ, the order of vanishing of L(E, s) at s = 1 is equal to the rank of the Mordell-Weil group E(ℚ).)

Via the 𝔮-adic Abel-Jacobi map, CH_{0}^{k/2}(M_{g})(ℚ) maps to H^{1}(ℚ, V′_{𝔮}(k/2)), and its image is contained in the subspace H_{f}^{1}(ℚ, V′_{𝔮}(k/2)), by 3.1 and 3.2 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. In fact, one could simply conjecture that the dimension of H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) is equal to the order of vanishing of L(g, s) at s = k/2. This would follow from the “conjectures” C_{r}(M) and C_{λ}^{i}(M) in Sections 1 and 6.5 of [Fo2].

Similarly, if L(f, k/2) ≠ 0 then we expect that H_{f}^{1}(ℚ, V_{𝔮}(k/2)) = 0, so that H_{f}^{1}(ℚ, A_{𝔮}(k/2)) coincides with the 𝔮-part of � .

Theorem 6.1. 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] is an irreducible representation of Gal(\overline{ℚ}/ℚ) and that for no prime p | N is f congruent modulo 𝔮 to a newform of weight k, trivial character and level dividing N/p. Suppose that, for all primes p | N, p ≢ −w_{p} (mod q), with p ≢ −1 (mod q) if p^{2} | N. (Here w_{p} is the common eigenvalue of the Atkin-Lehner involution W_{p} acting on f and g.) Then the 𝔮-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(k/2)) has 𝔽_{𝔮}-rank at least r.

Proof. 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 (note that T′_{𝔮}(k/2)/𝔮T′_{𝔮}(k/2) ≃ A′[𝔮](k/2)) 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. It follows that 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, 𝔮-torsion class γ ∈ H^{1}(ℚ, A_{𝔮}(k/2)).

Our aim is to show that res_{p}(γ) ∈ H_{f}^{1}(ℚ_{p}, A_{𝔮}(k/2)), for all (finite) primes p. We consider separately the cases p ∤ qN, p | N and p = q.

(1) p ∤ qN. Consider the I_{p}-cohomology of the short exact sequence

0 ⟶ A′[𝔮](k/2) ⟶ A′_{𝔮}(k/2) ⟶^{π} A′_{𝔮}(k/2) ⟶ 0,

where π is multiplication by a uniformising element of O_{𝔮}. 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 𝔮divisible. Therefore H^{1}(I_{p}, A′[𝔮](k/2)) (which, remember, is the same as H^{1}(I_{p}, A[𝔮](k/2))) injects into H^{1}(I_{p}, A′_{𝔮}(k/2)). It follows from the fact that d ∈ H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) that the image in H^{1}(I_{p}, A′_{𝔮}(k/2)) of the restriction of c is zero, hence that the restriction of c (to H^{1}(I_{p}, A′[𝔮](k/2)) ≃ H^{1}(I_{p}, A[𝔮](k/2))) is zero. Hence the restriction of γ to H^{1}(I_{p}, A_{𝔮}(k/2)) is also zero. By line 3 of p. 125 of [Fl1], 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)).

(2) p | N. First we 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)),

since this would imply that the natural map from H^{0}(I_{p}, V′_{𝔮}(k/2)) to H^{0}(I_{p}, A′_{𝔮}(k/2)) is surjective, but this may be done as in the proof of Lemma 4.3. It follows as above that the image of c ∈ H^{1}(ℚ, A[𝔮](k/2)) in H^{1}(I_{p}, A[𝔮](k/2)) is zero. Then res_{p}(c) comes from H^{1}(D_{p}/I_{p}, H^{0}(I_{p}, A[𝔮](k/2))), by inflationrestriction. 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 −w_{p}p^{−1}. Since p ≢ −w_{p} (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)).

(3) p = q.

Since q ∤ N is a prime of good reduction for the motive M_{g}, 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].) As already noted in the proof of Lemma 4.5, T_{𝔮} is the O_{λ}[Gal(\overline{ℚ}_{q}/ℚ_{q})]-module associated to the filtered module T_{dR} ⊗O_{λ}. Since also q > k, we may now prove, in the same manner as Proposition 9.2 of [Du1], that res_{q}(γ) ∈ H_{f}^{1}(ℚ_{q}, A_{𝔮}(k/2)).

□

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.)

In this paper we have encountered two technical problems which we dealt with in quite similar ways:

(1) dealing with the 𝔮-part of c_{p} for p | N;

(2) proving local conditions at primes p | N, for an element of 𝔮-torsion. If our only interest was in testing the Bloch-Kato conjecture at 𝔮, we could have made these problems cancel out, as in Lemma 8.11 of [DFG1], by weakening the local conditions. However, we have chosen not to do so, since we are also interested in the Shafarevich-Tate group, and since the hypotheses we had to assume are not particularly strong.

7. Examples and Experiments

7.1. Visible � Table 1. Table 1 on page 10 lists sixteen pairs of newforms f and g (of equal weights and levels) along with at least one prime q such that there is a prime 𝔮 | q with f ≡ g (mod 𝔮). In each case, ord_{s=k/2} L(g, k/2) ≥ 2 while L(f, k/2) ≠ 0. The notation is as follows. The first column contains a label whose structure is

[Level]k[Weight][GaloisOrbit]

This label determines a newform g = ∑ 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 the sequence of absolute values |Tr(a_{p}(g))| 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. The second column contains the degree of the field ℚ(…, a_{n}, …). The third and fourth columns contain f and its degree, respectively. The fifth column contains at least one prime q such that there is a prime 𝔮 | q with f ≡ g (mod 𝔮), and such that the hypotheses of Theorem 6.1 (except possibly r > 0) are satisfied for f, g, and 𝔮.

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

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

Figure 1. Visible �

g  deg(g)  f  deg(f) 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^{2}
657k4A  1  657k4C  7  5
657k4A  1  657k4G 12  5
681k4A  1  681k4D 30  59
684k4C  1  684k4K  4  7^{2}
95k6A  1  95k6D  9  31, 59
122k6A  1  122k6D  6  73
260k6A  1  260k6E  4  17

with L(g, 2) = 0. Because W_{127}(g) = g, the functional equation has sign +1, so L′(g, 2) = 0 as well. We also find a newform f ∈ 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(f, 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 we have checked the hypotheses of Theorem 6.1, so if r is the dimension of H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) then the 𝔮-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(k/2)) has 𝔽_{𝔮}-rank at least r.

Since ord_{s=k/2} L(g, s) ≥ 2, we expect that r ≥ 2. Then, since L(f, k/2) ≠ 0, we expect that the 𝔮-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(k/2)) is equal to the 𝔮-torsion subgroup of � . Admitting these assumptions, we have constructed the 𝔮-torsion in � predicted by the Bloch-Kato conjecture.

For particular examples of elliptic curves one can often find and write down rational points predicted by the Birch and Swinnerton-Dyer conjecture. It would be nice if likewise one could explicitly produce algebraic cycles predicted by the Beilinson-Bloch conjecture in the above examples. Since L′(g, k/2) = 0, Heegner cycles have height zero (see Corollary 0.3.2 of [Z]), so ought to be trivial in CH_{0}^{k/2}(M_{g}) ⊗ ℚ.

7.2. 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 MAGMA (see [BCP]).

Let g, f, and q be some data from a line of Table 1 and let N denote the level of g. We verified the existence of a congruence modulo q, that L(g, \frac{k}{2} ) = L′(g, \frac{k}{2} ) = 0 and L(f, \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_{g,𝔮} is irreducible by computing a set that contains all possible residue characteristics of congruences between g 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 g 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 possible congruences, where again we disregard the Fourier coefficients of index not coprime to N.

To verify that L(g, \frac{k}{2} ) = 0, we computed the image of the modular symbol e = X^{\frac{k}{2} −1}Y ^{\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(g, \frac{k}{2} ), so that e maps to 0 implies that L(g, \frac{k}{2} ) = 0. In a similar way, we verified that L(f, \frac{k}{2} ) ≠ 0. Next, we checked that W_{N}(g) = (−1)^{k/2}g which, because of the functional equation, implies that L′(g, \frac{k}{2} ) = 0. Table 1 is of independent interest because it includes examples of modular forms of weight > 2 with a zero at \frac{k}{2} that is not forced by the functional equation. We found no such examples of weights ≥ 8.

For the two examples 581k4E and 684k4K, the square of a prime appears in the q-column, meaning f and g are congruent mod 𝔮^{2} for some 𝔮 | q. In these cases, a modification of Theorem 6.1 will show that the 𝔮^{2}-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(k/2)) has O_{E}/𝔮^{2}-rank at least r.

7.3. Conjecturally nontrivial � . In this section we apply some of the results of Section 4 to compute lower bounds on conjectural orders of Shafarevich-Tate group of many modular motives. The results of this section suggest that � of modular motives is usually not explained by congruences at level N, which agrees with the observations of [CM1] and [AS]. For example, when k > 6 we find many examples of nontrivial � but no examples of nontrivial visible � .

For any newform f, let L(M_{f}, s) = ∏_{σ} L(f^{σ}, s) where f^{σ} runs over the Gal(\overline{ℚ}/ℚ)- conjugates of f. Let T be the complex torus ℂ^{d}/L, where the lattice L is defined by integrating integral modular symbols against the conjugates of f. Let Ω_{M_{f}} denote the volume of the −1 eigenspace T^{−} = \{z ∈ T : \overline{z} = −z\} for complex conjugation on T.

The following conjecture is probably not difficult to prove, but we haven’t given a proof, so we formally state it as a conjecture, then assume it.

Conjecture 7.1. If p ∤ Nk is a prime and p | \frac{L(M_{f} ,k/2)}{Ω_{k/2}}, then p | Norm (\frac{L(f,k/2)}{vol_{∞}}).

For the rest of this section, we officially assume the Bloch-Kato conjecture and Conjecture 7.1.

12 NEIL DUMMIGAN, WILLIAM STEIN, AND MARK WATKINS

Figure 2. Conjecturally nontrivial � (mostly invisible)

f  deg(f) B (� bound)  all odd congruence primes
127k4C∗ 17  43^{2}  43, 127
159k4E∗  8  23^{2}  3, 5, 11, 23, 53, 13605689
263k4B  39  41^{2}  263
269k4C  39  23^{2}  269
271k4B  39  29^{2}  271
281k4B  40  29^{2}  281
295k4C  16  7^{2}  3, 5, 11, 59, 101, 659, 70791023
299k4C  20  29^{2}  13, 23, 103, 20063, 21961
319k4C  19  17^{2}  3, 11, 23, 29, 37, 3181, 434348087
321k4C  16  13^{2}  3, 5, 107, 157, 12782373452377
95k6D∗  9  31^{2}·59^{2}  3, 5, 17, 19, 31, 59, 113, 26701
101k6B  24  17^{2}  101
103k6B  24  23^{2}  103
111k6C  9  11^{2}  3, 37, 2796169609
122k6D∗  6  73^{2}  3, 5, 61, 73, 1303196179
153k6G  5  7^{2}  3, 17, 61, 227
157k6B  34  251^{2}  157
167k6B  40  41^{2}  167
172k6B  9  7^{2}  3, 11, 43, 787
173k6B  39  71^{2}  173
181k6B  40  107^{2}  181
191k6B  46  85091^{2}  191
193k6B  41  31^{2}  193
199k6B  46  200329^{2}  199
47k8B  16  19^{2}  47
59k8B  20  29^{2}  59
67k8B  20  29^{2}  67
71k8B  24  379^{2}  71
73k8B  22  197^{2}  73
74k8C  6  23^{2}  37, 127, 821, 8327168869
79k8B  25  307^{2}  79
83k8B  27  1019^{2}  83
87k8C  9  11^{2}  3, 5, 7, 29, 31, 59, 947, 22877, 3549902897
89k8B  29  44491^{2}  89
97k8B  29  11^{2}·277^{2}  97
101k8B  33  19^{2}·11503^{2}  101
103k8B  32  75367^{2}  103
107k8B  34  17^{2}·491^{2}  107
109k8B  33  23^{2}·229^{2}  109
111k8C  12  127^{2}  3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187
113k8B  35  67^{2}·641^{2}  113
115k8B  12  37^{2}  3, 5, 19, 23, 572437, 5168196102449
117k8I  8  19^{2}  3, 13, 181
118k8C  8  37^{2}  5, 13, 17, 59, 163, 3923085859759909
119k8C  16  1283^{2}  3, 7, 13, 17, 109, 883, 5324191, 91528147213

f  deg(f) B (� bound)  all odd congruence primes
121k8F  6  71^{2}  3, 11, 17, 41
121k8G 12  13^{2}  3, 11
121k8H 12  19^{2}  5, 11
125k8D 16  179^{2}  5
43k10B 17  449^{2}  43
47k10B 20  2213^{2}  47
53k10B 21  673^{2}  53
55k10D  9  71^{2}  3, 5, 11, 251, 317, 61339, 19869191
59k10B 25  37^{2}  59
62k10E  7  23^{2}  3, 31, 101, 523, 617, 41192083
64k10K  2  19^{2}  3
67k10B 26  191^{2}·617^{2}  67
68k10B  7  83^{2}  3, 7, 17, 8311
71k10B 30  1103^{2}  71
19k12B  9  67^{2}  5, 17, 19, 31, 571
31k12B 15  67^{2}·71^{2}  31, 13488901
35k12C  6  17^{2}  5, 7, 23, 29, 107, 8609, 1307051
39k12C  6  73^{2}  3, 13, 1491079, 3719832979693
41k12B 20  54347^{2}  7, 41, 3271, 6277
43k12B 20  212969^{2}  43, 1669, 483167
47k12B 23  24469^{2}  17, 47, 59, 2789
49k12H 12  271^{2}  7

Let S be the set of newforms with level N and weight k satisfying either k = 4 and N ≤ 321, or k = 6 and N ≤ 199, or k = 8 and N ≤ 125, or k = 10 and N ≤ 72, or k = 12 and N ≤ 49. Compute a bound B on #� , for each newform f ∈ S, as follows:

(1) Let L_{1} be the rational number L(M_{f}, k/2)/Ω_{M_{f}}. If L_{1} = 0 let B = 1 and terminate.

(2) Let L_{2} be the part of L_{1} that is coprime to N · k!.

(3) Let L_{3} be the part of L_{2} that is coprime to p + 1 for every prime p such that p^{2} | N.

(4) Let L_{4} be the part of L_{3} coprime to any prime of congruence between f and a form of weight k and lower level.

(5) Let B be the part of L_{4} coprime to any prime of congruence between f and an Eisenstein series. (This eliminates residue characteristic of reducible representations.)

Proposition 4.7 implies that if ord_{p}(B) > 0, then ord_{p}(#� ) > 0.

We computed B for every newform in S. Because of Tamagawa numbers, there are many examples in which L_{3} is large, but B is not. For example, 39k4C has L_{3} = 19, but B = 1 because of a 19-congruence with a form of level 13; in this case we must have 19 | c_{13}.

The newforms for which B > 1 are given in Table 2. The second column of the table records the degree of the field generated by the Fourier coefficients of f. This third contains B. Let W be the intersection of the span of all conjugates of f with S_{k}(Γ_{0}(N), ℤ) and W^{⊥} the Petersson orthogonal complement of W in S_{k}(Γ_{0}(N), ℤ).

Then the fourth column contains the odd prime divisors of #(S_{k}(Γ_{0}(N), ℤ))/(W + W^{⊥}), which are exactly the possible primes of congruence for f. We place a ∗ next to the four entries of Table 2 that also occur in Table 1.

7.4. Other examples. We have some other examples where forms of different levels are congruent. However, Remark 5.3 does not apply, so that one of the forms could have an odd functional equation, and the other could have an even functional equation. For instance, we have a 13-congruence between g = 81k4A and f = 567k4L; here L(567k4L, 2) ≠ 0, while L(81k4, 2) = 0 since it has odd functional equation. Here f obviously fails the condition about not being congruent to a form of lower level, so in Lemma 4.3 it is possible that ord_{𝔮}(c_{7}(2)) > 0. In fact this does happen. Because V′_{𝔮} (attached to g of level 81) is unramified at p = 7, H^{0}(I_{p}, A[𝔮]) (the same as H^{0}(I_{p}, A′[𝔮])) is two-dimensional. As in (2) of the proof of Theorem 6.1, one of the eigenvalues of Frob_{p}^{−1} acting on this two-dimensional space is α = −w_{p}p^{(k/2)−1}, where W_{p}f = w_{p}f. The other must be β = −w_{p}p^{k/2}, so that αβ = p^{k−1}. Twisting by k/2, we see that Frob_{p}^{−1} acts as −w_{p} on the quotient of H^{0}(I_{p}, A[𝔮](k/2)) by the image of H^{0}(I_{p}, V_{𝔮}(k/2)). Hence ord_{𝔮}(c_{p}(k/2)) > 0 when w_{p} = −1, which is the case in our example here with p = 7. Likewise H^{0}(ℚ_{p}, A[𝔮](k/2)) is non-trivial when w_{p} = −1, so (2) of the proof of Theorem 6.1 does not work. This is just as well, since had it worked we would have expected ord_{𝔮}(L(f, k/2)/ vol_{∞}) ≥ 3, which computation shows not to be the case.

Here is an example where the divisibility between the levels is the other way round: a 7-congruence between g = 122k6A and f = 61k6B. In this case both L-functions have even functional equation, and we have L(122k6A, 3) = 0. In the proof of Theorem 6.1, we find a problem with the local condition at p = 2. The map from H^{1}(I_{2}, A′[𝔮](3)) to H^{1}(I_{2}, A′_{𝔮}(3)) is not necessarily injective, but its kernel is at most one-dimensional, so we still get the 𝔮-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(2)) having 𝔽_{𝔮}-rank at least 1 (assuming r ≥ 2), and thus get (probable) elements of � for 61k6B. In particular, these elements of � are invisible at level 61. When the levels are different we are no longer able to apply Theorem 2.1 of [FJ]. However, we still have the congruences of integral modular symbols required to make the proof of Proposition 5.1 go through. Indeed, as noted above, the congruences of modular forms were found by producing congruences of modular symbols. Despite these congruences of modular symbols, Remark 5.3 does not apply, since there is no reason to suppose that w_{N} = w_{N′}, where N and N′ are the distinct levels.

Finally, there are two examples where we have a form g with even functional equation such that L(g, k/2) = 0, and a congruent form f which has odd functional equation; these are a 23-congruence between g = 453k4A and f = 151k4A, and a 43-congruence between g = 681k4A and f = 227k4A. If ord_{s=2} L(f, s) = 1, it ought to be the case that dim(H_{f}^{1}(ℚ, V_{𝔮}(2))) = 1. If we assume this is so, and similarly that r = ord_{s=2}(L(g, s)) ≥ 2, then unfortunately the appropriate modification of Theorem 6.1 does not necessarily provide us with non-trivial 𝔮torsion in � . It only tells us that the 𝔮-torsion subgroup of H_{f}^{1}(ℚ, A_{𝔮}(2)) has 𝔽_{𝔮}-rank at least 1. It could all be in the image of H_{f}^{1}(ℚ, V_{𝔮}(2)). � appears in the conjectural formula for the first derivative of the complex L function, evaluated at s = k/2, but in combination with a regulator that we have no way of calculating.

Let L_{q}(f, s) and L_{q}(g, s) be the q-adic L functions associated with f and g by the construction of Mazur, Tate and Teitelbaum [MTT], each divided by a suitable canonical period. We may show that 𝔮 | L′_{q}(f, k/2), though it is not quite clear what to make of this. This divisibility may be proved as follows. The measures dμ_{f,α} and (a q-adic unit times) dμ_{g,α′} in [MTT] (again, suitably normalised) are congruent mod 𝔮, as a result of the congruence between the modular symbols out of which they are constructed. Integrating an appropriate function against these measures, we find that L′_{q}(f, k/2) is congruent mod 𝔮 to L′_{q}(g, k/2). It remains to observe that L′_{q}(g, k/2) = 0, since L(g, k/2) = 0 forces L_{q}(g, k/2) = 0, but we are in a case where the signs in the functional equations of L(g, s) and L_{q}(g, s) are the same, positive in this instance. (According to the proposition in Section 18 of [MTT], the signs differ precisely when L_{q}(g, s) has a “trivial zero” at s = k/2.)

7.5. Excluded data. We also found some examples for which the conditions of Theorem 6.1 were not met. We have a 7-congruence between 639k4B and 639k4H, but w_{71} = −1, so that 71 ≡ −w_{71} (mod 7). There is a similar problem with a 7congruence between 260k6A and 260k6E — here w_{13} = 1 so that 13 ≡ −w_{13} (mod 7). Finally, there is a 5-congruence between 116k6A and 116k6D, but here the prime 5 is less than the weight 6.

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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