Visible elements and congruences of modular forms

Neil Dummigan

February 2nd, 2001

1

Abstract We show that the congruence modulo 11 between the normalized cusp form Δ of weight 12 and the normalized cusp form of weight 2 and level 11 ‘descends’ to a congruence between forms of weights 13/2 and 3/2. Combining Waldspurger’s theorem with the Bloch-Kato conjecture we predict the existence of elements of order 11 in Selmer groups for certain quadratic twists of Δ. These are then constructed using rational points on twists of the elliptic curve X_{0}(11), assuming the Birch and Swinnerton-Dyer conjecture on the rank. Everything generalizes to forms of weights 2 + 10s in an 11-adic family, to congruences modulo higher powers of 11, and to other elliptic curves over ℚ of prime conductor p ≡ 3(mod4).

1 Introduction

The Tamagawa-number conjecture of Bloch and Kato [BK] is a very general conjecture on the special values (at integer points) of the L-functions arising from arithmetic geometry. Examples of such L-functions are those attached to modular forms. The connection with arithmetic geometry is through modular curves, and gives rise to the Galois representations famously associated with modular forms. In an earlier paper [Du1] we gained some evidence for the conjecture, in the special case of modular forms of level one and weight k, by comparing special values at different points in the critical range 1 ≤ s ≤ k − 1. Looking at ratios of critical values avoided the difficulty of determining a canonical period, by making it cancel. In the present paper we look mainly at the central critical point s = 3Dk/2, but consider ratios of the values at that point of various quadratic twists of the L-function attached to the modular form.

12000 Mathematics Subject Classification 11F33, 11F37, 11F67, 11G40. Keywords: modular form, L-function, Bloch-Kato conjecture, Shafarevich-Tate group.

The particular example on which we concentrate is the celebrated discriminant function

Δ(z) = 3Dη(z)^{24} = 3Dq ∏_{n=3D1}^{∞}(1 − q^{n})^{24} = 3D ∑_{=n=3D1}^{∞} τ(n)q^{n},

where q = 3De^{2πiz} and η(z) = 3Dq^{1/24} ∏(1 − q^{n}) is Dedekind’s eta function. Δ is the unique normalized cusp form of weight 12 and level one. Let E be the elliptic curve with minimal Weierstrass equation y^{2} + y = 3Dx^{3} − x^{2} − 10x − 20. This is in fact the modular curve X_{0}(11), so E is trivially ‘modular’. The associated modular form, the unique normalized cusp form of weight 2 on Γ_{0}(11), is

F (z) = 3Dη(z)^{2}η(11z)^{2} = 3Dq ∏(1 − q^{n})^{2} ∏(1 − q^{11n})^{2}.

It is evident from the infinite products that the q-expansions of Δ and F are congruent modulo 11. It is this congruence which acts as a lever for us to prise some information out of the motives attached to quadratic twists of Δ.

A question raised by Hida is whether congruences between integer weight newforms descend to congruences between half-integral weight Hecke eigenforms which correspond to them under the Shimura map [Shim],[K1],[K2]. Maeda [Mae] produced an example with congruences modulo 433, using newforms of weight k = 3D8 and level N = 3D52. Further examples involving congruences between cusp forms and Eisenstein series were considered in [Kob] and [G1]. In our case, we can associate with F and Δ explicit Hecke eigenforms g and h, of weights 3/2 and 13/2, on Γ_{0}(44) and Γ_{0}(4) respectively. The forms F and Δ have different weights, and we find that the congruence modulo 11 between F and Δ descends to a congruence modulo 11 between the Fourier coefficients not quite of g and h, but of g and h|T_{11}. Put another way, if g = 3D ∑ a_{n}q^{n} and h = 3D ∑ b_{n}q^{n} then a_{n} ≡ b_{11n}( mod 11) for all n. We prove the congruence using a theorem of Sturm [Stu], which shows in our case that it suffices to check the congruence for n ≤ 156. A corollary is Theorem 10 of [BDO], that 11|b_{11n} whenever n is a quadratic non-residue modulo 11. Their proof also uses Sturm’s theorem, but necessitates checking the divisibility for n ≤ 95832.

Let d, positive and coprime to 11, be such that −d is the discriminant of a quadratic field. Let E_{−d} be the quadratic twist of E associated to the quadratic character (\frac{−d}{n}). According to a theorem of Waldspurger [Wa], the central critical value L(E_{−d}, 1) is (more or less) proportional to the square of the d^{th} Fourier coefficient of g. If we restrict now to d which are quadratic residues modulo 11, the sign in the functional equation of L(E_{−d}, s) is positive, so L(E_{−d}, s) vanishes to even order at s = 3D1. If the d^{th} Fourier coefficient of g is zero then L(E_{−d}, s) vanishes to order at least two at s = 3D1. According to the Birch and Swinnerton-Dyer conjecture, the group of rational points E_{−d}(ℚ) then has rank at least 2. I am grateful to J. Cremona for checking in the first few cases that the rank really is 2.

Waldspurger’s theorem implies that the square of b_{11d} is more or less proportional to the central critical value of the twisted L-function, L(Δ, χ_{11d}, 6).

In those cases where a_{d} = 3D0, the aforementioned congruence implies that 11|b_{11d}. Take two different d and d′ as above, both quadratic residues modulo 11. Suppose that a_{d} = 3D0 but that a_{d′} ≠ 0(mod11). Then the ratio \sqrt{d′/d}L(Δ, χ_{11d}, 6)/L(Δ, χ_{11d′}, 6) is a rational number divisible by 11. If it is non-zero, then the Bloch-Kato conjecture, applied to these critical values, predicts that some Shafarevich-Tate group for Δ_{11d} has order divisible by 11, even by 11^{2}. (We already mentioned the cancellation of periods. Similarly, although we do not know the local fudge factors c_{p} appearing in the Bloch-Kato formula, we can show that their 11-parts do not change from one twist to another.)

Now, to provide some evidence for the Bloch-Kato conjecture, all we need to do is to produce an element of order 11 in the appropriate Selmer group for the motive attached to Δ_{11d}. Recall that since a_{d} = 3D0, L(E_{−d}, s) vanishes to order at least two at s = 3D1. The Birch and Swinnerton-Dyer conjecture would then imply that E_{−d}(ℚ) has rank at least two. Assuming this is true (it can be checked in particular cases), we get two independent elements of order 11 in a Selmer group for E_{−d}. The congruence of F and Δ implies that the mod 11 Galois representations attached to E and Δ are isomorphic. This allows us to transfer these elements of order 11 to some Galois cohomology group attached to Δ_{11d}. To show that they are in the Selmer group, as required, we would have to show that they satisfy the necessary local conditions at every prime p. For p ≠ 11 this is quite straightforward. For p = 3D11 we might be tempted to try to apply the theory of Fontaine and Lafaille, as in [Du2]. However, this does not apply when p < k, and here p = 3D11 while k = 3D12. Fortunately, it is a matter of simple linear algebra to show that some non-zero linear combination of the two classes must satisfy the local condition at 11, so we do get an element of order 11 in the Selmer group, as required. Note that the inequality p < k is no accident, since for two modular forms (of trivial character) to be congruent modulo p, the difference of their weights must be divisible by p − 1 [SwD].

In the rare event that L(Δ, χ_{11d}, 6) = 3D0, the Beilinson-Bloch generalization [B] of the Birch and Swinnerton-Dyer conjecture predicts the existence of a rational element of infinite order in the Chow group CH_{0}^{6} of the motive attached to Δ_{11d}. Assuming injectivity of the 11-adic Abel-Jacobi map, this would predict an element of order 11 in a Selmer group, which would again be given by the construction outlined above.

Let us reflect a little on what has occurred. The mod 11 congruence between F and Δ, when lifted to g and h, directly links the vanishing of L(E_{−d}, 1) with the divisibility by 11 of \sqrt{d′/d}L(Δ, χ_{11d}, 6)/L(Δ, χ_{11d′}, 6). Conjecturally, the former is associated with rational points on E_{−d}, while the latter is associated with elements of order 11 in a Selmer group for Δ_{11d}. The same mod 11 congruence, when interpreted in terms of Galois representations, provides the link between the rational points and the Selmer group.

The use of rational points on an elliptic curve to produce elements in a Selmer group, which are then transferred to a Selmer group for another motive using a congruence of Galois representations, is in the same spirit as the construction, by Cremona and Mazur [CM], of ‘visible’ likely elements in Shafarevich-Tate groups of elliptic curves. The construction used in [Du2] and [Du3] may be viewed in the same light. It is shown in [GP], by analytic methods, that a_{d} = 3D0 is only possible if 5 divides the class number of ℚ(\sqrt{−d}). We can construct ‘visible’ elements of order 5 in these class groups, starting from the rational points on E_{−d}, getting from them elements of 5-Selmer groups, then using the fact that E[5] ≃ ℤ/5ℤ ⊕ μ_{5} as a Galois module. The details are left to the reader.

So far we have been dealing with the case where d is a quadratic residue modulo 11, where the order of vanishing of L(E_{−d}, s) at s = 3D1 is even. Balog, Darmon and Ono [BDO] concentrated on the case where d is a quadratic non-residue modulo 11, and the order of vanishing is odd. They expressed the hope that their result (11|b_{11d}) could be explained in terms of the Bloch-Kato conjecture. Unfortunately, even if we calculated the canonical period, our ignorance of the 11-part of the fudge factor c_{11} would prevent us from doing this at present.

Most of this paper serves merely as an introduction and illustrative example for some general results stated in the last section. First, Δ can be replaced by any normalized Hecke eigenform of weight 2 + 10s in the Hida 11-adic family containing F. When 11^{t−1} divides s, we find that 11^{2t} divides ratios of Lvalues, and we can construct the predicted elements of order 11^{t} in Selmer groups. Then, more generally, E = 3DX_{0}(11) can be replaced by any elliptic curve of prime conductor p ≡ 3(mod4). Stevens’s Λ-adic Shintani lifting [Ste] is very useful for proving the congruences of half-integer weight forms, and could have been used in Section 3, instead of Sturm’s theorem.

2 The Shimura correspondence

Let N be a positive integer. As usual, let Γ_{0}(N) = 3D\{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) :

N|c}. Let k be an even positive integer. A modular form of weight (k + 1)/2 and character χ for Γ_{0}(4N) is a holomorphic function g on the completed upper half plane, satisfying

g(γz) = 3Dχ(d)j(γ, z)^{k+1}g(z)

for all γ = 3D (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(4N), where

j(γ, z) = 3Dϵ_{d}^{−1} · (\frac{c}{d}) · (cz + d)^{1/2},

ϵ_{d} = 3D1 or i according as d ≡ 1 or 3(mod4) and −π/2 < arg(cz + d)^{1/2} ≤ π/2. It has a q-expansion ∑_{n=3D0}^{∞} a_{n}q^{n}, where q = 3De^{2πiz}. Note that the square of a modular form of half-integral weight is not necessarily a modular form of integral weight, but its fourth power is.

Shimura [Shim] defined a correspondence which associates to a Hecke eigenform g = 3D ∑ a_{n}q^{n} of weight (k + 1)/2, character χ and level 4N a Hecke eigenform f of weight k, character χ^{2} and level dividing 4N. For primes l not dividing 4N, the eigenvalue of the Hecke operator T_{l} acting on f is the same as the eigenvalue of T_{l^{2}} acting on g. Given a square-free d, all the coefficients a_{dt^{2}} are determined by a_{d} and the coefficients of f. For a fixed square-free d, knowledge of all the a_{dt^{2}} determines the coefficients of f.

Let S_{(k+1)/2}^{+}(Γ_{0}(4N)) be the space of cusp forms (with trivial character) such that a_{n} = 3D0 unless (−1)^{k/2}n ≡ 0 or 1(mod4). This space is preserved by Hecke operators T_{l^{2}} for primes l not dividing 4N. Kohnen [K1],[K2] proved that a slightly modified version of the Shimura correspondence induces an isomorphism between S_{(k+1)/2}^{+}(Γ_{0}(4N)) and S_{k}(Γ_{0}(N)). In the introduction we specified cusp forms F ∈ S_{2}(Γ_{0}(11)) and Δ ∈ S_{12}(Γ_{0}(1)). Under Kohnen’s isomorphism they correspond to forms g = 3D ∑ a_{n}q^{n} ∈ S_{3/2}^{+}(Γ_{0}(44)) and h = 3D ∑ b_{n}q^{n} ∈ S_{13/2}^{+}(Γ_{0}(4)) respectively. Both are determined only up to scalar multiples.

Shintani [Shin] devised a way of starting with a form of even integer weight k and going back to a form of weight (k+1)/2 which maps to it under the Shimura correspondence. But in the present case, since S_{13/2}^{+}(Γ_{0}(4)) and S_{3/2}^{+}(Γ_{0}(44)) are one-dimensional, all we need to do is to produce non-zero elements of these spaces. The main example in [KZ] provides the formula

h(z) = 3D(60/2πi)(2G_{4}(4z)θ′(z) − G′_{4}(4z)θ(z)),

where G_{4}(z) = 3D \frac{1}{240} + ∑_{n=3D1}^{∞} σ_{3}(n)q^{n} (σ_{3}(n) = 3D ∑_{d|n,d>0} d^{3}) and θ(z) =

3D1+2 ∑_{n=3D1}^{∞} q^{n^{2}}. This formula is very suitable for the purpose of computing the Fourier coefficients b_{n}. Example 3.5 of [Shin] gives a useful formula for the Fourier coefficients of g: a_{n} = 3Dc_{4n} where

∑ c_{n}q^{n} = 3Dθ(11z)η(2z)η(22z).

3 The proof of the congruence

Theorem 3.1 Let g = 3D ∑ a_{n}q^{n} ∈ S_{3/2}^{+}(Γ_{0}(44)) and h = 3D ∑ b_{n}q^{n} ∈ S_{13/2}^{+}(Γ_{0}(4)) be the forms described above. Then a_{n} ≡ b_{11n}(mod11) for all n.

Proof. We need to show that the coefficients of g and h|T_{11} are congruent modulo 11. (Note that although h has level 4, we consider it at level 44, so that 11 divides the level and the Hecke operator T_{11} exists. Also, though h has trivial character, h|T_{11} has character (\frac{11}{·}). ) It is clearly equivalent to show the same thing for the coefficients of g^{2} and (h|T_{11})^{2}, or even for g^{4} and (h|T_{11})^{4}, which are modular forms of weights 6 and 26 respectively. For even r > 2 let E_{r} = 3D1− \frac{2r}{B_{r}} ∑_{n=3D1}^{∞} σ_{r−1}(n)q^{n} be the normalized Eisenstein series of weight r. A well known consequence of the Clausen-Von Staudt theorem is that for an odd prime number p > 3 the q-expansion of E_{p−1} is congruent to 1 mod p. Letting p = 3D11, multiplying g^{4} by E_{10}^{2} will not make any difference to its Fourier coefficients modulo 11. Now the forms g^{4}E_{10}^{2} and (h|T_{11})^{4} have the same weight k = 3D26, and are both forms on Γ_{0}(M) with M = 3D44.

This allows us to apply the theorem of Sturm [Stu] which guarantees that the congruence holds for all n as long as it holds for all n ≤ \frac{kM}{12} ∏_{p|M}(1 + \frac{1}{p}). In our case the bound is only 156, so the condition is easily checked using Maple. ■

4 Waldspurger’s theorem

The Shimura correspondence tells us relations among the Fourier coefficients a_{dt^{2}} of an eigenform g of half-integral weight, for a fixed square-free d, in terms of the coefficients of the Shimura lift f. It does not say anything about the relation between the a_{d} for different square-free values of d (or d/4). Waldspurger [Wa] proved a theorem to the effect that the ratios of the squares of these coefficients are more or less the ratios of the central critical values of the L-functions associated to f with quadratic twists. We state a refined version in a special case which covers what we need. This is Corollary 1 to Theorem 3 in [K3].

Theorem 4.1 Suppose that N is an odd, square-free positive integer and k is a positive even integer. Temporarily abandoning earlier names, let g = 3D ∑ b_{n}q^{n} ∈ S_{(k+1)/2}^{+}(Γ_{0}(4N)) correspond to f = 3D ∑ c_{n}q^{n} ∈ S_{k}(Γ_{0}(N) under Kohnen’s isomorphism, where f is a newform. Let d be a positive integer such that D := 3D(−1)^{k/2}d is the discriminant of a quadratic field. Let L(f, χ_{D}, s) be the analytic continuation of ∑_{n=3D1}^{∞} χ_{D}(n)c_{n}n^{−s}, where χ_{D}(n) = 3D(\frac{D}{n}). Let ⟨f, f⟩ be the Petersson norm and let ν(N) be the number of distinct prime divisors of N. Suppose that for every prime l|N, (\frac{D}{l}) = 3Dw_{l}, the eigenvalue for f with respect to the Atkin-Lehner involution W_{l}. Then

\frac{b_{d}^{2}}{⟨g, g⟩} = 3D2^{ν(N)} \frac{(\frac{k}{2} − 1)!}{π^{k/2}}d^{(k−1)/2} \frac{L(f, χ_{D}, k/2)}{⟨f, f⟩}.

Corollary 4.2 For two different d′ and d as above, supposing the conditions about (\frac{D}{l}) and (\frac{D′}{l}) are satisfied, and that L(f, χ_{D}, k/2) ≠ 0,

L(f, χ_{D′}, k/2)/L(f, χ_{D}, k/2) = 3D(d/d′)^{(k−2)/2}\sqrt{d/d′}b_{d′}^{=}2/b_{d}^{2}.

In the example with Δ of weight k = 3D12 and h of weight (k +1)/2 = 3D13/2, N = 3D1 so there are no conditions to check. In the example with F of weight k = 3D2 and g of weight (k + 1)/2 = 3D3/2, N = 3D11 and w_{11} = 3D − 1 so we require (\frac{−d}{11}) = 3D − 1, which is equivalent to (\frac{d}{11}) = 3D1. Actually, we shall only be using the above corollary in the example where k = 3D12.

5 Motives and Galois representations

Let f = 3D ∑ e_{n}q^{n} be a newform of weight k for Γ_{0}(N) (and let’s say trivial character and rational coefficients, since that is all we need). A special case of a theorem of Deligne [De1] implies the existence, for each prime l, of a continuous representation

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

(V_{l} is a two-dimensional vector space over ℚ_{l}), such that

1. ρ_{l} is unramified at p for all primes p not dividing lN;

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

Following Scholl[Sc], V_{l} may be constructed as the l-adic realisation of a Grothendieck motive M. There are also Betti and de Rham realisations V_{B} and V_{dR}, both 2-dimensional ℚ-vector spaces. For details of the construction see [Sc]. The de Rham realisation has a Hodge filtration V_{dR} = 3DF^{0} ⊃ F^{1} = 3D … = 3DF^{k−1} ⊃ F^{k} = 3D{0}. The Betti realisation V_{B} comes from singular cohomology, while V_{l} comes from étale l-adic cohomology. There are natural isomorphisms V_{B} ⊗ ℚ_{l} ≃ V_{l}. Using a basis for singular cohomology with ℤcoefficients, we get Gal(\overline{ℚ}/ℚ)-stable ℤ_{l}-modules T_{l} inside each V_{l}. Define A_{l} = 3DV_{l}/T_{l}. There are two kinds of twist we shall have to consider. There is the Tate twist V_{l}(j) (for an integer j), which amounts to multiplying the action of Frob_{p} by p^{j}. Then, for D the discriminant of a quadratic field, there is the quadratic twist V_{l}(χ_{D}), where V_{l} has been tensored with a one-dimensional space on which Gal(\overline{ℚ}/ℚ) acts via the quadratic character χ_{D}.

Following [BK], for p ≠ l let

H_{f}^{1}(ℚ_{p}, V_{l}(χ_{D}, j)) = 3D ker(H^{1}(D_{p}, V_{l}(χ_{D}, j)) → H^{1}(I_{p}, V_{l}(χ_{D}, j))).

Here 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 = 3Dl let

H_{f}^{1}(ℚ_{l}, V_{l}(χ_{D}, j)) = 3D ker(H^{1}(D_{l}, V_{l}(χ_{D}, j)) → H^{1}(D_{l}, V_{l}(χ_{D}, j) ⊗ B_{cris}))

(see [BK] for a definition of Fontaine’s ring B_{cris}). The subscript f stands for “finite part”. Let H_{f}^{1}(ℚ, V_{l}(χ_{D}, j)) be the subspace of elements of H^{1}(ℚ, V_{l}(χ_{D}, j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, V_{l}(χ_{D}, j)) for all primes p.

There is a natural exact sequence

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

Let H_{f}^{1}(ℚ_{p}, A_{l}(χ_{D}, j)) = 3Dπ_{∗}H_{f}^{1}(ℚ_{p}, V_{l}(χ_{D}, j)). Define the l-Selmer group H_{f}^{1}(ℚ, A_{l}(χ_{D}, j)) to be the subgroup of elements of H^{1}(ℚ, A_{l}(χ_{D}, j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, A_{l}(χ_{D}, j)) for all primes p. Define the Shafarevich-Tate group

\underline{III}(χ_{D}, j) = 3D ⊕_{l} H_{f}^{1}(ℚ, A_{l}(χ_{D}, j))/π_{∗}H_{f}^{1}(ℚ, V_{l}(χ_{D}, j)).

For 1 ≤ j ≤ k − 1 define the set of global points

Γ_{ℚ}(χ_{D}, j) = 3D ⊕_{l} H^{0}(ℚ, A_{l}(χ_{D}, j)).

6 The Bloch-Kato conjecture

In this section we assume further that f has level N = 3D1. Let j be an integer such that k/2 ≤ j ≤ k − 1. (Certainly then L(f, χ_{D}, j) is a critical value in the sense of [De2].) If j = 3Dk/2 we suppose we are in the case that L(f, χ_{D}, k/2) ≠ 0. The Bloch-Kato conjecture for the motive M(χ_{D}, j) predicts that

L(f, χ_{D}, j) = 3D(∏_{p} c_{p}(χ_{D}, j))vol_{∞}(χ_{D}, j)#\underline{III}(χ_{D}, j)/(#Γ_{ℚ}(χ_{D}, j)#Γ_{ℚ}(χ_{D}, k−j)).

The local factors vol_{∞}(χ_{D}, j) and c_{p}(χ_{D}, j) depend on the choice of a basis for V_{dR}, but their product does not. A careful treatment of real periods such as vol_{∞}(χ_{D}, j) is given in [De2]. Note that the definition of vol_{∞}(j) given in Section 4 of [Du1] is mistaken, what is defined there actually being 1/vol_{∞}(j). The calculation of the periods of Artin motives in terms of generalized Gauss sums, in Section 6 of [De2], yields the following lemma.

Lemma 6.1 vol_{∞}(χ_{D}, j) = 3Dvol_{∞}(j)/\sqrt{D}.

Define H_{f}^{1}(ℚ_{p}, T_{l}(χ_{D}, j)) to be the inverse image of H_{f}^{1}(ℚ_{p}, V_{l}(χ_{D}, j)) under the natural map from H^{1}(ℚ_{p}, T_{l}(χ_{D}, j)) to H^{1}(ℚ_{p}, V_{l}(χ_{D}, j)). The factor c_{p}(χ_{D}, j), which is 1 for all but finitely many p, is defined to be vol_{p}(χ_{D}, j) divided by the Euler factor (1 − χ_{D}(p)a_{p}p^{−j} + p^{k−1−2j}). Here, vol_{p}(χ_{d}, j) is the product of the size of ⊕_{l≠p}H_{f}^{1}(ℚ_{p}, T_{l}(χ_{D}, j)) and a certain measure of H_{f}^{1}(ℚ_{p}, T_{p}(χ_{D}, j)).

We need to look a bit more carefully at the definition of the p-part of vol_{p}(χ_{d}, j). Let D_{dR}(V_{p}) = 3DH^{0}(ℚ_{p}, V_{p}⊗B_{dR}), where B_{dR} is one of Fontaine’s rings [Fo]. This D_{dR}(V_{p}) is a filtered ℚ_{p}-vector space which, according to Fontaine’s de Rham conjecture, should be naturally isomorphic to V_{dR} ⊗ ℚ_{p} with its Hodge filtration.

Since p is a prime of good reduction for the motive M, V_{p} is a crystalline representation of Gal(\overline{ℚ}_{p}/ℚ_{p}), meaning D_{cris}(V_{p}) and V_{p} have the same dimension, where D_{cris}(V_{p}) := 3DH^{0}(ℚ_{p}, V_{p} ⊗ 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_{p})/F^{j}D_{dR}(V_{p}) ≃ H_{f}^{1}(ℚ_{p}, V_{p}(j)),

via their exponential map. Now if we admit the de Rham conjecture, then a choice of basis for V_{dR} will give a measure on the right-hand side, with respect to which |vol_{p}(j)|_{p}^{−1} is defined to be the volume of H_{f}^{1}(ℚ_{p}, T_{p}(j)).

Fortunately, 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]. In Section 7 of [Du1], we had a condition that p > k, because we needed a comparison theorem with ℤ_{p} coefficients rather than just ℚ_{p} coefficients. That condition is not necessary here, which is good, given that 11 < 12.

Lemma 6.2 Let q be an odd prime and D a quadratic discriminant. For any prime p ≠ q, we have ord_{q}(c_{p}(χ_{D}, j)) = 3D0.

Proof. As on p. 30 of [Fl1], for p ≠ q,

|c_{p}(χ_{D}, j)|_{q}^{−1} = 3D#H^{0}(ℚ_{p}, H^{1}(I_{p}, T_{q}(χ_{D}, j))_{tors})),

where I_{p} is the inertia group at p. If p does not divide D then T_{q}(χ_{D}, j) is a trivial I_{p}-module, so H^{1}(I_{p}, T_{q}(χ_{D}, j)) is torsion-free. In general, since q ≠ p,

H^{1}(I_{p}, T_{q}(χ_{D}, j)) ≃ T_{q}(χ_{D}, j)/(1 − τ)T_{q}(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_{q}(χ_{D}, j), hence H^{1}(I_{p}, T_{q}(χ_{D}, j)) is trivial (remember q ≠ 2). ■

Lemma 6.3 Suppose that D and D′ are two quadratic discriminants, both divisible by q, with χ_{D/ϵq}(q) = 3Dχ_{D′/ϵq}(q), where ϵ = 3Dχ_{−4}(q). Then c_{q}(χ_{D}, j) = 3Dc_{q}(χ_{D′}, j).

Proof. For all primes l, T_{l}(χ_{D}, j) and T_{l}(χ_{D′}, j) are exactly the same as modules for Gal(\overline{ℚ}_{q}/ℚ_{q}). Furthermore, the Euler factors (1 − χ_{D}(q)a_{q}q^{−j} + q^{k−1−2j}) and (1 − χ_{D′}(q)a_{q}q^{−j} + q^{k−1−2j}) are the same. For primes l ≠ q we have ord_{l}(vol_{q}(χ_{D}, j)) = 3D#H_{f}^{1}(ℚ_{q}, T_{l}(χ_{D}, j)), so it is a triviality that ord_{l}(vol_{q}(χ_{D}, j)) = 3Dord_{l}(vol_{q}(χ_{D′}, j)). For l = 3Dq,

(V_{dR}(χ_{D}) ⊗ ℚ_{q})/F^{j} ≃ H_{f}^{1}(ℚ_{q}, V_{q}(χ_{D}, j)),

and a choice of basis for V_{dR} determines the q-part of c_{q}(χ_{D}, j). If χ_{D/ϵq}(q) = 3Dχ_{D′/ϵq}(q) then replacing D by D′ introduces no essential change on either side, so c_{q}(χ_{D}, j) = 3Dc_{q}(χ_{D′}, j). ■

Remark 6.4 We will use only the ‘q-part’ of the above lemma.

Lemma 6.5 ord_{l}(#Γ_{ℚ}(χ_{D}, j)) = 3D0 unless the coefficients of f = 3D ∑ e_{n}q^{n} satisfy the congruence e_{p} ≡ χ_{D}(p)(p^{j} + p^{k−1−j})(modl) for all primes p ≠ l.

This follows from the interpretation of e_{p} as a trace of Frobenius. Note that in the case of interest to us, where f = 3DΔ, j = 3Dk/2 = 3D6 and l = 3D11, the congruence is not = satisfied for any D. See [SwD].

Now let us concentrate on the case where f = 3DΔ and j = 3Dk/2 = 3D6. Recall the forms F, g = 3D ∑ a_{n}q^{n} and h = 3D ∑ b_{n}q^{n} of weights 2, 3/2 and 13/2 respectively. Now b_{33} = 3D − 8968320 ≡ 2(mod11) so we can take d′ = 3D3 and have 11 not dividing b_{11d′}. When a_{d} = 3D0, the congruence a_{n} ≡ b_{11n}(mod11) implies that 11|b_{11d}. Combining the above lemmas with Theorem 4.1, we find the following.

Proposition 6.6 Consider the case k = 3D12, f = 3DΔ. Suppose d is a positive integer such that 11d is the discriminant of a quadratic field, d is a quadratic residue modulo 11, a_{d} = 3D0 and L(Δ, χ_{11d}, 6) ≠ 0. If the Bloch-Kato conjecture is true then 11^{2}|#\underline{III}(χ_{11d}, 6), so H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)) contains elements of order 11.

Remark 6.7 Via Flach’s generalization of the Cassels-Tate pairing [Fl2], if 11|#\underline{III}(χ_{11d}, 6) then 11^{2}|#\underline{III}(χ_{11d}, 6).

Even if L(Δ, χ_{11d}, 6) = 3D0, as explained in the introduction, we still expect that H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)) contains an element of order 11. Using Maple we checked that L(Δ, χ_{11d}, 6) ≠ 0, for all relevant d with 1 ≤ d ≤ 10, 000. See [OS] for what can be proved about non-vanishing.

As noted in the next section, the sign in the functional equation of L(Δ, χ_{11d}, s) is positive, so the order of vanishing at s = 3D6 is always even. Our task in the next section is to construct an element of order 11 in the Selmer group H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)).

7 Local conditions

Given a prime l, we have defined V_{l}, T_{l} and A_{l} for a modular form f of even weight k, and retained this notation for the special case f = 3DΔ, k = 3D12. For the modular form F of weight 2 and level 11 associated to the elliptic curve E = 3DX_{0}(11), we shall use the notation V′_{l}, T′_{l} and A′_{l}. The l-adic Tate module of E is then T′_{l}(1). We use the notation A′[l] for the l-torsion subgroup of A′_{l}.

Theorem 7.1 Suppose that d > 0 and 11d is the = discriminant of a quadratic field. Let r be the rank of the group of rational points E_{−d}(ℚ). If r > 1, the 11torsion subgroup of the Selmer group H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)) has 𝔽_{11}-rank at least r − 1.

Proof. As Gal(\overline{ℚ}/ℚ)-modules, E_{−d}[11] ≃ A′(χ_{−d}, 1)[11] ≃ A(χ_{−d}, 1)[11], the second isomorphism being thanks to the congruence between Δ and F, and the irreducibility of these modules. The fifth power of the 11-cyclotomic character is congruent mod 11 to the quadratic character of conductor 11, so this in turn is isomorphic to A(χ_{11d}, 6)[11].

The injection from E_{−d}(ℚ)/11E_{−d}(ℚ) to H^{1}(ℚ, E_{−d}[11]) then gives us an 𝔽_{11}-subspace of dimension r in H^{1}(ℚ, A(χ_{11d}, 6)[11]), which injects into H^{1}(ℚ, A_{11}(χ_{11d}, 6)) because, as already noted, H^{0}(ℚ, A_{11}(χ_{11d}, 6)) is trivial. Choose some element c ∈ H^{1}(ℚ, A(χ_{11d}, 6)[11]), coming from E_{−d}(ℚ)/11E_{−d}(ℚ), mapping to (say) γ ∈ H^{1}(ℚ, A_{11}(χ_{11d}, 6)). First we show that res_{p}(γ) ∈ H_{f}^{1}(ℚ_{p}, A_{11}(χ_{11d}, 6)), for all primes p ≠ 11, i.e. that res_{p}(γ) in H^{1}(I_{p}, A_{11}(χ_{11d}, 6)) is zero.

Looking at c ∈ H^{1}(ℚ, A′(χ_{−d}, 1)[11]) it maps to γ′ ∈ H_{f}^{1}(ℚ, A′_{11}(χ_{−d}, 1)), so res_{p}(γ′) in H^{1}(I_{p}, A′_{11}(χ_{−d}, 1)) is zero. Now A′_{11}(1) is unramified at p ≠ 11, so H^{0}(I_{p}, A′_{11}(χ_{−d}, 1)) is 11-divisible and H^{1}(I_{p}, A′(χ_{−d}, 1)[11]) injects into H^{1}(I_{p}, A′_{11}(χ_{−d}, 1)). Hence res_{p}(c) in H^{1}(I_{p}, A′(χ_{−d}, 1)[11]) = 3DH^{1}(I_{p}, A(χ_{11d}, 6)[11]) is zero, and its image res_{p}(γ) in H^{1}(I_{p}, A_{11}(χ_{11d}, 6)) is zero, as required.

It remains to deal with the local condition at p = 3D11. The exponential map of Bloch and Kato gives an isomorphism

V_{dR}(χ_{11d}) ⊗ ℚ_{11}/F^{6} ≃= H_{f}^{1}(ℚ_{11}, V_{11}(χ_{11d}, 6)).

Meanwhile, Section 3 of [BK] tells us that H_{f}^{1}(ℚ_{11}, V_{11}(χ_{11d}, 6)) is its own annihilator in H^{1}(ℚ_{11}, V_{11}(χ_{11d}, 6)) with respect to the Tate duality pairing. Hence H_{f}^{1}(ℚ_{11}, V_{11}(χ_{11d}, 6)) has codimension one in H^{1}(ℚ_{11}, V_{11}(χ_{11d}, 6)). It follows that our r-dimensional 𝔽_{11}-subspace of the 11-torsion in H^{1}(ℚ, A_{11}(χ_{11d}, 6)) has at least an (r − 1)-dimensional subspace landing in H_{f}^{1}((ℚ_{11}, A_{11}(χ_{11d}, 6)). ■

For a newform of weight k on Γ_{0}(N), and D the discriminant of a quadratic field, the sign in the functional equation of L(f, χ_{D}, s) is (−1)^{k/2}w_{N}χ_{D}(−N). For L(Δ, 11d, s) as above, the sign is always positive, since χ_{D}(−1) = 3D1 for the character associated to a real quadratic field. For our F of weight 2 and level 11 we have w_{11} = 3D − 1, so the sign in the functional equation of L(f, s) is positive. When we twist by −d as above, the sign becomes χ_{−d}(−11). Hence when (\frac{d}{11}) = 3D1 the sign is positive and the L-function vanishes to even order at s = 3D1. If also the coefficient a_{d} = 3D0 then Waldspurger’s theorem implies that L(E_{−d}, s) vanishes at s = 3D1, and we now know it must do so to order at least 2. According to the Birch and Swinnerton-Dyer conjecture, in this case r ≥ 2, so the above theorem provides the required element of order 11 in the Selmer group H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)).

The squarefree d < 550 such that d ≡ 3(mod4), (\frac{d}{11}) = 3D1 and a_{d} = 3D0 are d = 3D47, 103, 119, 159 and 455. Cremona has checked, for all these values of d, that the order of vanishing of L(E_{−d}, s) at s = 3D1 is precisely two, and that E_{−d}(ℚ) does have rank 2. Guerzhoy and Panchishkin [GP] found the simple rational points (−1, 1), (−3, 1) and (1, 1), of infinite order on E_{−47}, E_{−103} and E_{−119} respectively, with reference to Weierstrass equations −dy^{2} = 3D4x^{3} − 4x^{2} − 40x − 79.

The proof of Theorem 7.1 is easily adapted to show that if the 11-torsion in H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)) has dimension t then the 11-torsion in H_{f}^{1}(ℚ, A′_{11}(χ_{−d}, 1)) has dimension at least t − 1. In fact, we made the first part of the proof slightly longer than necessary just to ensure this reversibility. As noted in the introduction, since 11 < 12 we are unable to apply the theory of Fontaine and Lafaille to show that the 11-torsion in H_{f}^{1}(ℚ, A_{11}(χ_{11d}, 6)) and the 11-torsion in H_{f}^{1}(ℚ, A′_{11}(χ_{−d}, 1)) are isomorphic. This is actually a good thing, since if (\frac{d}{11}) = 3D − 1, so that the order of vanishing of L(E_{−d}, s) at s = 3D1 is odd, we would expect the dimensions of these 𝔽_{11}-vector spaces to differ in parity.

8 More general results

Let E be an elliptic curve of prime conductor p. Let f_{2} be the normalized Hecke eigenform of weight 2 and level p attached to E. Since E has multiplicative reduction at p, the form f_{2} is ordinary at p. In other words, p does not divide the p^{th} Fourier coefficient. Hence, according to [H], if k > 2 is an integer such that k ≡ 2(modp^{t−1}(p − 1)) then, if t is sufficiently large, there exists a normalized eigenform f_{k}, an oldform of weight k on Γ_{0}(p), such that f_{k} ≡ f_{2}(modp^{t}) as q-expansions. (In the example with p = 3D11, it happens that t = 3D1 is large enough.) There is a normalized newform F_{k} of level one such that f_{k}(z) = 3DF_{k}(z) − βF_{k}(pz), where β is the non-unit root of Frobenius.

Let ϵ = 3D ± 1, ν = 3D(−1)^{(p−1)/2} and let d be a positive integer such that ϵd is the discriminant of a quadratic field. The sign in the functional equation of L(E_{ϵd}, s) is −w_{p}χ_{ϵd}(−p). We shall be interested in those d such that this sign is positive, so that if L(E_{ϵd}, s) vanishes at s = 3D1, it does so to order at least two.

Consider the motive attached to F_{k}, with its V_{λ}, T_{λ} and A_{λ}, for prime ideals λ in the ring of integers of the field generated by the coefficients of F_{k}. These coefficients are naturally regarded as elements of ℤ_{p}, so there is a favoured split prime 𝔭 dividing p in the coefficient field.

Theorem 8.1 Suppose that d > 0 is coprime to p, and that ϵd is the discriminant of a quadratic field. Let r be the rank of the group of rational points E_{ϵd}(ℚ), and suppose that r > 1. Suppose that k ≡ 2(modp^{t−1}(p − 1)). If k − 2 is an odd multiple of ϕ(p^{t}), let S = 3DH_{f}^{1}(ℚ, A_{𝔭}(χ_{ϵνpd}, k/2)). Otherwise, let S = 3DH_{f}^{1}(ℚ, A_{𝔭}(χ_{ϵd}, k/2)). Then S has a subgroup isomorphic to (ℤ/p^{t}ℤ)^{r−1}.

This may be proved in essentially the same way as Theorem 7.1. Note that it follows from Hida’s construction that E[p^{t}](−1) and A[p^{t}] are isomorphic as Gal(\overline{ℚ}/ℚ)-modules, not just that the traces of Frobenius (i.e. the Fourier coefficients modulo p^{t}) are equal.

If L(E_{ϵd}, s) vanishes to order at least two at s = 3D1, then the conjecture of Birch and Swinnerton-Dyer predicts that r ≥ 2. If r = 3D2 then the above theorem provides a subgroup isomorphic to ℤ/p^{t}ℤ in a suitable Selmer group. Assuming that the Selmer group equals the Shafarevich-Tate group, it follows from [Fl2] that its order is at least p^{2t}.

Now we just have to content ourselves that this is as predicted by the Bloch-Kato conjecture, applied to a ratio \sqrt{d′/d}L(f_{k}, χ_{ϵd}, k/2)/L(f_{k}, χ_{ϵd′}, k/2) or \sqrt{d′/d}L(f_{k}, χ_{ϵνpd}, k/2)/L(f_{k}, χ_{ϵνpd′}, k/2). First, the representation of Gal(\overline{ℚ}/ℚ) on E[p] is irreducible, thanks to Theorem 4 of [Maz]. Hence factors such as Γ_{ℚ}(χ_{ϵd}, k/2) make no contribution. Second, it may be shown just as in Section 6 that fudge factors such as c_{q}(χ_{ϵd}, k/2) make no contribution. (There is a minor complication in that we must consider their orders at all the primes dividing p.) What we need is then provided by the proposition below, except we must impose the conditions p ≡ 3(mod4) and ϵ = 3D − 1. If p ≡ 1(mod4) and ϵ = 3D1 then the L-values in question would automatically be zero, thanks to a negative sign in the functional equation. If ϵ ≠ ν then the theorems quoted below do not apply.

Proposition 8.2 Suppose that p ≡ 3(mod4), with E/ℚ an = elliptic curve of conductor p. Let d ≡ d′(mod8) be positive, squarefree integers, coprime to p, with d ≡ 3(mod4) and −w_{p}χ_{−d}(−p) = 3D − w_{p}χ_{−d′}(−p) = 3D1. Suppose that L(E_{−d}, 1) = 3D0 but L(E_{−d′}, 1) ≠ 0. Suppose that k > 2 is an integer with k ≡ 2(modϕ(p^{t})), and t sufficiently large. Let s = 3D(k − 2)/ϕ(p^{t}). If s is even, then the rational ratio \sqrt{d′/d}L(f_{k}, χ_{−d}, k/2)/L(f_{k}, χ_{−d′}, k/2) (if defined) is divisible by p^{2(t−u)}. If s is odd, then \sqrt{d′/d}L(f_{k}, χ_{pd}, k/2)/L(f_{k}, χ_{pd′}, k/2) (if defined) is divisible by p^{2(t−u)}. Here, u is a non-negative integer which depends only on d′, not on k.

Sketch of proof. Let χ be the trivial character if s is odd, the Legendre symbol (\frac{·}{p}) if s is even. Let θ_{χ}(f_{k}) ∈ S_{(k+1)/2}(Γ_{0}(4p^{m}), χ^{∗}) be Shintani’s lifting [Shin],

where m = 3D(1 + (−1)^{s})/2 and χ^{∗}(n) = 3Dχ(n)(\frac{(−1)^{k/2}p}{n}). Stevens [Ste] constructed a Λ-adic Shintani lifting, p-adically interpolating the Shintani liftings of the various f_{k}. Here we consider only a sufficiently small open neighbourhood of k = 3D2 in his X(R). It follows from his Theorem 3.3 that, as q-expansions,

Ω_{k,χ}θ_{χ}(f_{k})/ω(f_{k})|T_{p}^{1−m} ≡ Ω_{2}θ(f_{2})/ω(f_{2})(modp^{t}).

Here, ω(f_{k}) and ω(f_{2}) are certain periods chosen so that θ_{χ}(f_{k})/ω(f_{k}) and θ(f_{2})/ω(f_{2}) are algebraic integers, or at least algebraic numbers, integral at primes dividing p. The factors Ω_{k,χ} and Ω_{2} are elements of ℚ_{p}.

For ω(f_{2}) we take 2πi times the canonical real period of the elliptic curve E. We choose E to be a strong Weil curve within its isogeny class. The rational number θ(f_{2})/ω(f_{2}) really is integral at p. This is because the coefficients of θ(f_{2}) are integer linear combinations of integrals of f_{2}(τ) dτ over paths joining Γ_{0}(p)-equivalent cusps. (Manin’s constant is coprime to p (Corollary 4.1 of [Maz]), so \frac{1}{2πi}f_{2}(τ) dτ is as good as the pullback of the canonical differential on E.)

The Ω_{k} in [Ste] are the same as those in [GS], where Ω_{2} = 3D1 for our choice of ω(f_{2}) (see [GS] between 0.8 and 0.9). Let θ(f_{2})/ω(f_{2}) = 3D ∑ a_{n}q^{n} and Ω_{k,χ}θ_{χ}(f_{k})/ω(f_{k}) = 3D ∑ b_{n}q^{n}. Then a_{n} ≡ b_{n}(modp^{t}) if s is even, while a_{n} ≡ b_{pn}(modp^{t}) if s is odd. Let u = 3Dord_{p}(a_{d′}) and make sure t is large enough not only for f_{k} to exist, but for t > u. Since a_{d} = 3D0, the congruences imply that p^{t−u} divides b_{d}/b_{d′}, or b_{pd}/b_{pd′} as appropriate.

Now the proposition follows directly from Corollaire 2 to Théorème 1 of [Wa], which, under our hypotheses, states that

(d/d′)^{(k−1)/2}L(f_{k}, χ_{−d}, k/2)/L(f_{k}, χ_{−d′}, k/2) = 3Db_{d}^{2}/b_{d=′}^{2}

or

(d/d′)^{(k−1)/2}L(f_{k}, χ_{pd}, k/2)/L(f_{k}, χ_{pd′}, k/2) = 3Db_{pd}^{2}/b =_{pd′}^{2},

as appropriate, in both cases assuming that the L-value in the denominator is non-zero. ■

Remark 8.3 In the p = 3D11 case, we could choose d′ so that u = 3D0. One might expect that this is typical.

We should mention the work of Guerzhoy [G2], who considers certain p-adic families of cusp forms and generalizes the ‘quadratic congruences’ of [BDO]. His approach is via two-variable p-adic L-functions, and can be used to prove congruences of the type we have used. If one is interested purely in special values rather than in half-integral weight modular forms, this approach has the merit that it allows one to by-pass Waldspurger’s theorem. Also, it should work when p ≡ 1(mod4), or if p ≡ 3(mod4) but we look at twists of E by real quadratic characters. In these cases, the L-function of the modular form of weight k ≡ 2(modϕ(p^{t})) will have to be twisted by the same character as that of E, and evaluated at the left-most critical point s = 3D1 rather than at the central point s = 3Dk/2.

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University of Sheffield, Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, U.K. E-mail:n.p.dummigan@shef.ac.uk