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

NEIL DUMMIGAN, WILLIAM STEIN, AND MARK WATKINS

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.

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 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 rank two. The rational points on the second elliptic curve produce classes in the common H^{1}(ℚ, E[m]). They expect 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 (with which are associated 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 some 𝔮, and L(g, s) vanishes to order at least 2 at s = k/2. We are able to find eleven examples (all with k = 4 and k = 6), and in each case 𝔮 appears in 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 between f 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: March 8th, 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 [CM] 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 predict the exact order of � . 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 cannot prove theoretically, but which are predicted by Bloch-Kato, given the existence of a non-degenerate alternating pairing on � [Fl1].

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}. 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))).

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

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)).

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

Define the set of global 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. 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}.

3. Canonical periods

From now on we assume 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 any 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 [DFG]. (See especially Sections 2.2 and 5.4 of [DFG].)

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 X_{0}(N) be the modular curve over ℤ[1/N] parametrising elliptic curves with cyclic N-isogenies. Let 𝔈 be the universal elliptic curve over X_{0}(N). Let 𝔈^{k−2} be the (k − 2)-fold fibre product of 𝔈 over X_{0}(N). Realising X_{0}(N) as the quotient Γ_{0}(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 welldefined 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}^{±} = 2πiΩ_{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 ∑_{n=1}^{∞} a_{n}n^{−s}, 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 that

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

Here, ± represents the parity of (k/2) − 1, and vol_{∞} is equal to (2πi)^{k/2}Ω_{±}. For l ≠ p, ord_{λ}(c_{p}(j)) is defined to be

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

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].) 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.)

Lemma 4.1. Let p ∤ N be a prime, 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 4.2. 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, and that, for all primes p | N, 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, with p any prime that exactly divides N. Then for any p | N, and j an integer, ord_{𝔮}(c_{p}(j)) = 0.

Proof. Bearing in mind that

ord_{𝔮}((1 − a_{p}p^{−j} + p^{k−1−2j})) = #H^{0}(ℚ_{p}, V_{𝔮}(j)^{I_{p}}/T_{𝔮}(j)^{I_{p}}),

and also the long exact sequence in I_{p}-cohomology arising from

0 ⟶ T_{𝔮}(j) ⟶ V_{𝔮}(j) ⟶^{π} A_{𝔮}(j) ⟶ 0,

it suffices to show that

dim H^{0}(I_{p}, A[𝔮](j)) = dim H^{0}(I_{p}, V_{𝔮}(j)).

If the dimensions differ then, given that f 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](j) 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. □

Lemma 4.3. Let 𝔮 | q be a prime of E such that q ∤ Nk! Then ord_{𝔮}(c_{q}) = 0.

Proof. It follows from the isomorphism at the end of Section 2.2 of [DFG] (an application 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_{λ} by the functor they call 𝕍. Given this, the lemma follows from Theorem 4.1(iii) of [BK]. □

Lemma 4.4. ord_{λ}(#Γ_{ℚ}(j)) = 0 if A[λ] is an irreducible representation of Gal(\overline{ℚ}/ℚ).

This follows trivially from the definition.

Putting together the above lemmas we arrive at the following:

Proposition 4.5. Let q ∤ N be an odd prime such that q > k. Let 𝔮 | q be a prime of E such that A[𝔮] is an irreducible representation of Gal(\overline{ℚ}/ℚ). 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. 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)/𝔞^{±}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_{0}(N). Let S_{Γ_{0}(N)}(k, R) := Hom_{Γ_{0}(N)}(D_{0}, P_{k−2}(R)) be the R-module of weight k modular symbols for Γ_{0}(N).

Via the isomorphism (8) in Section 1.5 of [V], ω_{f}^{±} corresponds to an element Φ_{f}^{±} ∈ S_{Γ_{0}(N)}(k, ℂ), and δ_{f}^{±} corresponds to an element Δ_{f}^{±} ∈ S_{Γ_{0}(N)}(k, O_{E}).

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_{0}(N)}(k, O_{E}). 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 with 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 4.5, 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}^{±}.

This is analogous to the remark at the end of Section 3 of [CM]. It 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 such that L(g, k/2) = 0, unless Maeda’s conjecture (that all the normalised cuspidal eigenforms are Galois conjugate) is false. See [CF].

6. Constructing elements of the Shafarevich-Tate group

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 (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, null-homologous, codimension k/2 algebraic cycles on the motive M_{g}, modulo rational equivalence. (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] 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 neither f nor g is congruent modulo 𝔮 to any newform of weight k, trivial character and level dividing N/p, with p any 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.

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 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, 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. The case p = q does not quite work, which is the reason for the “r − 1” in the statement of the theorem.

(1) 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)) (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 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 [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 must 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)),

but this may be done as in the proof of Lemma4.2. 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. res_{p}(c) then comes from H^{1}(D_{p}/I_{p}, H^{0}(I_{p}, A[𝔮](k/2))), by inflation-restriction. 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)).

(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].) 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.

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. Hence there is a subspace of H_{f}^{1}(ℚ, V′_{𝔮}(k/2)) of dimension at least r − 1, mapping to zero 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)) (in fact, mapping to zero).

□

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.)

Figure 1. Newforms Relevant to Theorem 6.1

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
95k6A  1  95k6D  9  31, 59
122k6A  1  122k6D  6  73

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 [DFG], 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. Eleven examples

Table 1 on page 9 lists eleven 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.

7.1. 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 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 |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.

7.2. 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

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

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 − 1.

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.

7.3. 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 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^{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(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) = g which, because of the functional equation, implies that L′(g, \frac{k}{2} ) = 0.

In the course of our search, we found 17 plausible examples and had to discard 6 of them because either the representation ρ_{f,𝔮} arose from lower level, was reducible, or p ≡ ±1 (mod q) for some p | N. Table 1 is of independent interest because it includes examples of modular forms of prime level and 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.

References

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

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

[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.

[CF] B. Conrey, D. Farmer, On the non-vanishing of L_{f} (s) at the center of the critical strip, preprint.

[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.

[DFG] F. Diamond, M. Flach, L. Guo, Adjoint motives of modular forms and the Tamagawa number conjecture, preprint. http://www.andromeda.rutgers.edu/˜ krm/liguo/lgpapers.html

[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. 8 (2001), 479–494.

[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.

[FJ] G. Faltings, B. Jordan, Crystalline cohomology and GL(2, ℚ), Israel J. Math. 90 (1995), 1–66.

[Fl1] M. Flach, A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990), 113–127.

[Fl2] 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.

[Fo1] 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.

[Fo2] J.-M. Fontaine, Valeurs spéciales des fonctions L des motifs, Séminaire Bourbaki, Vol.

1991/92. Astérisque 206 (1992), Exp. No. 751, 4, 205–249.

[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, p-adic Abel-Jacobi maps and p-adic heights. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 367–379, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.

[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.

[V] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–419.

[Z] S. Zhang, Heights of Heegner cycles and derivatives of L-series, Invent. Math. 130 (1997), 99–152.

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