SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH-TATE GROUPS

NEIL DUMMIGAN

Abstract. We use Zagier’s method to compute the critical values of the symmetric square L-functions of the six cuspidal eigenforms of level one with rational coefficients. According to the Bloch-Kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalised Shafarevich-Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa’s congruences for Siegel modular forms of degree two.

1. Introduction

In [BK], Bloch and Kato formulated a general conjecture on the values at integer points of the L-functions attached to motives. Previous conjectures of Deligne, Beilinson and Bloch had predicted these values only up to rational multiples. Bloch and Kato provided strong evidence for their conjecture in the case of the Riemann zeta function, and for the L-functions of complex-multiplication elliptic curves at s = 2. Evidence is also provided by the work of Kolyvagin and others on the Birch and Swinnerton-Dyer conjecture for abelian varieties (at s = 1), and by the work of Flach, Taylor and Wiles [Fl1],[Fl2],[W],[TW] (see also [HTU]) on the symmetric square motives attached to modular forms of weight two (at s = 2). See also [Gu],

[Han] and [Har] for results concerning Hecke L-functions.

The “critical values” which are the subject of Deligne’s conjecture [De2] perhaps offer a good opportunity for testing the Bloch-Kato conjecture, since no complicated regulators get in the way. The value of a motivic L-function at a (non-central) critical point is conjecturally just a local volume at infinity (determinant of a period matrix times a power of 2πi) times some non-zero rational number (if the motive has rational coefficient field). Separating out the local volume and the rational number could be a problem. However, the local volume at infinity is the same (up to a power of 2πi) at different critical points of the same parity, and it should cancel when we take the ratio of L-values at those points. Removing the power of π, evidence for the Bloch-Kato conjecture should survive in the factorization of the rational ratio. This is exactly what happens for the L-functions attached to the cuspidal eigenforms on SL_{2}(ℤ) with rational coefficients (weights k = 12, 16, 18, 20, 22 and 26), as described in [Du].

Date: December 2nd, 1999.

1991 Mathematics Subject Classification. 11F67, 11G99, 11F33, 11F80.

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

In these cases the Bloch-Kato conjecture predicts the existence of non-trivial elements in certain Shafarevich-Tate groups, but the prospects for finding such elements appear bleak. In the present paper we find that in certain cases we have much better luck with the symmetric square L-functions attached to these modular forms.

The critical values (right of the central point) are at the points s = r + k − 1 for odd r with 1 ≤ r ≤ k − 1. They may be calculated by a method due to Zagier [Za]. Each is equal to a power of π times the norm of the cusp form f with respect to the Petersson inner product, times a non-zero rational number. Large primes occurring in the numerators of these rational numbers should be the orders of elements in the associated Shafarevich-Tate groups.

A striking fact which appears to have no elementary explanation is that when k = 18, 22 or 26 (so k/2 is odd) and r = k/2, the irregular prime divisors of the Bernoulli number B_{k} are just such large prime divisors. Allowing an assumption or two, we construct the predicted elements of Shafarevich-Tate groups using Nekovár’s work on Heegner cycles [Ne1], together with the Galois-theoretic interpretation of Ramanujan-style congruences [Se2] (see also [SwD]). For this we need to check that the first derivative of the p-adic L-function attached to f does not vanish at s = k/2. Actually, we are only able to check this non-vanishing in the cases k = 22, p = 131 or 593, due to the enormous size of the computation that would be necessary for the larger primes p = 43867 and 657931.

We briefly discuss what ought to happen for higher symmetric powers of modular forms, assuming not only Bloch-Kato, but also the Beilinson-Bloch conjecture [B], which relates vanishing at the central point to the existence of certain algebraic cycles. There is presently a single piece of computational evidence for our suspicions, concerning the symmetric fourth power of the discriminant form. We present computational evidence of a similar phenomenon for the tensor product L-function associated to the forms of weights 12 and 26.

SYMMETRIC SQUARE L-FUNCTIONS 3

Associated to f is its Klingen-Eisenstein series [f], a non-cuspidal Siegel modular eigenform of degree two. Kurokawa [Ku1] conjectured that the Hecke eigenvalues of [f] are congruent to those of some cuspidal Siegel modular eigenform F, of degree two, modulo certain primes dividing the critical value for r = k − 1. Kurokawa proved a congruence modulo 71^{2} when k = 20 and Mizumoto [Mi1] proved the general conjecture. These primes modulo which there is a congruence should, as large prime divisors of critical values, be the orders of elements in certain Shafarevich-Tate groups. The well-known connection between the deformation theory of Galois representations and Selmer groups for symmetric square motives suggests that the existence of the desired element should be a fairly direct consequence of the existence of the congruence. If the Galois representation attached to F by Weissauer [We] satisfies a few likely conditions then we can use a construction based on work of Urban [Ur].

Hence, in the r = k − 1 case we have a direct explanation leading from divisibility of a critical value, via a congruence, to an element in a Shafarevich-Tate group. By contrast, in the r = k/2 case we can construct an element of the Shafarevich-Tate group, but have to just look and see that the critical value is divisible by the Eisenstein prime in accord with the Bloch-Kato conjecture.

2. Calculating the critical values

For the basics on modular forms see [?], for example. Let f be a normalised Hecke eigenform of weight k for SL_{2}(ℤ). Let f = ∑_{n=1}^{∞} a_{n}q^{n}, where q = e^{2πiz}, for z in the upper half plane, and a_{1} = 1. We now assume that k = 12, 16, 18, 20, 22 or 26. In these cases f is unique and the Fourier coefficients a_{n} are rational integers. For example, when k = 12, f is the discriminant function

Δ = ∑ τ(n)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{24}.

Associated with f is the L-function L_{f}(s) = ∑ a_{n}n^{−s} for ℜ(s) sufficiently large, continuable to a holomorphic function on the whole complex plane by means of the integral formula

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

It satisfies the functional equation

Λ(s) = (−1)^{k/2}Λ(k − s).

Since f is a Hecke eigenform, the L-function has an Euler product

L_{f}(s) = ∏_{p}(1 − a_{p}p^{−s} + p^{k−1−2s})^{−1},

the product taken over all prime numbers. For each p let α_{p} and β_{p} be the roots of the polynomial x^{2} − a_{p}x + p^{k−1}. Then the symmetric square L-function D_{f}(s) attached to f is defined by the Euler product

D_{f}(s) = ∏_{p}[(1 − α_{p}^{2}p^{−s})(1 − β_{p}^{2}p^{−s})(1 − α_{p}β_{p}p^{−s})]^{−1}.

The Euler product converges only for ℜ(s) sufficiently large, but there is again a holomorphic continuation to the whole complex plane (with a functional equation relating D_{f}(s) and D_{f}(2k − 1 − s)). This was proved first by Shimura [Sh], and later by Zagier [Za] using a different method.

Let (f, f) be the norm of f with respect to the Petersson inner product, and for odd r with 1 ≤ r ≤ k − 1 define

D_{f}^{∗}(r + k − 1) = D_{f}(r + k − 1)/(f, f)π^{2r+k−1}.

These are rational numbers and we now describe the calculation of these numbers, which is justified by the contents of [Za]. See especially formulas (5), (24) and (28) of that paper.

Let p_{k,r}(t, m) be the coefficient of x^{k−r−1} in (1 − tx + mx^{2})^{−r}. For fixed k and r let c_{0} = p_{k,r}(0, 1), c_{1} = p_{k,r}(1, 1) + p_{k,r}(−1, 1) and c_{2} = p_{k,r}(2, 1) + p_{k,r}(−2, 1). For D ≡ 0 or 1(mod4) let χ_{D} be the quadratic character associated to the quadratic order of discriminant D. Let L(s, χ_{D}) be the associated Dirichlet L-function. Let ζ(s) be the Riemann zeta function.

For r = 1 we have D_{f}^{∗}(k) = 2^{2k−1}/(k − 1)!. For odd r ≥ 3 we have

D_{f}^{∗}(r + k − 1) = −(4^{r+k−2}(k − r − 1)!/(r + k − 2)!(k − 2)!)β

where

β = c_{0}L(1 − r, χ_{−4}) + c_{1}L(1 − r, χ_{−3}) + c_{2}ζ(1 − 2r)

if r < k − 1, but for r = k − 1,

β = c_{0}L(1 − r, χ_{−4}) + c_{1}L(1 − r, χ_{−3}) + (c_{2} + 2k/B_{k})ζ(1 − 2r).

Recall that L(1 − r, χ) = −B_{r,χ}/r and ζ(1 − 2r) = −B_{2r}/2r, where, for a character χ of conductor m, the generalised Bernoulli number is B_{r,χ} = m^{r−1} ∑_{a=1}^{m} χ(a)B_{r}(a/m). The Bernoulli polynomials are defined by ∑ B_{n}(x)t^{n}/n! = te^{tx}/(e^{t} − 1), and the Bernoulli numbers are B_{n} = B_{n}(0).

These calculations were performed using the Maple computer package. The results are given in the next section.

3. The results

Here are the values of D_{f}^{∗}(r + k − 1) = D_{f}(r + k − 1)/(f, f)π^{2r+k−1} for odd r with 1 ≤ r ≤ k − 1.

k = 12

r = 1 2^{15}/3^{4}.5^{2}.7.11
3  2^{17}/3^{5}.5^{2}.7^{2}.11.13
5  2^{16}/3^{7}.5^{3}.7^{2}.11.13
7  2^{19}/3^{9}.5^{4}.7^{2}.11.13.17
9  2^{23}/3^{8}.5^{5}.7^{4}.11.13.17.19
11 2^{24}/3^{9}.5^{4}.7^{2}.11.13.17.19.23.691

k = 16

r = 1 2^{20}/3^{6}.5^{3}.7^{2}.11.13
3  2^{18}.179/3^{7}.5^{3}.7^{3}.11.13^{2}.17
5  2^{22}/3^{8}.5^{3}.7^{2}.11.13^{2}.17.19
7  2^{22}.23^{2}/3^{11}.5^{5}.7^{3}.11^{2}.13^{2}.17.19
9  2^{21}.2243/3^{13}.5^{5}.7^{3}.11^{2}.13^{2}.17.19.23
11 2^{24}.839/3^{12}.5^{8}.7^{5}.11^{3}.13^{2}.17.19.23
13 2^{28}.373/3^{14}.5^{8}.7^{5}.11^{3}.13^{2}.17.19.23
15 2^{30}/3^{13}.5^{6}.7^{2}.11.13^{2}.17.19.23.29.31.3617

k = 18

r = 1 2^{20}/3^{6}.5^{3}.7^{2}.11.13.17
3  2^{20}/3^{8}.5^{3}.7^{3}.13.17.19
5  2^{22}/3^{10}.5^{4}.7^{3}.11^{2}.17.19
7  2^{22}/3^{9}.5^{4}.7^{3}.11^{2}.13.17.19.23
9  2^{21}.43867/3^{12}.5^{7}.7^{5}.11^{2}.13^{2}.17.19.23
11 2^{24}.1951/3^{17}.5^{7}.7^{4}.11^{2}.13^{2}.17.19.23
13 2^{28}.19501/3^{18}.5^{9}.7^{5}.11^{2}.13^{2}.17.19.23.29
15 2^{30}.541.2879/3^{17}.5^{7}.7^{7}.11^{4}.13^{3}.17.19.23.29.31
17 2^{32}/3^{17}.5^{5}.7^{4}.11^{2}.13.17.19.23.29.31.43867

k = 20

r = 1 2^{23}/3^{8}.5^{3}.7^{2}.11.13.17.19
3  2^{25}/3^{10}.5^{4}.7^{3}.11.17^{2}.19
5  2^{27}/3^{12}.5^{4}.7^{2}.11.13.17^{2}.19.23
7  2^{26}.2593/3^{12}.5^{7}.7^{3}.11^{2}.13^{2}.17^{2}.19.23
9  2^{28}.8831/3^{15}.5^{7}.7^{5}.11^{2}.13^{2}.17^{2}.19.23
11 2^{27}.304793977/3^{19}.5^{8}.7^{5}.11^{3}.13^{3}.17^{2}.19.23.29
13 2^{25}.40706077/3^{20}.5^{9}.7^{5}.11^{2}.13^{2}.17^{2}.19.23.29.31
15 2^{28}.9385577/3^{19}.5^{9}.7^{7}.11^{3}.13^{2}.17^{2}.19.23.29.31
17 2^{32}.439367/3^{19}.5^{11}.7^{5}.11^{4}.13^{3}.17^{2}.19.23.29.31
19 2^{32}.71^{2}/3^{18}.5^{9}.7^{3}.11^{2}.13^{2}.17^{2}.19.23.29.31.37.283.617

k = 22

r = 1 2^{25}/3^{9}.5^{4}.7^{3}.11.13.17.19
3  2^{28}/3^{10}.5^{5}.7^{2}.11^{2}.17.19^{2}.23
5  2^{26}.59/3^{1}1.5^{6}.7^{3}.11^{2}.13.17.19^{2}.23
7  2^{28}.239/3^{15}.5^{8}.7^{3}.11^{2}.13^{2}.19^{2}.23
9  2^{28}.25537/3^{16}.5^{8}.7^{4}.11^{2}.13^{2}.17.19^{2}.23.29
11 2^{28}.131.593/3^{17}.5^{9}.7^{5}.11.13^{2}.17.19^{2}.23.29.31
13 2^{28}.2436904891/3^{20}.5^{10}.7^{5}.11^{4}.13^{2}.17^{2}.19^{2}.23.29.31
15 2^{32}.98513941/3^{22}.5^{10}.7^{6}.11^{4}.13^{2}.17^{2}.19^{2}.23.29.31
17 2^{34}.545715463/3^{21}.5^{12}.7^{7}.11^{4}.13^{2}.17^{2}.19^{2}.23.29.31.37
19 2^{37}.281.286397/3^{21}.5^{10}.7^{7}.11^{4}.13^{4}.17^{3}.19^{2}.23.29.31.37
21 2^{37}.61.103/3^{21}.5^{8}.7^{4}.11^{4}.13^{2}.17.19^{2}.23.29.31.37.41.131.593

k = 26

r = 1 2^{29}/3^{10}.5^{6}.7^{3}.11^{2}.13.17.19.23
3  2^{30}/3^{13}.5^{6}.7^{4}.11^{2}.13^{2}.23^{2}
5  2^{33}/3^{15}.5^{6}.7^{3}.11^{2}.13^{2}.19.23^{2}.29
7  2^{32}.3373/3^{16}.5^{8}.7^{4}.11^{3}.13^{2}.17.23^{2}.29.31
9  2^{31}.3308551/3^{17}.5^{8}.7^{6}.11^{3}.13^{2}.17^{2}.19.23^{2}.29.31
11 2^{30}.6560341/3^{20}.5^{10}.7^{5}.11^{3}.13^{2}.17^{2}.19.23^{2}.29.31
13 2^{35}.1559.657931/3^{22}.5^{10}.7^{6}.11^{4}.13^{2}.17^{2}.19^{2}.23^{2}.29.31.37
15 2^{32}.83.1681092571/3^{22}.5^{9}.7^{8}.11^{4}.13^{4}.17^{2}.19^{2}.23^{2}.29.31.37
17 2^{30}.1097.1249037.129901/3^{25}.5^{12}.7^{6}.11^{5}.13^{4}.17^{2}.19^{2}.23^{2}.29.31.37.41
19 2^{33}.78750222771431/3^{27}.5^{11}.7^{8}.11^{4}.13^{4}.17^{2}.19^{2}.23^{2}.29.31.37.41.43
21 2^{38}.47.160217.4157.54377/3^{27}.5^{14}.7^{10}.11^{5}.13^{4}.17^{2}.19^{2}.23^{2}.29.31.37.41.43
23 2^{39}.4598642018203/3^{27}.5^{12}7^{7}.11^{6}.13^{4}.17^{3}.19^{3}.23^{2}.29.31.41.43.47
25 2^{41}.163.187273/3^{26}.5^{10}.7^{7}.11^{4}.13^{2}.17^{2}.19.23^{2}.29.31.37.41.43.47.657931

4. An observation

Ramanujan’s famous congruence τ(n) ≡ σ_{11}(n)(mod 691) generalises to a_{n} ≡ σ_{k−1}(n)(mod p), where f = ∑ a_{n}q^{n} is the unique normalised cusp form (on SL_{2}(ℤ)) of weight k = 12, 16, 18, 20, 22 or 26 and p > k is a prime divisor of the Bernoulli number B_{k}. Such p are irregular primes. We shall refer to them as Eisenstein primes, since the q-expansion of f is congruent to that of an un-normalized Eisenstein series, modulo p. See [SwD] for a discussion of congruences, and [Ma] for a proof of the above congruences using periods of f. The Eisenstein primes are p = 691 (when k = 12), p = 3617 (when k = 16), p = 43867 (when k = 18), p = 283 or 617 (when k = 20), p = 131 or 593 (when k = 22) and p = 657931 (when k = 26).

Surveying the numbers in the previous section, one notices that for k = 18, 22 or 26, the numerator of D_{f}^{∗}((k/2) + k − 1) is divisible by the Eisenstein prime divisors of B_{k}. Looking at the formulas used to calculate D_{f}^{∗}(r + k − 1), there appears to be no elementary explanation for this observation. In fact, when we repeat the calculation with the inapplicable value k = 30, we do not find an appropriate irregular prime in the numerator, suggesting that there can be no elementary explanation we have overlooked.

Since r must be odd, it is natural that we should be looking at those values of k for which k/2 is odd, but the significance of the k/2 is that it is the central point of symmetry for the functional equation of L_{f}(s). It is precisely when k/2 is odd that the functional equation forces L_{f}(s) to have odd order of vanishing at s = k/2, and in particular to vanish there. We shall see later how this, together with the Ramanujan-style congruences, may be used to explain the observation.

5. Galois representations

Let f = ∑ a_{n}q^{n} be one of the normalised eigenforms already introduced. 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 ≠ l

2. If Frob_{p} is an arithmetic Frobenius element at p then the characteristic polynomial of Frob_{p}^{−1} acting on V_{l} is x^{2} − a_{p}x + p^{k−1}.

For N ≥ 5 let M_{N} be the modular scheme over ℤ[1/N] parametrising elliptic curves, each with a given basis for the group of points of order N. Let X_{N} be the universal elliptic curve over M_{N}. Let \overline{M_{N}} be the compactification of M_{N} and let \overline{X_{N}} be the universal generalised elliptic curve over \overline{M_{N}}. Let \overline{X_{N}}^{k−2} be the (k − 2)-fold fibre

product of \overline{X_{N}} over \overline{M_{N}} and let X = \overline{\overline{X_{N}}}^{k−2} be Deligne’s canonical desingularisation. X is proper and smooth over ℤ[1/N], though not geometrically irreducible.

We shall now fix N = 5. Following Scholl[Sc], V_{l} may be constructed as the l-adic realisation of a Grothendieck motive M. V_{l} = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℚ_{l}), where P_{f} is a suitable projector, in our case in the group ring of the automorphism group of X. Let V_{l}(j) = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℚ_{l}(j)), the Tate twist as a Galois representation. For each prime l let T_{l}(j) = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℤ_{l}(j))/tors. Let A_{l}(j) = V_{l}(j)/T_{l}(j).

For the symmetric square motive let V′_{l} = Sym^{2}(V_{l}), T′_{l} = Sym^{2}(T_{l}) and A′_{l} = V′_{l}/T′_{l}. These have rank 3. We sometimes drop the subscript when it is convenient to do so. For an integer j of the form r + k − 1 or k − r, with r odd and 1 ≤ r ≤ k − 1 define the set of global points Γ_{ℚ}(j) = ⊕_{l}H^{0}(ℚ, A′_{l}(j)).

Following [BK], for p ≠ l let

H_{f}^{1}(ℚ_{p}, V′_{l}(j)) = ker(H^{1}(D_{p}, V′_{l}(j)) → H^{1}(I_{p}, V′_{l}(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 = l let

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

There is a natural exact sequence

0 ⟶ T′_{l}(j) ⟶ V′_{l}(j) ⟶^{p} A′_{l}(j) ⟶ 0.

Let H_{f}^{1}(ℚ_{p}, A′_{l}(j)) = p_{∗}H_{f}^{1}(ℚ_{p}, V′_{l}(j)). Define the l-Selmer group H_{f}^{1}(ℚ, A′_{l}(j)) to be the subgroup of elements of H^{1}(ℚ, A′_{l}(j)) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, A′_{l}(j)) for all primes p. Define the Shafarevich-Tate group

\underline{III}(j) = ⊕_{l}H_{f}^{1}(ℚ, A′_{l}(j))/p_{∗}H_{f}^{1}(ℚ, V′_{l}(j))

Of course, we could also define all these things with V′_{l}, T′_{l} and A′_{l} replaced by V_{l}, T_{l} and A_{l}.

6. The Bloch-Kato conjecture

Let j be of the form r + k − 1 with r odd and 1 ≤ r ≤ k − 1. The Bloch-Kato conjecture for the motive Sym^{2}M(j) predicts that

D_{f}(j) = (∏_{p} c_{p}(j))vol_{∞}(j)#\underline{III}(j)/(#Γ_{ℚ}(j)#Γ_{ℚ}(2k − 1 − j)).

The c_{p}(j) and vol_{∞}(j) are defined similarly to those in Section 4 of [Du]. The Hodge filtration of the deRham realisation of the motive Sym^{2}M has F^{0} = Sym^{2}(P_{f}H_{dR}^{k−1}(X)), F^{1} = ... = F^{k−1} = P_{f}H_{dR}^{k−1}(X)P_{f}H^{0}(X, Ω^{1}), F^{k} = ... = F^{2k−2} = P_{f}H^{0}(X, Ω^{1}) ⊗ P_{f}H^{0}(X, Ω^{1}), F^{2k−1} = {0}.

When the coefficients are in ℤ_{p} with p > k, we showed in Section 7 of [Du] that P_{f}H^{0}(X, Ω^{1}) and P_{f}H^{1}(X, O_{X}) are torsion-free. It follows that the subfactors of the above Hodge filtration are also torsion-free (for ℤ_{p} coefficients with p > k). Since also the length of the Hodge filtration is 2k − 1, it then follows as in Section 8 of [Du] that c_{p}(j) = 1 for all primes p > 2k, by an application of Faltings’ comparison theorem [Fa] and Theorem 4.1(iii) of [BK]. Also, each c_{p}(j) (for p < 2k) is a power of p.

Since F^{k} = ... = F^{2k−2} and k ≤ j ≤ 2k − 2 with all the j of the same parity, it follows as in Section 4 of [Du] that the vol_{∞}(j) for different j are the same, up to an appropriate power of 2πi.

If the Bloch-Kato conjecture is true, then if we look at the prime factorisation of D_{f}^{∗}(j)/D_{f}^{∗}(j′), with j ≠ j′, any primes p > 2k must be accounted for by elements of order p in Shafarevich-Tate groups and/or groups of global points, for j and j′. Now suppose that a prime p > 2k occurs in the numerator of precisely one of the D_{f}^{∗}(j)’s. Then according to the Bloch-Kato conjecture, either \underline{III}(j) contains an element of order p or, for all other critical j′, p divides #Γ_{ℚ}(j′)#Γ_{ℚ}(2k − 1 − j′). But the ptorsion subgroup A′[p] cannot have so many twists with Galois-fixed lines, since it has at most three distinct composition factors as a Galois module. Therefore the Bloch-Kato conjecture predicts that there must be an element of order p in \underline{III}(j). In the above tables of values of D_{f}^{∗}(j), we marked in boldface those primes p for which we will construct the predicted elements of \underline{III}(j).

7. Heegner cycles

For a detailed description of the Heegner cycles of codimension k/2 on X, see [Ne2]. They project to Heegner divisors on the modular curve M_{N}, and involve products of graphs of endomorphisms of CM elliptic curves. Via the p-adic Abel-Jacobi map, a Heegner cycle gives rise to a class in H^{1}(ℚ, V_{p}(k/2)), (for any prime p), and Nekovár shows in [Ne2] that it is in fact in the subspace H_{f}^{1}(ℚ, V_{p}(k/2)).

Now suppose that k = 18, 22 or 26. The vanishing of the complex L-function L_{f}(s) at s = k/2 implies the vanishing of the p-adic L-function L_{p,f}(s) at s = k/2. (For the construction of this p-adic L-function see [MTT].) According to Theorem C of [Ne1], if the first derivative L′_{p,f}(s) does not vanish at s = k/2 then H_{f}^{1}(ℚ, V_{p}(k/2)) is one-dimensional, generated by the classes of Heegner cycles. We would like to check this non-vanishing for Eisenstein primes p. From now on, p will be one of these primes (p = 43867 when k = 18, 131 or 593 when k = 22 and 657931 when k = 26).

We refer to [MTT] for more details on the p-adic L-function and calculations with modular symbols which we will not explain fully here. What we need to show is that

∫_{ℤ_{p}^{∗}} x^{r} log_{p}(x)dμ_{f,α} ≠ 0,

where r = (k/2) − 1 (which is even), α is a root of X^{2} − a_{p}X + p^{k−1} chosen so that ord_{p}(α) < k − 1, log_{p} is the p-adic logarithm and dμ_{f,α} is a certain p-adic measure. Manin [Ma] showed that integrals such as this one converge when ord_{p}(α) = 0. Such an α exists in the ordinary case, when p does not divide a_{p}. In [AV] and [V] the condition is relaxed to ord_{p}(α) < k − 1 (see [MTT]), and there always exists such an α, but we are in the ordinary case anyway since a_{p} ≡ σ_{k−1}(p) ≡ 1(modp).

To cut a long story short, calculating with modular symbols and approximating the integrand by a step function locally constant on discs of radius p^{−2}, it suffices to show that

∑_{a,b=1}^{p−1} a^{2r}b[m_{1}^{r} + m_{2}^{r} + ... + 1^{r}] ≠ 0(modp).

Here we have used sequences of integers a_{0} > a_{1} > ... > a_{t} = 0 and m_{0} > m_{1} > ... > m_{t} = 1 (t depends on a and b) with m_{0} = p^{2}, 1 ≤ a_{0} < p^{2}, a_{0} ≡ a^{p}(1 + bp)(modp^{2}), a_{j}m_{j+1} ≡ 1(modm_{j}) and a_{j+1} = (a_{j}m_{j+1} − 1)/m_{j}. Alternatively we could take the m_{j} to be the denominators of the convergents to the continued fraction for a^{p}(1 + bp)/p^{2}.

This drastic simplification of the appropriate Riemann sum is made possible by the fact that p divides the even period ratios for f (see the table in [Ma]), a fact which was one of the main concerns of [Du], and is the basis of that proof of the Ramanujan-style congruences to which we have already referred.

We ran a simple Maple program to evaluate this sum modulo p. When k = 22 and p = 131 the answer came out to 12, after 129 seconds of computer time. When k = 22 and p = 593 it came out to 167, after 83 minutes of computer time. For p = 43867 or 657931 the computation would have taken millions of minutes, so we did not start it.

In any case, assume that we have a non-zero class c ∈ H^{1}(ℚ, A[p](k/2)) coming from the class of a Heegner cycle in H_{f}^{1}(ℚ, V_{p}(k/2)). Here A[p] is the p-torsion subgroup of A_{p}. (We know this assumption to be true when k = 22, p = 131 or 593.)

8. Ramanujan congruences and Shafarevich-Tate groups

For p as above, there is a congruence a_{n} ≡ σ_{k−1}(n)( mod p), for all n ≥ 1. It follows that for prime l ≠ p, the trace of Frob_{l}^{−1} acting on A[p] is 1 + l^{k−1}. It then follows from the Brauer-Nesbitt theorem that A[p] contains a Galois submodule isomorphic to either 𝔽_{p} or the twist 𝔽_{p}(1 − k). In [Du], assuming the latter led to an explanation (via Bloch-Kato) of the occurrence of Eisenstein primes in the numerators of even period ratios. We shall make the same assumption and see that it serves our purposes once again. This consistency strongly suggests that the assumption is in fact true. Now if A[p] has a submodule isomorphic to 𝔽_{p}(1 − k) then A′[p] has a submodule isomorphic to A[p](1−k). Twisting, A′[p]((k/2)+k −1) has a submodule isomorphic to A[p](k/2).

The element Frob_{l}^{−1} acts as l^{1−k−(k/2)} on the quotient Q, so H^{0}(ℚ, Q) is trivial and H^{1}(ℚ, A[p](k/2)) injects into H^{1}(ℚ, A′[p]((k/2) + k − 1)), sending the Heegner cycle class c to a non-zero class c′.

There is an exact sequence

0 ⟶ A′[p]((k/2) + k − 1) ⟶ A′((k/2) + k − 1) ⟶^{p}
A′((k/2) + k − 1) ⟶ 0

and H^{0}(ℚ, A′((k/2) + k − 1)) = 0 (since k > 2 so k − 1 ≠ k/2, and the composition factors of A′[p] are 𝔽_{p}, 𝔽_{p}(1−k) and 𝔽_{p}(2−2k)), so H^{1}(ℚ, A′[p]((k/2)+k −1)) injects into H^{1}(ℚ, A′((k/2)+k−1)), and we get a non-zero class d ∈ H^{1}(ℚ, A′((k/2)+k−1)). Since (k/2)+k−1 is a non-central critical point, conjecturally H_{f}^{1}(ℚ, V′((k/2)+k−1)) should be trivial. Admitting this assumption, to show we have constructed an element of order p in \underline{III}((k/2)+k−1) we just have to show that d ∈ H_{f}^{1}(ℚ, A′((k/2)+k−1)).

Notice that the occurrence in A′[p] of a submodule isomorphic to 𝔽_{p}(2 − 2k) fits well with Bloch-Kato and the Eisenstein prime in the denominator of D_{f}^{∗}(2k −2) (see Section 3). Of course, the presence of that Eisenstein prime has a simple explanation since there is a Bernoulli number in the calculation.

Notice also that, since (for k = 18, 22, 26) the numerator of D_{f}^{∗}((k/2)+k −1) is not divisible by the square of the Eisenstein prime p, there should not be any elements of order p in \underline{III}(k/2) (for M, not Sym^{2}(M)). Kolyvagin’s method [Ne2] is unable to confirm this for such exceptional p.

9. Checking the local conditions

Proposition 9.1. For all primes l ≠ p, res_{l}(d) ∈ H_{f}^{1}(ℚ_{l}, A′((k/2) + k − 1)).

Proof. We know that res_{l}(c) ∈ H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A[p](k/2)), so that res_{l}(c′) ∈ H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A′[p]((k/2) + k − 1)) and then res_{l}(d) ∈ H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A′((k/2) + k − 1)). But it follows from the fact that A′ is unramified at l that

H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A′((k/2) + k − 1)) = H_{f}^{1}(ℚ_{l}, A′((k/2) + k − 1)),

so we have what we want (see the proof of Theorem 3 of [Fl3]). ■

Incidentally, one may show exactly as in Lemma 4.1 of [Ne2] that H^{1}(ℚ_{l}, T′((k/2)+ k−1)) = H^{1}(ℚ_{l}^{ur}/ℚ, T′((k/2)+k−1)), where ℚ_{l}^{ur} is the maximal unramified extension of ℚ_{l}. Since T′((k/2) + k − 1) is unramified at l ≠ p,

H^{1}(ℚ_{l}^{ur}/ℚ, T′((k/2) + k − 1)) ≃ T′((k/2) + k − 1)/(1 − Frob_{l})T′((k/2) + k − 1),

which is finite by the Weil conjectures. It follows that H^{1}(ℚ_{l}, V′((k/2) + k − 1)) = 0, hence H_{f}^{1}(ℚ_{l}, A′((k/2) + k − 1)) is actually trivial.

Proposition 9.2. res_{p}(d) ∈ H_{f}^{1}(ℚ_{p}, A′((k/2) + k − 1)).

Proof. In Lemma 4.4 of [BK], a cohomological functor {h^{i}}_{i≥0} is constructed on the Fontaine-Lafaille category of filtered Dieudonné modules over ℤ_{p}. h^{i}(D) = 0 for all i ≥ 2 and all D, and h^{i}(D) = Ext^{i}(1_{F D}, D) for all i and D, where 1_{F D} is the “unit” filtered Dieudonné module.

Let D = P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) and D′ = Sym^{2}(D). By Lemma 4.5 (c) of [BK], (and the application of [Fa] described in Section 8 of [Du]),

h^{1}(D) ≃ H_{e}^{1}(ℚ_{p}, T),

where

H_{e}^{1}(ℚ_{p}, T) = ker(H^{1}(ℚ_{p}, T) → H^{1}(ℚ_{p}, V)/H_{e}^{1}(ℚ_{p}, V))

and

H_{e}^{1}(ℚ_{p}, V) = ker(H^{1}(ℚ_{p}, V) → H^{1}(ℚ_{p}, B_{cris}^{f=1} ⊗ V)).

(Note that D and D′ are torsion-free, by Section 7 of [Du].)

For an integer j let D(j) be D with the Hodge filtration shifted by j. Then h^{1}(D(j)) ≃ H_{e}^{1}(ℚ_{p}, T (j)), etc. By Corollary 3.8.4 of [BK],

H_{f}^{1}(ℚ_{p}, V (j))/H_{e}^{1}(ℚ_{p}, V (j)) ≃ D(j) ⊗ ℚ_{p}/(1 − f)D(j) ⊗ ℚ_{p},

where f is the Frobenius operator on crystalline cohomology. By 1.2.4(ii) of [Sc], and the Weil conjectures, H_{e}^{1}(ℚ_{p}, V (j)) = H_{f}^{1}(ℚ_{p}, V (j)), since j ≠ (k − 1)/2. Similarly H_{e}^{1}(ℚ_{p}, V′(j)) = H_{f}^{1}(ℚ_{p}, V′(j)) unless j = k − 1.

We have then h^{1}(D(k/2)) ≃ H_{f}^{1}(ℚ_{p}, T (k/2)) and h^{1}(D′((k/2) + k − 1)) ≃ H_{f}^{1}(ℚ_{p}, T′((k/2) + k − 1)). The exact sequence in the middle of page 366 of [BK] gives us a commutative diagram.

h^{1}(D(k/2)) ⟶^{p} h^{1}(D(k/2)) ⟶ h^{1}(D(k/2)/pD(k/2))
↓  ↓  ↓
H^{1}(ℚ_{p}, T (k/2)) ⟶ H^{1}(ℚ_{p}, T (k/2)) ⟶ H^{1}(ℚ_{p}, A[p](k/2))

The vertical arrows are all inclusions and we know that the image of h^{1}(D(k/2)) in H^{1}(ℚ_{p}, T (k/2)) is exactly H_{f}^{1}(ℚ_{p}, T (k/2)). The top right horizontal map is surjective since h^{2}(D(k/2)) = 0.

The Heegner cycle class c ∈ H^{1}(ℚ_{p}, A[p](k/2)) is in the image of H_{f}^{1}(ℚ_{p}, T (k/2)) (by [Ne2]) and therefore is in the image of h^{1}(D(k/2)/pD(k/2)). Recall that A′[p]((k/2) + k − 1) has a Galois submodule isomorphic to A[p](k/2). By the fullness of the Fontaine-Lafaille functor [FL] (see Theorem 4.3 of [BK]), D′((k/2)+k −1)/pD′((k/2)+k −1) has a subobject isomorphic to D(k/2)/pD(k/2).

It follows that the class c′ ∈ H^{1}(ℚ_{p}, A′[p]((k/2) + k − 1)) is in the image of h^{1}(D′((k/2) + k − 1)/pD′((k/2) + k − 1)) by the vertical map in the exact sequence analogous to the above. Since the map from h^{1}(D′((k/2) + k − 1)) to h^{1}(D′((k/2) + k − 1)/pD′((k/2) + k − 1)) is surjective, c′ lies in the image of H_{f}^{1}(ℚ_{p}, T ((k/2) + k − 1)). From this it follows that d ∈ H_{f}^{1}(ℚ_{p}, A′((k/2) + k − 1)), as desired. ■

10. Higher symmetric powers

Let f be one of the cuspidal eigenforms we are concerned with in this paper (level one, rational coefficients). For n ≥ 1 one may define an n^{th} symmetric power Lfunction

L_{n,f}(s) := ∏_{p} ∏_{i=0}^{n}[(1 − α_{p}^{i}β_{p}^{n−i}p^{−s})]^{−1}.

The Euler product converges for Re(s) > n(k − 1)/2 + 1 but conjecturally there is an analytic continuation to the whole complex plane, with a functional equation relating the values at s and (n − 1)(k − 1) + k − s. This is known for n ≤ 3, the case n = 3 being due to Garrett [Ga], who shows that the sign in the functional equation is −1.

This forces the symmetric cube L-function L_{3,f}(s) to vanish at the central point s = (k−1)+k/2. According to the conjecture of Beilinson and Bloch [B], there should consequently be a rationally defined, null-homologous algebraic cycle, of infinite order in the ((k − 1) + k/2)^{th} Chow group of the symmetric cube of the motive attached to f. An explicit construction of a candidate for such a cycle is easily obtained by generalising the modified diagonal cycle considered in [GS] (where k = 2).

The Abel-Jacobi image of this cycle should give us (for any prime p), a non-zero element of H_{f}^{1}(ℚ, Sym^{3}(A)(k − 1 + (k/2)). Admitting this, in the case where p is an Eisenstein prime (for any k), arguments analogous to those of Sections 8 and 9 will then give us a non-zero element of H_{f}^{1}(ℚ, Sym^{4}(A)(2(k − 1) + (k/2))), which ought to be isomorphic to the appropriate Shafarevich-Tate group.

The right-of-centre critical points for L_{4,f}(s) are of the form s = 2(k − 1) + r with even 2 ≤ r ≤ k − 2. Therefore when k/2 is even (i.e. for k = 12, 16 or 20), the Bloch-Kato conjecture leads us to expect that the ratio of L_{4,f}(2(k − 1) + (k/2)) to any other critical value is an appropriate power of π times a rational number with p in the numerator.

For k = 12 (f = Δ), highly accurate computations of approximate critical values for L_{4,f}(s) were made, using over a thousand terms of the Euler product. The results of these computations, made by G. Köckritz and R. Schillo on an IBM 370/168, were reported in [Za]. The ratios of the approximate critical values turned out to be extremely close to powers of π times fairly simple rational numbers. It is very nice to see that in these rational numbers, the factor of p = 691 is there, exactly where we expect it to be. See the entry for s = 28 in the table at the end of Section 1 of [Za].

In general, when n is odd the critical points are s = ((n − 1)/2)(k − 1) + r with integer 1 ≤ r ≤ k − 1. When n is even the right-of-centre critical points are even integers of the form s = (n/2)(k − 1) + r with 1 ≤ r ≤ k − 1. Thus, r is odd when n ≡ 2(mod4), but even when n ≡ 0(mod4).

For any k = 12, 16, 18, 20, 22 or 26 and Eisenstein p, we seem to have an element of order p in the Shafarevich-Tate group \underline{III}(2(k − 1) + (k/2)) for Sym^{4}(M). Using Flach’s generalisation of the Cassels-Tate pairing [Fl3], we can reflect this across the central point to get an element of order p in \underline{III}(k − 1 + (k/2)). Then, using twice the Ramanujan congruence trick of Section 8, we (hopefully) get an element of order p in \underline{III}(3(k − 1) + (k/2)) for Sym^{6}(M). Clearly, repetition of this process should give elements of order p in \underline{III}((n/2)(k − 1) + (k/2)) for Sym^{n}(M), for any even n ≥ 2. Therefore, when n/2 and k/2 have the same parity we expect the ratio of L_{n,f}((n/2)(k − 1) + (k/2)) to the other critical values to be an appropriate power of π times a rational number with p dividing the numerator.

Using the table in 5.3 of [De2], we find that for odd n, the sign in the functional equation of L_{n,f}(s) is predicted to be −1 precisely when n ≡ 3( mod 8), or n ≡ 1( mod 8) and k/2 is odd, or n ≡ 5(mod8) and k/2 is even. Hence when n ≡ 5(mod8) or n ≡ 7(mod8) we cannot be led to our expectations concerning divisibility of critical values for Sym^{n+1}(M) more directly by using cycles for Sym^{n}(M).

11. A tensor product L-function

We have no experimental evidence for the above when n > 4 or k > 12. A recurrence relation described in [Wa] allows the first N coefficients of Δ to be computed in time which grows like N^{3/2}. But for n = 8 and k = 12, too many terms of the Euler product are required to get the necessary accuracy. For k > 12, Δ must be multiplied by Eisenstein series, and the time to get the first N coefficients grows like N^{2}. This makes it too difficult to get enough terms of the Euler product, even when n = 4. (Using all primes less than N=3000 was not enough.) However, there is one more place to look for evidence of a similar phenomenon, where it turns out that the computational difficulties are not too great.

Consider the forms of weights k = 12 and k′ = 26 and form their tensor product L-function

L(s) := ∏_{p}[(1 − αα′p^{−s})(1 − αβ′p^{−s})(1 − βα′p^{−s})(1 − ββ′p^{−s})]^{−1},

with the obvious notation for roots of Hecke polynomials.

The critical points for the associated motive are integers s such that k − 1 < s ≤ k′ − 1. We have chosen the only values of k and k′ such that s = k − 1 + (k′/2) is critical. Using Heegner cycles for k′ = 26 and the Ramanujan congruence for k = 12, we would expect that π^{2}L(24)/L(25) is a rational number with p = 691 in the numerator.

Using Maple to multiply together all terms of the Euler product for primes less than 1000, we found an approximation whose continued fraction has partial quotients

[9, 1, 6, 1, 3, 1, 1, 22253370239262, ...].

This suggests that the exact value has continued fraction [9, 1, 6, 1, 3, 2]. This rational number is precisely 691/70.

Actually, there is another method which will recover this rational number 691/70 without any need for approximation. One uses the identity

L(s) = ζ(2s + 2 − k − k′)D(s, f_{k}, f_{k′}),

where ζ is the Riemann zeta function and

D(s, f_{k}, f_{k′}) = ∑_{n=1}^{∞} a_{n}b_{n}n^{−s}

is the Rankin convolution. Now when s = (k′/2) + k − 1 we find 2s + 2 − k − k′ = k, and ζ(k) gives us the Eisenstein prime factor we are looking for. To show that it is not cancelled by a factor in the value of the Rankin convolution at another point, one may use Proposition 6 in §5 of [Za].

For the symmetric square,

D_{f}(s) = (ζ(2s + 2 − 2k)/ζ(s + 1 − k)) ∑_{n=1}^{∞} a_{n}^{2}n^{−s},

but a similar explanation is not possible, otherwise the values of the Riemann zeta function would put other irregular primes into various ratios of critical values, and we have seen that they do not occur. (The difference is that a theta series, as well as an Eisenstein series, occurs in the Rankin-Selberg integral for ∑_{n=1}^{∞} a_{n}^{2}n^{−s}.)

12. Klingen-Eisenstein series and Kurokawa’s congruences

Let f be one of the modular forms considered in this paper. Associated to f is its Klingen-Eisenstein series [f], a non-cuspidal Siegel modular eigenform of weight k and degree two. For us, the important fact about the relation between f and [f] is that, if L_{[f]}(s) is the L-function naturally associated with the eigenform [f], then L_{[f]}(s) = L_{f}(s)L_{f}(k − 2 − s).

Kurokawa proved [Ku1] that when k = 20 there exists a cuspidal Siegel eigenform F, of weight k and degree two, such that the Hecke eigenvalues of F and [f] are congruent modulo 71^{2}. Mizumoto proved [Mi1] that when k = 22 there exists a cuspidal Siegel eigenform F, of weight k and degree two, such that the Hecke eigenvalues of F and [f] are congruent modulo 61.103. In both cases, the Fourier coefficients and Hecke eigenvalues of F are all rational. In the case k = 26 we do not know the multiplicity-one result which would be needed in order to apply Mizumoto’s general theorem. The congruences result from the appearence of D_{f}(2k − 2) in the denominator of an expression for the Fourier coefficients of [f] [Mi2].

Let p be 71 (when k = 20) or 61 or 103 (when k = 22), and let F be as above. Thanks to Weissauer [We] we know there is a continuous representation

θ_{p} : Gal(\overline{ℚ}/ℚ) → Aut(W_{p})

(W_{p} is a four-dimensional vector space over ℚ_{p}), such that 1. θ_{p} is unramified at l for all primes l ≠ p;

2. If Frob_{l} is a Frobenius element at l then the characteristic polynomial of Frob_{l}^{−1} acting on W_{p} is the degree four polynomial with roots reciprocal to those of the polynomial associated to the l^{th} Euler factor of L_{F} (s).

(This θ_{p} exists for any prime p, not just our special choices.)

The congruence between the Hecke eigenvalues of F and f, and the equality L_{[f]}(s) = L_{f}(s)L_{f}(k − 2 − s), force us to the conclusion that the reduction modulo p of any Galois-invariant ℤ_{p}-lattice in W_{p} has composition factors isomorphic to A[p] and A[p](2 − k). Note that these factors are irreducible, since p is not a congruence prime for f, and that we have used the Brauer-Nesbitt theorem.

We make two assumptions about θ_{p} which we hope are reasonable.

1. θ_{p} is absolutely irreducible.

2. With respect to some symplectic form on W_{p}, the image of θ_{p} is contained in

the group of symplectic similitudes GSp_{4}(ℚ_{p}). The second assumption is only possible because p is an “exceptional” prime for F.

Proposition 12.1. Admitting the above assumptions, there exists a Galois-stable ℤ_{p} lattice S_{p} in W_{p} such that if B := W_{p}/S_{p} then there is a non-split exact sequence of 𝔽_{p}[Gal(\overline{ℚ}/ℚ)]-modules

0 ⟶ A[p](2 − k) ⟶^{i} B[p] ⟶^{π} A[p] ⟶ 0.

Moreover, letting s be a section for π as a map of 𝔽_{p}-vector spaces, there exists a skew-symmetric form ϕ on B[p] such that, for all g ∈ Gal(\overline{ℚ}/ℚ), and v, w ∈ B[p], ϕ(g.v, g.w) = χ(g)ϕ(v, w). Here, χ is the (2k − 3)^{rd} power of the p-cyclotomic character. Further, i(A[p](2 − k)) and s(A[p]) are isotropic subspaces for ϕ. There exists a basis {v_{1}, v_{2}} for A[p] with dual basis {w_{1}, w_{2}} for A[p](k−1) such that, when considered as bases for i(A[p](2−k)) and s(A[p]) respectively, we have ϕ(v_{i}, w_{j}) = δ_{ij}.

All this follows from Proposition 1.1 of [Ur]. Now, following Urban (Section 3.4 of [Ur]) we define a cocycle C : Gal(\overline{ℚ}/ℚ) → Hom_{𝔽_{p}}(A[p], A[p](2 − k)) by

C(g)(w) = i^{−1}(s(w) − g.s(g^{−1}.w)).

C is not a coboundary since the above exact sequence is non-split. The choice of section s does not affect the cohomology class of C. Since A[p] is dual to A[p](k − 1) as a Galois module, we have a non-zero element c ∈ H^{1}(ℚ, A[p] ⊗ A[p](1)) (because (k − 1) + (2 − k) = 1). In fact, just as in 3.4 of [Ur] (proof of Fact 2), c ∈ H^{1}(ℚ, (Sym^{2}A[p])(1)) = H^{1}(ℚ, A′[p](1)).

Proposition 12.2. c gives us a non-zero class d ∈ H^{1}(ℚ, A′(1)).

Proof. We need to show that H^{0}(ℚ, A′(1)) is trivial, so that the map from H^{1}(ℚ, A′[p](1)) to H^{1}(ℚ, A′(1)) is injective. It suffices to prove that H^{0}(ℚ, (A[p] ⊗ A[p])(1)) is trivial. Bearing in mind that A[p] is dual to A[p](k − 1), an element of H^{0}(ℚ, (A[p]⊗A[p])(1)) gives a Galois-equivariant map from A[p] to A[p](2−k), which must be trivial since these are non-isomorphic irreducible 𝔽_{p}[Gal(\overline{ℚ}/ℚ)]-modules. ■

Proposition 12.3. For any prime l ≠ p, res_{l}(d) ∈ H_{f}^{1}(ℚ_{l}, A′(1)).

Proof. For any prime l ≠ p, the restriction of c to H^{1}(I_{l}, A′[p](1)) is trivial, where I_{l} is an inertia group at l. This simply follows from the fact that A[p](2 − k), B[p] and A[p] are trivial 𝔽_{p}[I_{l}]-modules, so the exact sequence giving rise to c splits as a sequence of 𝔽_{p}[I_{l}]-modules. Hence res_{l}(c) ∈ H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A′[p](1)) so res_{l}(d) ∈ H^{1}(ℚ_{l}^{ur}/ℚ_{l}, A′(1)), which is the same as H_{f}^{1}(ℚ_{l}, A′(1)) since A′ is unramified at l (see the proof of Theorem 3 of [Fl3]). ■

Though we have shown that d satisfies the local conditions at all primes l ≠ p, we are unable to deal with the local condition at p. But if d does give us an element of order p in \underline{III}(1) for Sym^{2}(M), then [Fl3] allows us to reflect across the central point k − \frac{1}{2} and get an element of order p in \underline{III}(2k − 2). Recall that the existence of such an element was suggested by the Bloch-Kato conjecture and the results in Section 3.

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Merton College, Oxford, OX1 4JD, U.K.

  E-mail address: dummigan@maths.ox.ac.uk