Günter Harder

Congruences between modular forms of genus 1 and of genus 2

According to the general conjectures of Birch, Swinnerton-Dyer, Bloch, Kato, Deligne, Beilinson the so-called critical values of the L-function L(M, ν) of a motive M have arithmetical consequences for the underlying motive M. I want to explain how we can use the data from van der Geer’s talk to give some very strong evidence for these conjectures in the case that M = M(f) is the Scholl motive attached to a modular cusp form.

Let f be a cuspidal modular form of weight k for SL_{2}(ℤ). We assume that it is an eigenform for the Hecke algebra and for simplicity we assume that its Fourier coefficients are rational integers

f(q) = q + a_{2}q^{2} + a_{3}q^{3} + ...

This probably reduces us to the finitely many cases k = 12, 14, 16, 18, 20, 22, 26, but the conjectures below can be formulated also for arbitrary eigenforms f, then the formulation is more complicated. We can attach a motive M(f) to this modular form. The L-function attached to this motive is of course the classical Hecke L-series of f (modified by a Γfactor):

L(f, s) = L(M(f), s) = \frac{Γ(s)}{(2π)^{s}} ∑ \frac{a_{n}}{n^{s}} = \frac{Γ(s)}{(2π)^{s}} ∏_{p} \frac{1}{1 − a_{p}p^{−s} + p^{2k−1−2s}} .

It is holomorphic in the entire s-plane and satisfies the functional equation L(f, s) = L(k − s). By a theorem of Manin we know that we can define two periods Ω_{−}(f), Ω_{+}(f) such that for ν ∈ [k − 1, k − 2, …, 1]

L^{∗}(f, ν) = \frac{L(f, ν)}{Ω_{ϵ(ν)}(f)} ∈ ℚ.

A priori the periods Ω_{−}(f), Ω_{+}(f) are only defined up to an element in ℚ^{×}. But in [Ha-Ms, sect. 2] it is shown how we can use the theory of Whittaker models and integrality considerations to determine them up to an element in ℤ^{×} = {±1}. With these normalized periods we can show that the numbers L^{∗}(f, ν) have bounded denominators, actually they are even in ℤ except for the value L^{∗}(f, k − 1) = L^{∗}(f, 1), whose denominator is the numerator of ζ(1 − k).

To a pair k, ν with ν ∈ [k − 1, …, k/2] we determine l, m by k = l + m + 4, ν = l + 3. We consider the coefficient system 𝕍_{l,m} on A_{2}(ℂ) = Γ\ℍ_{2} as in van der Geer’s talk (Γ = Sp(4, ℤ) = Sp_{2}(ℤ)). We think of this coefficient system as obtained from a rational representation of GSp_{2} and we choose a Γ -invariant lattice V_{l,m} inside 𝕍_{l,m}. Such a lattice is unique up to homothety if we tensorize by a ring R = ℤ[1/(2 · 3 · . . . · r)] where we invert the “small” primes < max(l − m, m). The coefficient system V_{l,m} ⊗ R is again called V_{l,m}.

I want to formulate two conjectures. I say that a prime is large if it is > k. We consider the cohomology sequence (see [Ha-Ms, sect. 1], )

→ H_{c}^{3}(Γ\ℍ_{2}, V_{l,m}) → H^{3}(Γ\ℍ_{2}, V_{l,m}) → H^{3}(∂(Γ\ℍ_{2}), V_{l,m}) → .

This is a sequence of modules for the Hecke algebra. This Hecke algebra is essentially generated by Hecke operators

T_{p}^{(1)} ≃ ( \begin{matrix}p & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , T_{p}^{(2)} ≃ ( \begin{matrix}p^{2} & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

where p runs over all primes.

Under our assumption that k ≤ 26 we know that the modular form f defines a direct summand Rf ⊂ H^{3}(∂(Γ\ℍ_{2}), V_{l,m}), on which the two Hecke operators T_{p}^{(1)}, T_{p}^{(2)} act by the eigenvalues p^{m+1}+a_{p}+p^{l+2} and a_{p}(p^{m+1}+p^{l+2}), respectively. If we are in the regular case l > m > 0, the theory of Eisenstein series and the classical argument of Manin-Drinfeld give us a canonical lifting

Eis : Rf → H^{3}(Γ\ℍ_{2}, V_{l,m} ⊗ ℚ).

But in general Eis(Rf) ∉ Im(H^{3}(∂(Γ\ℍ_{2}), V_{l,m}) → H^{3}(∂(Γ\ℍ_{2}), V_{l,m} ⊗ ℚ), the denominator δ(f) is the smallest integer > 0 which multiplies Eis(f) into this image.

Now I can formulate :

Conjecture A: Let f be as above, let ν ∈ [k − 1, …, k/2]. If a large prime ℓ divides L^{∗}(f, ν) and if in addition ℓ does not divide L^{∗}(f, ν − 1) and ζ_{ℓ}(l − m + 1) then ℓ|δ(f).

This conjecture implies by a standard argument

Conjecture B: Under the same assumptions on f,ν and ℓ there exists a Siegel modular form F ∈ S_{l−m,m+3} (see v.d. Geer) which is an eigenform for the Hecke operators such that for all primes p

λ^{(1)}(F, p) ≡ p^{m+1} + a_{p} + p^{l+2} mod ℓ, λ^{(2)}(F, p) ≡ a_{p}(p^{m+1} + p^{l+2}) mod ℓ.

Conjecture B can now be checked numerically using the work of C. Faber and G. v.d. Geer.

The interesting modular forms are given by the forms f_{22} and f_{26} of weight 22 and 26, respectively. We find the following divisibilities by large primes

41|L^{∗}(f_{22}, 14), 43|L^{∗}(f_{26}, 23), 97|L^{∗}(f_{26}, 21), 29|L^{∗}(f_{26}, 19).

Hence we must find Siegel modular eigenforms in the spaces S_{4,10}, S_{20,5}, S_{16,7}, S_{12,9}. It turns out that all these spaces are of dimension 1, and the computer program of C. Faber and v.d.Geer produces the eigenvalues for the Hecke operators for all primes p ≤ 37. And indeed the congruences are true in all these cases. In fact, we also find that there are no other congruences modulo large primes (ℓ > k) if we choose other values of ν and k = 22 and 26.

If the Siegel modular form F is the generator of one of the above spaces we can again attach a motive M(F) and this gives us for all ℓ a 4-dimensional Galois representation

ρ(F) : Gal(ℚ̄/ℚ) → GL(M(F)_{et,ℓ}).

The cup product defines an alternating pairing < , > on M(F)_{et,ℓ} and hence the image of the Galois group is contained in a symplectic group. Then Conjecture A implies that after reduction mod ℓ the Galois module M(F)_{et,ℓ} ⊗𝔽_{ℓ} contains an invariant submodule 𝔽_{ℓ}(−ℓ− 2). Its orthogonal complement Y is of dimension 3 over 𝔽_{ℓ} and the module Y/𝔽_{ℓ}(−ℓ − 2) is M(f)_{et,ℓ}. The quotient M(F)_{et,ℓ} ⊗ 𝔽_{ℓ}/Y →^{∼}𝔽_{ℓ}(−m − 1). The semi-simplification of M(F)_{et,ℓ} ⊗ 𝔽_{ℓ}→^{∼}𝔽_{ℓ}(−ℓ − 2) ⊕ M(f)_{et,ℓ} ⊕ 𝔽_{ℓ}(−m − 1). The semi-simplification generates the field extension of ℓ− division points ℚ(M(f)_{et,ℓ}[ℓ]) and the action of Gal(ℚ̄/ℚ) on Y produces an unramified extension of this last field. It is visible in the matrix representation

ρ(F)|Y (σ) = ( \begin{matrix}α^{−l−2} & u(σ) & v(σ) \\ 0 & a(σ) & b(σ) \\ 0 & c(σ) & d(σ) \end{matrix} ) .

Here a, b, c, d are the matrix coefficients of ρ(f) ⊗ 𝔽_{ℓ}. One has to prove that we really get an extension, i.e. there are elements σ in the Galois group with ρ(f) ⊗ 𝔽_{ℓ}(σ) = Id but (u(σ), v(σ)) ≠ 0. Then we get an extension of ℚ(M(f)_{et,ℓ}[ℓ]) of the form predicted by the Birch, Swinnerton-Dyer, Bloch, Kato, Beilinson, Deligne conjectures.

For more details I refer to the very provisorial preprints on my home-page, which can be found under www.math.uni-bonn.de. These preprints are in the folder eisenstein in my ftp directory.

[Ha-koll] A congruence between a Siegel and an elliptic modular form

[Ha-Ms] Modular symbols and special values of automorphic L-functions

[Ha-MM] Modular construction of mixed motives II