Cohomology of Local Systems and Siegel Modular Forms for Genus 2

Gerard van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam

Mathematische Arbeitstagung June 13-19, 2003 Max-Planck-Institut für Mathematik, Bonn, Germany

This is a report on joint work with Carel Faber.

Let ρ : GL(2, ℂ) → V be a finite-dimensional representation. A (genus 2) Siegel modular form of weight ρ is a holomorphic function f : ℍ_{2} → V satisfying

f((Aτ + B)(Cτ + D)^{−1}) = ρ(Cτ + D)f(τ)

for all (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(4, ℤ) and all τ ∈ ℍ_{2}, the Siegel upper half space of de-

gree 2. If ρ is irreducible, say ρ = Sym^{j} ⊗ det^{k} of the standard representation then we say that f has weight (j, k). We let M_{j,k} be the vector space of modular forms of weight (j, k) and S_{j,k} be the subspace of cusp forms. Since the representations of GL(2, ℂ) form a ring we can form the ring

R := ⊕_{j,k}M_{j,k}

of (vector valued) Siegel modular forms. It contains the subring R^{0} := ⊕_{k}M_{0,k} of classical Siegel modular forms. Igusa determined generators for R^{0}, but for the larger ring R very little is known. Satoh and Ibukiyama determined generators for the R^{0}-module ⊕_{k}M_{2,k}, but that is about it. But at least there is a formula for the dimension of M_{j,k}, due to Tsushima.

In the genus 1 case one can get information on modular forms using the moduli A_{1} (= j-line) of elliptic curves. Let π : X_{1} → A_{1} be the universal elliptic curve. On A_{1} we have a local system 𝕎 := R^{1}π_{∗}ℚ of rank 2. We put 𝕎^{k} := Sym^{k}𝕎. Then the Eichler-Shimura isomorphism tells us that

H_{c}^{1}(A_{1}, 𝕎^{k} ⊗ ℂ) ≅ ℂ ⊕ S_{k+2} ⊕ \overline{S_{k+2}}.

Since A_{1} is defined over ℤ we can get information on this cohomology by calculating the traces of Frobenius on the ℓ- adic variant of the local system 𝕎^{k} over a finite field 𝔽_{q}. Since knowledge of the cohomology H_{c}^{1}(A_{1}, 𝕎^{k}) for all k is equivalent to knowing the cohomology of M_{1,n}, the moduli of n-pointed elliptic curves, for all n (with trivial coefficients), we also could count #M_{1,n}(𝔽_{q}) and thus get information about cusp forms. For example, one finds the formula

#\overline{M_{1,11}}(𝔽_{p}) = p^{11} + 2037p^{10} + 213677p^{9} + … + 2037p + 1 − τ(p),

where τ(p) is the p-th Fouriercoefficient of the normalized cusp form Δ ∈ S_{12} of weight 12 and \overline{M_{1,11}} denotes the Deligne-Mumford compactification.

We define the motivic Euler characteristic e_{c}(X) of a variety X as

e_{c}(X) := ∑(−1)^{i}[H_{c}^{i}(X)],

where the class [H_{c}^{i}(X)] is taken in K_{0} of mixed Hodge structures or of effective Chow motives. The formula above then becomes

e_{c}(\overline{M_{1,11}}) = L^{11} + 2037L^{10} + … + 2037L + 1 − S[12],

where we now write L for the Tate motive h^{2}(ℙ^{1}) and S[k] for the motive of cusp forms of weight k on SL(2, ℤ). This motive (shown to exist by Scholl) has rank 2 dim S_{k} and weight k − 1.

We consider the map H_{c}^{∗}(A, 𝕎^{k}) → H^{∗}(A_{1}, 𝕎^{k}). The image is denoted by H_{!}^{∗}(A_{1}, 𝕎^{k}) and is called interior cohomology. It satisfies Poincaré duality.

For k even, k ≥ 2 we now can write e_{c}(A_{1}, 𝕎^{k}) = −S[k + 2] − 1 and e_{!}(A_{1}, 𝕎^{k}) = −S[k + 2].

We now go to genus 2. Let A_{2} be the moduli stack of principally polarized abelian surfaces. It contains the moduli M_{2} of genus 2 curves as an open part. Moreover, A_{2} comes with an universal abelian variety π : X_{2} → A_{2} and with a standard local system 𝕍 := R^{1}π_{∗}ℚ of rank 4. A pair of integers (l, m) with l ≥ m ≥ 0 determines an irreducible representation of Sp(4, ℚ) and in a similar way a local system 𝕍_{l,m} over A_{2}. For example, 𝕍 = 𝕍_{1,0} and 𝕍_{1,1} is the primitive part of ∧^{2}𝕍.

We are interested in the motivic Euler characteristic

e_{c}(A_{2}, 𝕍_{l,m}) = ∑_{i}(−1)^{i}[H_{c}^{i}(A_{2}, 𝕍_{l,m})].

Because of the hyperelliptic involution this vanishes if l + m is odd.

What we did was this. We compiled for every finite field 𝔽_{q} of cardinality q ≤ 37 (q ≠ 27) a list of Weil polynomials (characteristic polynomial of Frobenius on H_{et}^{1}(X, ℚ_{ℓ})) occuring for a curve X of genus 2 defined over 𝔽_{q} and we determined their frequencies as the curve X ran through M_{2}(𝔽_{q}). Moreover, we complemented this by also treating the curves consisting of two elliptic curves. Since A_{2} is a stack we need to take into account the order of the automorphism group of the objects that we count.

Independently, we also counted #M_{2,n}(𝔽_{p}) for all primes p ≤ 233 and all n. All this required quite some computer time, to put it mildly.

Since we are really interested in the genus 2 forms we must first identify the parts that come from genus 1. Therefore, we define the Eisenstein cohomology by

e_{Eis}(A_{2}, 𝕍_{l,m}) := e_{c}(A_{2}, 𝕍_{l,m}) − e_{!}(A_{2}, 𝕍_{l,m}).

For genus 1 we have e_{Eis}(A_{1}, 𝕎^{k}) = −1 for k ≥ 2 and even. Eisenstein cohomology for g ≥ 2 was studied by Harder, Schwermer and Pink. On the basis of our calculations we could guess a formula for the Eisenstein cohomology and we now think we can prove the following.

Theorem 1 For l > m > 0 the Euler characteristic e_{Eis}(A_{2}, 𝕍_{l,m}) of the Eisenstein cohomology is

−S[l + 3] − s_{l+m+4}L^{m+1} + S[m + 2] + s_{l−m+2} · 1 + \begin{cases}1 & l even \\ 0 & else \end{cases} ,

with s_{n} = dim S_{n}.

Faltings proved that H^{3}(A_{2}, 𝕍_{l,m}) and H_{c}^{3}(A_{2}, 𝕍_{l,m}) possess a Hodge filtration of weight ≥ l + m + 3 and ≤ l + m + 3; H_{!}^{3}(A_{2}, 𝕍_{l,m}) thus gets a Hodge filtration

(0) ⊂ F^{l+m+3} ⊂ F^{l+2} ⊂ F^{m+1} ⊂ F^{0} = H_{!}^{3}(A_{2}, 𝕍_{l,m}).

Moreover, if the weight (l, m) is regular (i.e. l > m > 0) then H_{!}^{i}(A_{2}, 𝕍_{l,m}) = (0) for i ≠ 3.

The Hodge step F^{l+m+3} has an interpretation in terms of Siegel modular forms

F^{l+m+3} ≅ S_{l−m,m+3}.

But there is still one more contribution from g = 1 that we must pin down. It stems from the endoscopic lift. Endoscopic lifting for the symplectic group has been studied by various people, like Kudla, Rallis, Weissauer. On the basis of our calculations we can make a guess for this contribution. The experts on endoscopic lifing should be able to prove this.

Conjecture 1 For (l, m) regular the endoscopic contribution is given by

e_{endo}(A_{2}, 𝕍_{l,m}) = −s_{l+m+4}S[l − m + 2]L^{m+1}.

The remainder now comes from genus 2 Siegel modular forms in a “true sense”:

S[l − m, m + 3] := H_{!}^{3}(A_{2}, 𝕍_{l,m}) − H_{endo}^{3}(A_{2}, 𝕍_{l,m})

should be a motive of rank 4 dim S_{l−m,m+3} and with a Hodge filtration with steps l + m + 3, l + 2, m + 1, 0. In relevant cases we can check this conjecture.

Our results enable us, assuming the conjecture on the endoscopic contribution, to calculate the traces λ(p) of the Hecke operator T (p) on S_{j,k} for all p ≤ 37.

Let us give an example. The first case which is interesting is S_{6,8}; here the dimension is 1. At our request Ibukiyama constructed a non-zero form in S_{6,8} as a theta series in 16 variables. But for g = 2 it is very difficult to calculate Hecke eigenvalues from Fourier coefficients.

We can calculate the characteristic polynomials at the primes p = 2 and p = 7. These are

X^{4} − 204800X^{2} + 274877906944

and

X^{4} + 107822000 X^{3} + 5939548912704750 X^{2}
+ 1229051676677303024546000 X
+ 129934811447123020117172145698449.

Finally, I want to illustrate this by giving some Hecke eigenvalues on the 1-dimensional spaces S_{8,8} and S_{12,6}.

\begin{matrix}p & λ(p) on S_{8,8} & λ(p) on S_{12,6} \\ 2 & 2^{6} · 3 · 7 & −2^{4} · 3 · 5 \\ 3 & −2^{3} · 3^{2} · 89 & 2^{3} · 3^{5} · 5 · 7 \\ 5 & −2^{2} · 3 · 5^{2} · 13^{2} · 607 & 2^{2} · 3 · 5^{2} · 7 · 79 · 89 \\ 7 & 2^{4} · 7 · 109 · 36973 & −2^{4} · 5^{2} · 7 · 119633 \\ 11 & 2^{3} · 3 · 4759 · 114089 & 2^{3} · 3 · 23 · 2267 · 2861 \\ 13 & −2^{2} · 13 · 17 · 109 · 3404113 & 2^{2} · 5 · 7 · 13 · 50083049 \\ 17 & 2^{2} · 3^{2} · 17 · 41 · 1307 · 168331 & −2^{2} · 3^{2} · 5 · 7 · 13 · 47 · 14320807 \\ 19 & −2^{3} · 5 · 74707 · 9443867 & −2^{3} · 5 · 7^{3} · 19 · 2377 · 35603 \end{matrix}

References

[1] Carel Faber, Gerard van der Geer: Sur la cohomologie des Systèmes Locaux sur les Espaces des Modules des Courbes de Genre 2 et des Surfaces

     Abéliennes. math.AG/0305094.