# A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a Shimura variety to a sub-Shimura variety. However I formulate the question for the special case I am interested in.

Let $A_2$ be the moduli stack of principally polarized abelian surfaces. To the irreducible finite dimensional representation of $\mathrm{Sp}(4)$ of highest weight $a \geq b \geq 0$ we attach a a local system $V_{a,b}$ on $A_2$.

Suppose $(a,b) \neq (0,0)$. One can prove that $H^4_c(A_2,V_{a,b})$ vanishes unless $a=b$ is even, in which case $H^4_c(A_2,V_{2k,2k})$ is pure of Tate type and of the same dimension as the space of cusp forms of weight $4k+4$ for $\mathrm{SL}(2,\mathbf Z)$. The map $H^4_c \to H^4_{(2)}$ to the $L^2$-cohomology is an isomorphism. In terms of automorphic representations, these cohomology classes can be described as follows: for any level 1 cusp form $\pi$ on $\mathrm{GL}(2,\mathbf A)$ of weight $4k+4$ we consider the unique irreducible quotient of $$\mathrm{Ind}_{P(\mathbf A)}^{\mathrm{GSp}(4,\mathbf A)} \left( \vert \cdot \vert^{1/2} \pi \otimes \vert \cdot \vert^{-1/2} \right)$$ where $P$ denotes the Siegel parabolic subgroup (whose Levi factor is $\mathrm{GL}(2) \times \mathrm{GL}(1)$); this is a discrete automorphic representation for $\mathrm{GSp}(4)$ which contributes a Tate type class to the $L^2$-cohomology in degrees $2$ and $4$.

There is a map $\mathrm{Sym}^2(A_1) \hookrightarrow A_2$ given by taking a pair of elliptic curves to their product. We can also restrict $V_{a,b}$ to $\mathrm{Sym}^2(A_1)$. By determining the branching formula for $\mathrm{SL}(2)^2 \rtimes S_2 \subset \mathrm{Sp}(4)$ we find that the trivial local system occurs as a summand in the restriction of $V_{a,b}$ to $\mathrm{Sym}^2(A_1)$ if and only if $a=b$ is even, in which case it appears with multiplicity $1$. So $H^4_c(\mathrm{Sym}^2(A_1),V_{2k,2k})$ is also pure of Tate type but $1$-dimensional. Again we could think about $L^2$-cohomology and it would not make a difference.

MAIN QUESTION: Is the restriction map $H^4_c(A_2,V_{2k,2k}) \to H^4_c(\mathrm{Sym}^2(A_1),V_{2k,2k})$ nonzero for $k \geq 2$?

Any ideas or pointers at all would be appreciated. I am very ignorant about automorphic representations, Shimura varieties etc. and I am naively hoping that there exists some general method for answering question of this form.

This question arose from the paper http://arxiv.org/abs/1210.5761 . A positive answer would imply that all even cohomology of $\mathcal{\overline{M}}_{2,n}$ is tautological for $n < 20$, and that the Gorenstein conjecture fails on $\mathcal{\overline{M}}_{2,20}$.

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