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 $\mathcal{\overline{M}}\_{2,20}$.



A frustrating cohomology class on the moduli of abelian surfaces
The answer to the main question above is indeed positive: the restriction map is nonzero for all $k \geq 2$. This was proved in Section 5 of my paper *Tautological rings of spaces of pointed genus two curves of compact type.* Compos. Math. 152 (2016), no. 7, 1398–1420.


The idea of the calculation is that these are Eisenstein cohomology classes for the Siegel parabolic subgroup, which corresponds to the 0-dimensional boundary strata in the Baily-Borel compactification. Therefore one can reduce to a calculation on a deleted neighborhood of the 0-dimensional part of the boundary. Such a deleted neighborhood can be described as a fiber bundle over a smaller locally symmetric space. In the end one ends up changing the problem from considering the submanifold $A\_1 \times A\_1 \subset A\_2$ to considering the submanifold inside $A\_1$ given by the image of the imaginary axis. This is equivalent to reformulating the problem as an assertion about classical modular symbols.


