## Departmental PhD Thesis Exam – Shervin Shahrokhi Tehrani

Everyone welcome.  Refreshments will be served in the Math Lounge before the exam.

Monday, July 30, 2012, 2:10 p.m., in BA 6183, 40 St. George Street

PhD Candidate: Shervin Shahrokhi Tehrani

Thesis Title: Non-holomorphic cuspidal automorphic forms of $GSp(4;\mathbb{A}$)
and the Hodge structure of Siegel threefolds
http://www.math.toronto.edu/shervin/thesis.pdf

Thesis Abstract:

Let $\mathbb{V}(\lambda)$ denote a local system of weight $\lambda$  on $X = A_{2,n}(\mathbb{C})$, where $X$ is the moduli space
of principle polarized abelian varieties of genus $2$ over $\mathbb{C}$ with fixed $n$-level structure. The
inner cohomology of $X$ with coefficients in $\mathbb{V}(\lambda)$, $H_!^3 (X,\mathbb{V} (\lambda))$, has a Hodge filtration of weight $3$. Each term of this Hodge filtration can be presented as space of cuspidal automorphic representations of genus $2$. We consider the purely non-holomorphic part of $H_{\rm{End}^s}^3 (X,\mathbb{V}(\lambda))$.

First of all we show that there is a non-zero subspace of $H_{\rm{End}^s}^3 (X,\mathbb{V}(\lambda))$ denoted by $V_\Theta(K)$, where $K$ is an open compact subgroup of $GSp(4,\mathbb{A})$, such that elements of $V_\Theta(K)$ are obtained by the global theta lifting of cuspidal automorphic representations of $GL(2)\times GL(2)/\mathbb{G}_m$. This means that there is a non-zero part of $H_{\rm{End}^s}^3 (X,\mathbb{V}(\lambda))$ which is endoscopic.

Secondly, we consider the local theta correspondence and fi nd an explicit answer for the level of lifted cuspidal automorphic representations to $GSp(4, F)$ over a non-archimedean local field F. Therefore, we can present an explicit way for finding a basis for $V_\Theta(K)$ for a fi xed level structure$K$.

There is a part of the Hodge structure that only contributes in $H_!^{(3,0)} (X,\mathbb{V}(\lambda))\oplus H_!^{(0,3)} (X,\mathbb{V}(\lambda))$. This part is endoscopic and coming from the Yoshida lift from $O(4)$.

Finally, in the case $X = A_2$, if $e_{\rm {endo}}(A2,\mathbb{V}(\lambda))$  denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge fi ltration), then we have

$$e_{\rm {endo}}(A_2,\mathbb{V}(\lambda)) = -s_{\lambda_1+\lambda_2+4} S [ \lambda_1 – \lambda_2 + 2]\mathbb{L}^{\lambda_2 +1},$$
where  $\lambda = (\lambda_1;, \lambda_2)$  and  is far from walls. Here $S[k]$ denotes the motive corresponded to $S_k$, the space of cuspidal automorphic forms of weight $k$ and trivial level, and $s_k = \dim (S_k)$.