##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Abhishek Saha

#### California Institute of Technology

## L-functions for GSp(4) X GL(2) and their critical values

##### Abstract:

If M is an arithmetic or geometric object, one can often attach to it a complex analytic function L(M,s). This is called the L-function of M and provides a powerful tool to study its various properties. We will consider the case when M= (F,g) where F is a Siegel modular form of genus two and g a classical modular form. In this setup we prove the following result: for s lying in a certain set of so called critical points, the corresponding values L(M,s) are algebraic numbers up to certain period integrals and behave nicely under automorphisms. This is predicted by an old conjecture of Deligne on motivic L-functions. The main tool used in our proof is an integral representation of the L-function involving the pullback of an Eisenstein series defined on a unitary group.

Host: Wee Teck Gan

### February 12, 2009

### 1:00 PM

### AP&M 7321

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