##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Luis Lomeli

#### University of Oklahoma

## The Langlands-Shahidi method for the classical groups over function fields and the Ramanujan conject

##### Abstract:

The Langlands-Shahidi method provides us with a constructive way of studying automorphic L-functions. For the classical groups over function fields we will present recent results that allow us to obtain applications towards global Langlands functoriality. This is done via the Converse Theorem of Piatetski-Shapiro, which we can apply since our automorphic L-functions have meromorphic continuation to rational functions and satisfy a functional equation. We lift globally generic cuspidal automorphic representations of a classical group to an appropriate general linear group. Then, we express the image of functoriality as an isobaric sum of cuspidal automorphic representations of general linear groups, where the symmetric and exterior square automorphic L-functions play a technical role. As a consequence, we can use the exact Ramanujan bounds of Laurent Lafforgue for GL(N) to prove the Ramanujan conjecture for the classical groups. Our results are currently complete for the split classical groups under the assumption that characteristic p is different than two.

Host: Claus Sorensen

### November 14, 2013

### 2:00 PM

### AP&M 7321

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