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# $\ell$-adic trace functions in analytic number theory

## Introductory Workshop: Analytic Number Theory February 06, 2017 - February 10, 2017

February 07, 2017 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
Location: MSRI: Simons Auditorium
Tags/Keywords
• Kloosterman sums

• Kloosterman sheaves

• monodromy

• moments of L-functions

• arithmetic functions in arithmetic progressions

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

#### $\Ell$-Adic Trace Functions In Analytic Number Theory

Abstract

Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$-adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)-Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$-adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.

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