##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Francesc Fit\'e

#### Universitat Politecnica de Catalunya

## Sato-Tate groups and Galois endomorphism modules in genus 2

##### Abstract:

The (general) Sato-Tate Conjecture for an abelian variety A of dimension g defined over a number field k predicts the existence of a compact subgroup ST(A) of the unitary symplectic group USp(2g) that is supposed to govern the limiting distribution of the normalized Euler factors of A at the primes where it has good reduction. For the case g=1, there are 3 possibilities for ST(A) (only 2 of which occur for k=Q). In this talk, I will give a precise statement of the Sato-Tate Conjecture for the case of abelian surfaces, by showing that if g=2, then ST(A) is limited to a list of 52 possibilities, exactly 34 of which can occur if k=Q. Moreover, I will provide a characterization of ST(A) in terms of the Galois-module structure of the R-algebra of endomorphisms of A defined over a Galois closure of k. This is a joint work with K. S. Kedlaya, V. Rotger, and A. V. Sutherland

Host: Kiran Kedlaya

### March 1, 2012

### 1:00 PM

### AP&M 7218

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