##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Francesc Fite

#### University of Duisburg-Essen

## Fields of definition of CM elliptic $k$-curves and Sato-Tate groups of abelian surfaces

##### Abstract:

Let $A$ be an abelian variety defined over a number field $k$ that is isogenous over an algebraic closure to the power of an elliptic curve $E$. If $E$ does not have CM, by results of Ribet and Elkies concerning fields of definition of $k$-curves, $E$ is isogenous to an elliptic curve defined over a polyquadratic extension of $k$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato-Tate groups of abelian surfaces: First, we show that 18 of the 34 possible Sato-Tate groups of abelian surfaces over $\mathbb{Q}$, only occur among at most 51 $\overline{\mathbb{Q}}$-isogeny classes of abelian surfaces over $\mathbb{Q}$; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato-Tate groups of abelian surfaces. This is a joint work with Xevi Guitart. Preparatory talk: In the preparatory talk I plan to review very briefly basic definitions concerning abelian varieties necessary to introduce (in the main talk) the notion of abelian $k$-variety. I will also present the (general) Sato-Tate conjecture and show how it motivates the problem considered in the main talk.

Hosts: Alina Bucur and Kiran Kedlaya

### March 10, 2016

### 1:00 PM

### AP&M 7321

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