##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Grzegorz Banaszak

#### Adam Mickiewicz University in Poznan

## The algebraic Sato-Tate group and Sato Tate conjecture

##### Abstract:

Let $K$ be a number field and let $A$ be an abelian variety over $K$. In an effort of proper setting of the Sato-Tate conjecture concerning the equidistribution of Frobenius elements in the representation of the Galois group $G_K$ on the Tate module of $A$, one of attempts is the introduction of the algebraic Sato-Tate group $AST_{K}(A)$. Maximal compact subgroups of $AST_{K}(A)(\mathbb{C})$ are expected to be the key tool for the statement of the Sato-Tate conjecture for $A$. At the lecture, following an idea of J-P. Serre, an explicit construction of $AST_{K}(A)$ will be presented based on P. Deligne's motivic category for absolute Hodge cycles. I will discuss the arithmetic properties of $AST_{K}(A)$ along with explicit computations of $AST_{K}(A)$ for some families of abelian varieties. I will also explain how this construction extends to absolute Hodge cycles motives in the Deligne's motivic category for absolute Hodge cycles. This is joint work with Kiran Kedlaya.

Hosts: Cristian Popescu and Kiran Kedlaya

### April 30, 2015

### 2:00 PM

### AP&M 7321

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