## Number theory seminar in memory of F. Momose

### Computing Sato-Tate distributions of low genus curves.

#### Ponents

Andrew V. Sutherland

#### Resum

The generalized Sato-Tate conjecture predicts that the distribution
of normalized Euler factors of an abelian variety $A$ of dimension $g$ converges
to the distribution of characteristic polynomials in a certain compact Lie group
$ST_A$ (the Sato-Tate group of $A$) that is subgroup of $USp(2g)$ (the group of
$2g \times 2g$ complex matrices that are both unitary and symplectic). We have
developed a suite of computational tools to very efficiently compute, in the case
that $A$ is the Jacobian of a curve of genus $g \leq 2$, statistics that allow one to
to provisionally determine $ST_A$ , as well as techniques for then proving that this
provisional identification is correct. In this talk I will describe some of these
tools, which played a key role in the recently completed classification of the $52$
Sato-Tate groups that arise in genus $2$. Time permitting, I will also discuss the
prospects in genus $3$, where some preliminary work has begun. This is joint
work with Francesc Fité, Victor Rotger, and Kiran Kedlaya.

#### Fitxers

No hi ha fitxers per descarregar