Sato-Tate distributions of twists of the Fermat and the Klein quartics

Abstract: I will start by reviewing the Sato-Tate conjecture and its generalisations. I will focus on the Sato-Tate distributions and computational aspects. After reviewing the elliptic curves case and the genus 2 case I will move to my results on genus 3 with F. Fité and A. Sutherland. In this common work we determine the Sato-Tate groups and the Sato-Tate distributions of the twists of the Fermat and Klein quartics, the two quartics with the largest automorphism group. This produces 60 different Sato-Tate distributions in genus 3, which are already enough to see new phenomenons: for instance in genus 3 the individual distribution of the coefficients of the normalized Euler factor do not determine the Sato-Tate distribution.

number theory

Audience: researchers in the topic

( paper )

Around Frobenius distributions and related topics II