Sato-Tate distributions and Galois endomorphism modules in genus 2

TL;DR

This paper classifies the possible Sato-Tate groups for genus 2 abelian surfaces over number fields, relates them to Galois module structures of endomorphisms, and provides explicit examples with numerical verification.

Contribution

It provides a complete classification of Sato-Tate groups for genus 2 abelian surfaces and links these to Galois endomorphism modules, with explicit curve examples.

Findings

Limited to 55 possible Sato-Tate groups up to conjugacy.

At most 52 groups occur as Sato-Tate groups for suitable surfaces.

Numerical data matches theoretical Sato-Tate distribution expectations.

Abstract

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato-Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato-Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the R-algebra generated by endomorphisms of A_Qbar (the Galois type), and establish a matching with the classification of Sato-Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato-Tate groups for suitable A and k, of which 34 can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over Q whenever possible), and observe numerical agreement with the expected Sato-Tate distribution by comparing moment statistics.