Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1

TL;DR

This paper establishes the limiting distribution of Euler factors for certain abelian surfaces and proves the Sato-Tate Conjecture for Jacobians of specific twisted curves, revealing all possible Sato-Tate groups in these cases.

Contribution

It introduces the twisting Sato-Tate group and proves the Sato-Tate Conjecture for Jacobians of Q-twists of two specific curves, expanding understanding of Sato-Tate groups.

Findings

Determined the limiting distribution of normalized Euler factors for certain abelian surfaces.

Proved the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1.

Identified all 18 possible Sato-Tate groups for these Jacobians.

Abstract

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.