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

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 $\overline{\mathbb{Q}}$-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 $\mathbb{Q}$-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 $\mathbb{Q}$. 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 $\overline{\mathbb{Q}}$-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.