We determine the limiting distribution of the normalized Euler factors of an abelian surface defined over a number field when is -isogenous to the square of an elliptic curve defined over with complex multiplication. As an application, we prove the Sato–Tate conjecture for Jacobians of -twists of the curves and , which give rise to 18 of the 34 possibilities for the Sato–Tate group of an abelian surface defined over . With twists of these two curves, one encounters, in fact, all of the 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.
"Sato–Tate distributions of twists of $y^2=x^5-x$ and $y^2=x^6+1$." Algebra Number Theory 8 (3) 543 - 585, 2014. https://doi.org/10.2140/ant.2014.8.543