The twisting Sato-Tate group of the curve $y^2=x^8-14x^4+1$

Abstract

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2=x^8-14x^4+1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2=x^8-14x^4+1$. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all $\mathbb{Q}$-twists of the curve $y^2=x^8-14x^4+1$.

34 pages, comments welcome