#### Vol. 8, No. 3, 2014

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

### Francesc Fité and Andrew V. Sutherland

Vol. 8 (2014), No. 3, 543–585
##### 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 $\overline{ℚ}$-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 $ℚ$-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 $\overline{ℚ}$-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.

##### Keywords
Sato–Tate, hyperelliptic curves, abelian surfaces, twists
##### Mathematical Subject Classification 2010
Primary: 11M50
Secondary: 14K15, 14G10, 11G20, 11G10