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

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 -twists of the curves y2 = x5 x and y2 = x6 + 1, 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 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.

Sato–Tate, hyperelliptic curves, abelian surfaces, twists
Mathematical Subject Classification 2010
Primary: 11M50
Secondary: 14K15, 14G10, 11G20, 11G10
Received: 20 November 2012
Revised: 22 August 2013
Accepted: 23 September 2013
Published: 31 May 2014
Francesc Fité
Fakultät für Mathematik
Bielefeld Universität
P.O. Box 100131
D-33501 Bielefeld
Andrew V. Sutherland
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave.
Cambridge, MA 02139
United States