Sato–Tate distributions of twists of y2 = x5 − x and y2 = x6 + 1

Fité, Francesc; Sutherland, Andrew

doi:10.2140/ant.2014.8.543

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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

DOI: 10.2140/ant.2014.8.543

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 $\bar{ℚ}$ -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 $\bar{ℚ}$ -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

Milestones

Received: 20 November 2012

Revised: 22 August 2013

Accepted: 23 September 2013

Published: 31 May 2014

Authors

Francesc Fité
Fakultät für Mathematik Bielefeld Universität P.O. Box 100131 D-33501 Bielefeld Germany

Andrew V. Sutherland
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139 United States

Vol. 8, No. 3, 2014