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2014 Sato–Tate distributions of twists of $y^2=x^5-x$ and $y^2=x^6+1$
Francesc Fité, Andrew Sutherland
Algebra Number Theory 8(3): 543-585 (2014). 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 ¯-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=x5x 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.

Citation

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Francesc Fité. Andrew Sutherland. "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

Information

Received: 20 November 2012; Revised: 22 August 2013; Accepted: 23 September 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1303.14051
MathSciNet: MR3218802
Digital Object Identifier: 10.2140/ant.2014.8.543

Subjects:
Primary: 11M50
Secondary: 11G10 , 11G20 , 14G10 , 14K15

Keywords: abelian surfaces , hyperelliptic curves , Sato–Tate , twists

Rights: Copyright © 2014 Mathematical Sciences Publishers

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