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

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

2012-03-07

arxiv.org/abs/1203.1476v2

Mathematics

Number Theory

minor edits, 42 pages

Scientific paper

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 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 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.

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