Sato-Tate distributions of twists of \(y^2=x^5-x\) and \(y^2=x^6+1\)
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Publication:2509411
DOI10.2140/ANT.2014.8.543zbMATH Open1303.14051arXiv1203.1476OpenAlexW3101956835MaRDI QIDQ2509411
Author name not available (Why is that?)
Publication date: 27 July 2014
Published in: (Search for Journal in Brave)
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 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.
Full work available at URL: https://arxiv.org/abs/1203.1476
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