Images of 2-adic representations associated to hyperelliptic Jacobians

JOURNAL OF NUMBER THEORY

卷:151

Article

Yelton, Jeffrey

关键词: Galois group;    Elliptic curve;    Hyperelliptic curve;    Jacobian variety;    Tate module;   

DOI  :  10.1016/j.jnt.2014.10.020

来源: Elsevier

【 摘 要 】

Text. Let k be a subfield of C which contains all 2-power roots of unity, and let K = k(alpha(1), alpha(2), ... , alpha 2(g+1)), where the alpha(i)'s are independent and transcendental over k, and g is a positive integer. We investigate the image of the 2-adic Galois action associated to the Jacobian J of the hyperelliptic curve over K given by y(2) = Pi(2g+1)(i=1) (x - alpha(i)). Our main result states that the image of Galois in Sp(T-2(J)) coincides with the principal congruence subgroup Gamma (2) Sp(T-2(J)). As an application, we find generators for the algebraic extension K(J[4])/K generated by coordinates of the 4-torsion points of J. Video. For a video summary of this paper, please visit http://youtu.be/VXEGYxA6N8w. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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