Images of 2adic representations associated to hyperelliptic Jacobians
Abstract
Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$power roots of unity, and let $K = k(\alpha_{1}, \alpha_{2}, ... , \alpha_{2g + 1})$, 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 $y^{2} = \prod_{i = 1}^{2g + 1} (x  \alpha_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $\Gamma(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generators for the algebraic extension $K(J[4]) / K$ generated by coordinates of the $4$torsion points of $J$.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2668
 Bibcode:
 2014arXiv1410.2668Y
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 This paper is adapted from section 2 of my preprint at arXiv:1310.6447