Images of 2-adic 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 $\Gamma (2)⊲\mathrm{Sp}(T_2(J))$. As an application, we find generators for the algebraic extension $K(J[4])/K$ generated by coordinates of the $4$-torsion points of $J$.

arXiv commentThis paper is adapted from section 2 of my preprint at arXiv:1310.6447