Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobians

Abstract

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^2=f(x)(x-\lambda )$ for some irreducible monic polynomial $f\in {\mathcal{O}}_K[x]$ of discriminant $\Delta$ and some element $\lambda \in {\mathcal{O}}_K$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda ))$ but not $(2\Delta )$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_2(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_2(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(\lambda ))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^2=f(x)(x-\lambda )(x-{\lambda }^{\prime })$ for distinct elements $\lambda ,{\lambda }^{\prime }\in K$. We then show that the hypothesis in the former statement holds for almost all $\lambda \in {\mathcal{O}}_K$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

arXiv comment16 pages, 6 sections, 1 figure, 23 references. Only a few typos have been corrected in this update, which is the exact version of the paper that has now been published in Mathematische Nachrichten