Constructing hyperelliptic curves with surjective Galois representations

Abstract:

In this paper we show how to explicitly write down equations of hyperelliptic curves over $\mathbb {Q}$ such that for all odd primes $\ell$ the image of the mod $\ell$ Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the $\ell$-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod $\ell$ Galois representations.

The main result of the paper is the following. Suppose $n=2g+2$ is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than $n$ (this hypothesis is expected to hold for all $g\neq 0,1,2,3,4,5,7,$ and $13$). Then there is an explicit $N\in \mathbb {Z}$ and an explicit monic polynomial $f_0(x)\in \mathbb {Z}[x]$ of degree $n$, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $\operatorname {Gal}(\mathbb {Q}(J[\ell ])/\mathbb {Q})\cong \operatorname {GSp}_{2g}(\mathbb {F}_\ell )$ for all odd primes $\ell$ and $\operatorname {Gal}(\mathbb {Q}(J[2])/\mathbb {Q})\cong S_{2g+2}$, whenever $f(x)\in \mathbb {Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod {N}$ and with no roots of multiplicity greater than $2$ in $\overline {\mathbb {F}}_p$ for any $p\nmid N$.