Constructing hyperelliptic curves with surjective Galois representations

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-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 l 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 appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in $\mathbb{Z}$[x]$ofdegreen,suchthattheJacobian$J$ of every curve of the form $y^2=f(x)$ has $Gal($\mathbb{Q}$(J[l])/$\mathbb{Q}$)\cong GSp_{2g}($\mathbb{F}$l)$foralloddprimesland$Gal($\mathbb{Q}$(J[2])/$\mathbb{Q}$)\cong S{2g+2}$,whenever$f(x)\in$\mathbb{Z}$[x]$ismonicwith$f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in ${\bar{\mathbb{F}}}_p$ for any p not dividing N.