  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 $f_0(x)\in
\mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the
form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong
GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and
$Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, 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
not dividing N.
