The integral monodromy of hyperelliptic and trielliptic curves

Abstract

We compute the \integ/\ell and \integ_\ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the \integ/\ell monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group \sp_{2g}(\integ/\ell). We prove that the \integ/\ell monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group \su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])