 
Summary: The integral monodromy of hyperelliptic and trielliptic
curves
Jeffrey D. Achter & Rachel Pries
Abstract
We compute the Z/ and Z monodromy of every irreducible component of the mod
uli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that
the Z/ monodromy of the moduli space of hyperelliptic curves of genus g is the sym
plectic group Sp2g(Z/ ). We prove that the Z/ monodromy of the moduli space of
trielliptic curves with signature (r, s) is the special unitary group SU(r,s)(Z/ Z[3]).
MSC 11G18, 14D05, 14H40
keywords monodromy, hyperelliptic, trigonal, moduli, Jacobian
1 Introduction
If C S is a relative smooth proper curve of genus g 1 over an irreducible base,
then the torsion of the relative Jacobian of C encodes important information about the
family. Suppose is invertible on S, and let s S be a geometric point. The fundamental
group 1(S, s) acts linearly on the fiber Pic0
(C)[ ]s
= (Z/ )2g, and one can consider the
mod monodromy representation associated to C:
CS, : 1(S, s)  Aut(Pic0
