Finiteness of hyperelliptic and superelliptic curves with CM Jacobians

Abstract

In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\ge 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.

References in corpus (2)

• The Oort conjecture on Shimura curves in the Torelli locus of curves
• A note on Shimura subvarieties in the hyperelliptic Torelli locus