Finiteness of hyperelliptic and superelliptic curves with CM Jacobians

TL;DR

This paper proves that only finitely many superelliptic curves of fixed genus greater than or equal to 8 have CM Jacobians, using geometric and Higgs bundle techniques related to Shimura varieties.

Contribution

It establishes the finiteness of superelliptic curves with CM Jacobians for genus ≥8, advancing the understanding of the Coleman-Oort conjecture.

Findings

Finiteness of superelliptic curves with CM Jacobians for genus ≥8

Application of Shimura subvarieties and Higgs bundles in the proof

Supports the Coleman-Oort conjecture in the superelliptic case

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 $g\geq 8$ 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.