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arXiv · 1611.08477

Finiteness of hyperelliptic and superelliptic curves with CM Jacobians

Ke Chen·Xin Lu·Kang Zuo
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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.

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Ke Chen, Xin Lu, Kang Zuo. 2016-11-25. Finiteness of hyperelliptic and superelliptic curves with CM Jacobians. https://arxiv.org/abs/1611.08477

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