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 ≥ 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.

Distribuation of CM points of an infinite series of complete Calabi-Yau moduli spaces

“…So if n ≥ 5, then g ≥ 9. Note the main result in [1] asserts that if g ≥ 8, then up to isomorphism, there exist at most finitely many superelliptic curves of genus g with CM Jacobian. Then the n ≥ 5 cases follows from this result and the following claim.…”