Some evidence for the Coleman-Oort conjecture

TL;DR

This paper provides evidence supporting the Coleman-Oort conjecture by showing that, up to genus 100, all known families of Galois coverings that could produce counterexamples are the only ones, thus reinforcing the conjecture.

Contribution

The authors prove that no new families of Galois coverings, beyond known cases, exist for genera up to 100, strengthening the evidence for the Coleman-Oort conjecture.

Findings

No new counterexamples for g ≤ 100 found

Existing classified families are the only ones up to genus 100

Theoretical and computational methods exclude additional cases

Abstract

The Coleman-Oort conjecture says that for large $g$ there are no positive-dimensional Shimura subvarieties of $\mathsf{A}_g$ generically contained in the Jacobian locus. Counterexamples are known for $g\leq 7$. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: a) the Galois group is cyclic, b) it is abelian and the family is 1-dimensional, and c) $g\leq 9$. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for $g\leq 100$ there are no other families than those already known.