Some evidence for the Coleman–Oort conjecture

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 $$g\le 7$$ . 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, or c) $$g\le 9$$ . By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for $$g\le 100$$ there are no other families than those already known.