The Oort conjecture on Shimura curves in the Torelli locus of curves

TL;DR

This paper proves the Oort conjecture for specific types of Shimura curves in the Torelli locus of curves, establishing non-existence results for large genus and hyperelliptic cases, with implications for Jacobian decomposability.

Contribution

It confirms the Oort conjecture for Mumford type Shimura curves and certain isogenous abelian varieties, and for hyperelliptic curves of high genus, advancing understanding of Shimura curves in Torelli loci.

Findings

No Shimura curves of Mumford type in Torelli locus for g>11

No Shimura curves parameterizing certain abelian varieties for g>11

No Shimura curves in hyperelliptic Torelli locus for g>7

Abstract

Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.