On a question of Ekedahl and Serre

TL;DR

This paper investigates the Ekedahl-Serre problem in the context of weakly special subvarieties in Siegel moduli space, proposing conjectures, providing examples, and establishing bounds on genera of certain curves.

Contribution

It formulates new questions and conjectures relating Ekedahl-Serre and Coleman-Oort problems, and offers bounds on genera of curves with specific Jacobian properties.

Findings

Upper bound on genera of curves with Jacobians isogenous to products of elliptic curves with Sato-Tate distribution

Weakly special subvarieties meet the Torelli locus only finitely often

Proposes conjecture linking Ekedahl-Serre and Coleman-Oort questions

Abstract

In this paper we study various aspects of the Ekedahl-Serre problem. We formulate questions of Ekedahl-Serre type and Coleman-Oort type for general weakly special subvarieties in the Siegel moduli space, propose a conjecture relating these two questions, and provide examples supporting these questions. The main new result is an upper bound of genera for curves over number fields whose Jacobians are isogeneous to products of elliptic curves satisfying the Sato-Tate equidistribution, and we also refine previous results showing that certain weakly special subvarieties only meet the open Torelli locus in at most finitely many points.