Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

TL;DR

This paper classifies Galois covering families of curves that produce Shimura subvarieties in the moduli space of abelian varieties, identifying new examples and establishing bounds on genus based on the base curve's genus.

Contribution

It provides a complete classification of such Galois covering families satisfying a key condition for genera up to 9, including new examples and genus bounds.

Findings

Identified 6 families of Galois coverings with $g'=1$ and $g=2,3,4$.

Discovered 2 new Shimura subvarieties arising from these families.

Proved that for $g' eq 0$, the genus $g$ is bounded above by $6g'+1$.

Abstract

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.