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

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 $g' \ge 1$ 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 $g \ge 2,4$ and for $g' > 2$ and $g \le 9$. In Frediani et al. Shimura varieties in the Torelli locus via Galois coverings, arXiv:1402.0973 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 Frediani et al. Shimura varieties in the Torelli locus via Galois coverings, arXiv:1402.0973. Finally we prove that if a family satisfies this sufficient condition with $g'\ge 1$, then $g \le 6g'+1$.