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 $f:C\to C^{\prime }$ where $C^{\prime }$ is a smooth complex projective curve of genus $g^{\prime }\ge 1$ 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 $g^{\prime }=1,2$ and for all $g\ge 2,4$ and for $g^{\prime }>2$ and $g\le 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g^{\prime }=0$. Here we find 6 families of Galois coverings, all with $g^{\prime }=1$ and $g=2,3,4$ and we show that these are the only families with $g^{\prime }=1$ 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 $g^{\prime }\ge 1$, then $g\le 6g^{\prime }+1$.

arXiv comment18 pages, to appear in Geometriae Dedicata