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

By P. Frediani, Matteo Penegini, Paola Porru. 2014

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We study Shimura subvarieties of $$\mathsf {A}_g$$Ag obtained from families of Galois coverings $$f: C \rightarrow C'$$f:C→C′ where $$C'$$C′ is a smooth complex projective curve of genus $$g' \ge 1$$g′≥1 and $$g= g(C)$$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 $$\mathsf {A}_g$$Ag for $$g' =1,2$$g′=1,2 and for all $$g \ge 2,4$$g≥2,4 and for $$g' > 2$$g′>2 and $$g \le 9$$g≤9. In Frediani et al. Shimura varieties in the Torelli locus via Galois coverings, arXiv:1402.0973 similar computations were done in the case $$g'=0$$g′=0. Here we find 6 families of Galois coverings, all with $$g' = 1$$g′=1 and $$g=2,3,4$$g=2,3,4 and we show that these are the only families with $$g'=1$$g′=1 satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $$\mathsf {A}_g$$Ag, while the other examples arise from certain Shimura subvarieties of $$\mathsf {A}_g$$Ag already obtained as families of Galois coverings of $$\mathbb {P}^1$$P1 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 $$g'\ge 1$$g′≥1, then $$g \le 6g'+1$$g≤6g′+1.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/69a7964e99607e2dacab719349c010168c7653e9