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 \(g \le 6g'+1\).
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It is a pleasure to thank E. Colombo, A. Ghigi, G.P. Pirola and C. Gleissner for stimulating discussions.
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The first author was partially supported by PRIN 2012 MIUR ”Moduli, strutture geometriche e loro applicazioni” and by FIRB 2012 ”Moduli spaces and applications” . The third author was partially supported by PRIN 2010 MIUR “Geometria delle Varietà Algebriche”. The three authors were partially supported by INdAM (GNSAGA).
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Frediani, P., Penegini, M. & Porru, P. Shimura varieties in the Torelli locus via Galois coverings of elliptic curves. Geom Dedicata 181, 177–192 (2016). https://doi.org/10.1007/s10711-015-0118-0
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DOI: https://doi.org/10.1007/s10711-015-0118-0