On Shimura curves generated by families of Galois G-covers of curves

2023/11/23 by Mohajer, Abolfazl
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.13829

Abstract

In this paper we prove there are no families of cyclic \Z_n-covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus T_g of abelian varieties with g≥ 8 when n has a proper prime factor p≥ 7. This non-existence is also shown for families of \Z_n-covers of curves of any genus s and when n has a large enough prime number p (depending on s). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.

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