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Principal polarizations and Shimura data for families of cyclic covers of the projective line

Banff International Research Station for Mathematical Innovation and Discovery

Consider a family of degree m cyclic covers of the projective line, with any number of branch points and inertia type. The Jacobians of the curves in this family are abelian varieties having an automorphism of order m with a prescribed signature. For each such family, the signature determines a PEL-type Shimura variety. Under a condition on the class number of m, we determine the Hermitian form and Shimura datum of the component of the Shimura variety containing the Torelli locus. For the proof, we study the boundary of Hurwitz spaces, investigate narrow class numbers of real cyclotomic fields, and build on an algorithm of Van Wamelen about principal polarizations on abelian varieties with complex multiplication. This is joint work with Li, Mantovan, and Tang.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Colorado State University

2021-03-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77415

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/