The term 'degenerate' is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this talk we will examine degeneracy through these different but related lenses, specializing to Jacobians of hyperelliptic curves of the form y^2=x^m−1. We will explore the various forms of degeneracy for some examples, each illustrating different phenomena that can occur.
An Exploration of Degeneracy in Abelian Varieties of Fermat Type
Heidi Goodson, Brooklyn College, City University of New YorkAuthors: Heidi Goodson
2023 AWM Research Symposium
Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]