Papers › An Exploration of Degeneracy in Abelian Varieties of Fermat Type
An Exploration of Degeneracy in Abelian Varieties of Fermat Type
Heidi Goodson
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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 article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form y²=xᵐ-1. We prove that the Jacobian of the curve is degenerate whenever m is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.
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