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arXiv · 2211.03909

An Exploration of Degeneracy in Abelian Varieties of Fermat Type

Heidi Goodson
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Abstract

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 $y^2=x^m-1$. 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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Heidi Goodson. 2022-11-07. An Exploration of Degeneracy in Abelian Varieties of Fermat Type. https://doi.org/10.1080/10586458.2024.2362953

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