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An Exploration of Degeneracy in Abelian Varieties of Fermat Type
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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- Award ID(s):
- 2201085
- PAR ID:
- 10524855
- Publisher / Repository:
- Experimental Mathematics
- Date Published:
- Journal Name:
- Experimental Mathematics
- ISSN:
- 1058-6458
- Page Range / eLocation ID:
- 1 to 17
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1080/10586458.2024.2362953
