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Degeneracy and Sato-Tate groups of \\(y^2=x^{p^2}-1\\)
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Degeneracy and Sato-Tate groups of \\(y^2=x^{p^2}-1\\)
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Paper
Degeneracy and Sato-Tate groups of \\(y^2=x^{p^2}-1\\)
2025
Overview
We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves \\(C_{p^2}: y^2=x^{p^2}-1\\), where \\(p\\) is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.
Publisher
Cornell University Library, arXiv.org
Subject
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