Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Santiago Arango-Pineros; Deewang Bhamidipati; Soumya Sankar

doi:10.1093/imrn/rnae148

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Santiago Arango-Pineros^*, Deewang Bhamidipati, Soumya Sankar

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

Given a g-dimensional abelian variety A over a finite field F_q, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most g. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre-Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre-Frobenius groups that occur for g ≤ 3. We also give a partial classification for simple ordinary abelian varieties of prime dimension g≥3.

Original language	English
Pages (from-to)	11989-12020
Number of pages	32
Journal	International Mathematics Research Notices
Volume	2024
Issue number	16
Early online date	8 Jul 2024
DOIs	https://doi.org/10.1093/imrn/rnae148
Publication status	Published - Aug 2024

Bibliographical note

Publisher Copyright:
© 2024 The Author(s). Published by Oxford University Press. All rights reserved.

Funding

We would like to thank David Zureick-Brown, Kiran Kedlaya, Francesc Fit\u00E9, Brandon Alberts, Edgar Costa, and Andrew Sutherland for useful conversations about this paper. We thank Yuri Zarhin for providing us with useful references, and Hendrik Lenstra for pointing out one of the missing cases in Section 4 . We would also like to thank Everett Howe for helping us with a missing piece of the puzzle in Theorem 3.2.1. This project started as part of the Rethinking Number Theory workshop in 2021. We thank the organizers of the workshop for giving us the opportunity and space to collaborate, and the funding sources for the workshop: AIM, the Number Theory Foundation, and the University of Wisconsin-Eau Claire Department of Mathematics. We are also grateful to Rachel Pries for her guidance at the beginning of the workshop, which helped launch this project. Finally, we thank the anonymous referee for the elucidating and pertinent suggestions that improved the exposition and results in the paper.

Funders	Funder number
University of Wisconsin-Eau Claire Department of Mathematics

Keywords

Characteristic-polynomials
Isogeny classes
Number
Surfaces

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10.1093/imrn/rnae148

rnae148Final published version, 4.92 MBLicence: Taverne

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Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract

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Keywords

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