Frobenius distributions of low dimensional abelian varieties over finite fields
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909324494766080 |
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| author | Arango-Piñeros, Santiago Bhamidipati, Deewang Sankar, Soumya |
| author_facet | Arango-Piñeros, Santiago Bhamidipati, Deewang Sankar, Soumya |
| contents | Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, 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 $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g>3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02237 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Frobenius distributions of low dimensional abelian varieties over finite fields Arango-Piñeros, Santiago Bhamidipati, Deewang Sankar, Soumya Number Theory Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, 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 $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g>3$. |
| title | Frobenius distributions of low dimensional abelian varieties over finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.02237 |