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Abelian varieties over 𝔽 2 of prescribed order
Kiran S. Kedlaya1
1 Department of Mathematics, University of California San Diego, La Jolla, CA 92093, United States of America
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 43-58.

We prove that for every positive integer m, there exist infinitely many simple abelian varieties over 𝔽 2 of order m. The method is constructive, building on the work of Madan–Pal in the case m=1 to produce an explicit sequence of Weil polynomials giving rise to abelian varieties over 𝔽 2 of order m. This sequence itself depends on the choice of a suitable generalized binary representation of m; by making careful choices of this representation, we can ensure that the resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence of suitable irreducible factors.

Nous démontrons que pour tout entier positif m, il existe une infinité de variétés abéliennes simples sur 𝔽 2 d’ordre m. La méthode est constructive, se basant sur le travail de Madan–Pal pour le cas m=1 pour produire une suite explicite de polynômes de Weil donnant lieu à des variétés abéliennes sur 𝔽 2 d’ordre m. La suite elle-même dépend du choix d’une représentation binaire généralisée de m; par des choix soigneux de cette représentation, nous nous assurons que les polynômes qui en résultent ont des polygones de Newton 2-adiques garantissant l’existence de facteurs irréductibles convenables.

Received:
Revised:
Published online:
DOI: 10.5802/pmb.55

Kiran S. Kedlaya 1

1 Department of Mathematics, University of California San Diego, La Jolla, CA 92093, United States of America
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Abelian varieties over $\mathbb{F}_2$ of prescribed order},
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Kiran S. Kedlaya. Abelian varieties over $\mathbb{F}_2$ of prescribed order. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 43-58. doi : 10.5802/pmb.55. https://pmb.centre-mersenne.org/articles/10.5802/pmb.55/

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