Abelian varieties over $\mathbb{F}_2$ of prescribed order

Kiran S. Kedlaya

doi:10.5802/pmb.55

Abelian varieties over

𝔽_{2}

of prescribed order

Kiran S. Kedlaya ¹

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.

Résumé
Abstract

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.

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.

Reçu le : 2021-09-13
Révisé le : 2022-08-05
Publié le : 2024-04-22

DOI : 10.5802/pmb.55

Affiliations des auteurs :

Kiran S. Kedlaya ¹

¹ Department of Mathematics, University of California San Diego, La Jolla, CA 92093, United States of America

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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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Algèbre et Théorie des Nombres