Abelian varieties over \mathbbF₂ of prescribed order

2021/07/26 by Kiran S. Kedlaya, Kedlaya, Kiran S.
Computer Science · Mathematics · #11G10 #11G25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2107.12453

openalex publication_date 2021/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for every positive integer m, there exist infinitely many simple abelian varieties over \mathbbF₂ 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 \mathbbF₂ 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 the resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence of suitable irreducible factors.

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• Abelian varieties of prescribed order over finite fields
• Geometric decomposition of abelian varieties of order 1
• Weil polynomials of abelian varieties over finite fields with many rational points
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