Weil polynomials of abelian varieties over finite fields with many rational points

2021/01/29 by Elena Berardini, Berardini, Elena, Alejandro J. Giangreco Maidana +1 · 1 citation
Computer Science · Mathematics · #11G10 #14G05 #14G15 #14K02 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2101.12664

openalex publication_date 2021/01/29 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider the finite set of isogeny classes of g-dimensional abelian varieties defined over the finite field \mathbbF_q with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal number of rational points is unique, for any prime even power q big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on g. In dimension 3, we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power q.

Cited by

• Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension g>1

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