Estimates for the number of rational points on simple abelian varieties over finite fields

Borys Kadets*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let A be a simple Abelian variety of dimension g over the field Fq. The paper provides improvements on the Weil estimates for the size of A(Fq). For an arbitrary value of q we prove (⌊(q-1)2⌋+1)g≤#A(Fq)≤(⌈(q+1)2⌉-1)g holds with finitely many exceptions. We compute improved bounds for various small values of q. For instance, the Weil bounds for q= 3 , 4 give a trivial estimate # A(Fq) ≥ 1 ; we prove # A(F3) ≥ 1. 359 g and # A(F4) ≥ 2. 275 g hold with finitely many exceptions. We use these results to give some estimates for the size of the rational 2-torsion subgroup A(Fq) [2] for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.

Original languageEnglish
Pages (from-to)465-473
Number of pages9
JournalMathematische Zeitschrift
Volume297
Issue number1-2
DOIs
StatePublished - Feb 2021
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

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