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

Abstract

Let $A$ be a simple abelian variety of dimension $g$ over the field ${\mathbb{F}}_q$. The paper provides improvements on the Weil estimates for the size of $A({\mathbb{F}}_q)$. For an arbitrary value of $q$ we prove $(\lfloor (\sqrt{q}-1{)}^2\rfloor +1{)}^g⩽A({\mathbb{F}}_q)⩽(\lceil (\sqrt{q}+1{)}^2\rceil -1{)}^g$ 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 $A({\mathbb{F}}_q)⩾1$; we prove $A({\mathbb{F}}_3)⩾1.359^g$ and $A({\mathbb{F}}_4)⩾2.275^g$ hold with finitely many exceptions. We use these results to describe all abelian varieties over finite fields that have no new points in some finite field extension.

arXiv commentCorrected typos in the list of exceptional polynomials in Table 3