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

Let A be a simple Abelian variety of dimension g over the field F_q. The paper provides improvements on the Weil estimates for the size of A(F_q). 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(F_q) ≥ 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(F_q) [2] for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.