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 Fq. The paper provides improvements on the Weil estimates for the size of A(Fq). For an arbitrary value of q we prove ( (v q - 1)2 + 1)g = # A(Fq) = ( (v 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) = 1.359g and # A(F4) = 2.275g 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.