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

Abstract

We consider the finite set of isogeny classes of [Formula: see text]-dimensional abelian varieties defined over the finite field [Formula: see text] with endomorphism algebra being a field. We prove that the class within this set whose varieties have the maximal number of rational points is unique, for any prime even power [Formula: see text] 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 [Formula: see text]. In dimension [Formula: see text], we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power [Formula: see text].