Serre's theorem and measures corresponding to abelian varieties over finite fields

We study measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. Many years ago J.-P. Serre used a beautiful number-theoretic argument to prove the theorem limiting the set of measures that can actually occur on families of abelian varieties. For many years this theorem has not been published. First we present this theorem and its proof. Then we show that for jacobians of curves other methods characterize this set better, at least when the cardinality of the ground field is an even power of a prime. We are however very far from describing completely the set of measures corresponding to abelian varieties. In the appendix written by Yulia Kotelnikova, she proves that in the case of positive asymptotically exact families of Weil systems (in particular, in the case of asymptotically exact families of curves) Serre's theorem is true not only for polynomials $H(z) \in {\mathbb Z} [z]$ but for any $H(z) \in {\mathbb C} [z]$ with the absolute value of the leading coefficient at least $1$.