Measures Corresponding to Curves and Abelian Varietes

We study measures corresponding to families of abelian varieties over a finite field. These measures play an important role in our 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. It happens 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.