  We study probability measures corresponding to families of abelian varieties
over a finite field. These measures play an important role in the Tsfasman-
Vladuts theory of asymptotic zeta-functions defining completely the limit
zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on
conjugate algebraic integers, described the possible set of measures than can
correspond to families of abelian varieties over a finite field. The problem
whether all such measures actually occur was left open. Moreover, Serre
supposed that not all such measures correspond to abelian varieties (for
example, the Lebesgue measure on a segment). Here we settle Serre's problem
proving that Serre conditions are sufficient, and thus describe completely the
set of measures corresponding to abelian varieties.
