Complete Description of Measures Corresponding to Abelian Varieties over Finite Fields

TL;DR

This paper characterizes the probability measures associated with families of abelian varieties over finite fields, confirming Serre's conditions are sufficient to describe all such measures, thus completing the classification.

Contribution

It proves that Serre's conditions fully characterize measures corresponding to abelian varieties over finite fields, resolving an open problem.

Findings

Serre's conditions are sufficient for measures to correspond to abelian varieties.

Complete classification of measures associated with abelian varieties over finite fields.

Resolution of an open problem posed by Serre.

Abstract

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.