Abstract

Let A be an Abelian variety over a finite field Fq. We are interested in knowing the distribution of the groups A(Fq) of rational points on A as we run over all varieties defined over Fq. In particular, we want to show that they are in general not too "split". For the case of dimension 1 (elliptic curves) and dimension 2 (Abelian surfaces), there are some theoretical results due to David and her collaborators, but the general case is open.

We are interested in Abelian Varieties of dimension 3. We use Rybakov's criterion, which relates the existence of a given abstract group as the group of points of some Abelian variety to properties of the characteristic polynomial of the variety. We can use it to derive precise properties and then we use the fact that some sequence of monomials of five variables is uniformly distributed modulo one to obtain stronger results that will hold with probability one.

By Weyl's criterion, equidistribution follows by bounding exponential sums, and in order to do so, we will use a combination of different methods. We are particularly interested in Van Der Corput's lemma. It has a continuous version that exhibits the decay of oscillatory integrals and a discrete version that gives a bound for exponential sums. We will see the relation between these two versions and how they apply to the original problem of Abelian varieties.