Vladuts characterized in 1999 the set of finite fields k such that all elliptic curves defined over k have a cyclic group of rational points. Under the conjecture of infinitely many Mersenne primes, this set is infinite. 

In this talk, I study the question in higher dimension. Precisely, I prove that there is no such fields. This is related with the existence of some totally real algebraic integers having some arithmetic properties.

I am going to present a result about cyclicity of maximal abelian varieties as well, and how this is related to some totally positive algebraic integers.

In both problems, open questions about totally real algebraic integers arise, some of which are addressed from an algorithm point of view.