Arnaud Plessis

Lehmer's problem (1933) asks if there exists a constant c > 1 such that the (logarithmic, absolute) Weil height of every nonzero algebraic number x is bounded from below by c [Q(x) : Q]^{-1} , except if x is a root of the unity.  In 1979, Dobrowolski proved that it is true up to epsilon. 

In this talk, we are going to study algebraic fields L which have the Bogomolov Property, that is when the Weil height of every nonzero element of L which is not a root of the unity can be bounded from below by an universal constant. Next, we will give other kind of "Lehmer's problem" according with these results. 

 Finally, we will state a Lehmer's problem (due to David) for abelian varieties and I will explain the difficulties about this one.