When small points have their good reasons

The height of an algebraic number is a real-valued function that measures the ``arithmetic complexity'' of the number. While numbers of height zero are well understood, many questions remain open regarding numbers of small height. For example, a key question is whether a given infinite algebraic extension of the rationals contains numbers of arbitrarily small (but non-zero) height.
The talk will focus on the case where the answer is positive and, in particular, on the following question: in fields where small points can ``obviously'' be found, do these points have always ``good reasons'' to be small?
For instance, the field generated over the rationals by all roots of 2 contains some obvious points of very small height (0, small fractional powers of 2 multiplied by roots of unity). Does it contain other small points? A very particular case of a conjecture of Rémond suggests that the answer is no. Rémond’s conjecture more generally concerns the saturated closure of subgroups of finite ranks in tori and abelian varieties defined over number fields. It remains widely open and generalizes several important problems, such as Lehmer’s conjecture. Recently, Pottmeyer established a necessary group-theoretical condition for the conjecture to hold and proved it in the case of tori. I will present joint work with G. A. Dill, where we extend this result by showing that the condition is also satisfied for split semi-abelian varieties.

This is a joint work with G. A. Dill.