Estimates for the Artin conductor

By using geometry of numbers, Minkowski showed that there exists a constant C such that if D_K is the discriminant of a number field K, then |D_K|>C^[K:Q]. In 1978, from the existence of  infinite class field towers, Martinet constructed  sequences of number fields of growing degree and  bounded root discriminant.

It is natural to ask if it is possible to extends the previous results to the Artin conductor.

In this talk, we will review the previous results and in particular let us see the existence of irreducible Artin characters of growing degree with bounded root conductors.