[GS-NT] On the inverse Galois problem with restricted ramification 2

ABSTRACT

In this series of talks, I will consider various strong versions of the inverse Galois problem with added conditions on ramified primes. I will first discuss various versions of "minimal ramification problems" and review known results and conjectures. Here "minimal ramification" can be understood as the problem to minimize the number of primes ramifiying in an extension with a given Galois group, or also as the problem to minimize the ramification indices. I will also discuss relations to conjectures and heuristics about the distribution of Galois groups (such as the Malle-Bhargava heuristics) and the distribution of class groups (such as the Cohen-Lenstra heuristics). I will furthermore present new results on Galois realizations with restricted ramification indices, obtained in joined work with D.Neftin and J.Sonn. We use two essentially different approaches, one coming from Shavarevich's method for solvable group, and another one coming from geometric Galois theory and specialization of Galois coverings. Finally, I will present some results on so-called intersective polynomials. The notion of intersective polynomials, that is, integer polynomials with a root in every Q_p, but not in Q, was introduced by Sonn. Their construction is related to restricted-ramification problems. I will present conjectures and new results on the existence of such polynomials for prescribed Galois groups. This last talk in particular will be very easily accessible without a lot of theoretical background.