Carlo Pagano (Concordia Unversity
Title: On Malle's conjecture for nilpotent groups.
Abstract: Malle's conjecture is a strengthening of the Galois inverse problem, asking for an asymptotic formula for the count of G-extensions with bounded ramification invariant (discriminant, conductor etc). Yet, even in the case of nilpotent groups, where the Galois inverse problem was solved by Shafarevich, Malle's conjecture is as of now wide open, with the exception of abelian groups (Wright) and a few sporadic families of exceptions.
I will discuss past and ongoing joint work in progress with Peter Koymans on this conjecture, for various orderings. I will explain how the problem is sensitive to the ordering of the fields, and the connection between the discriminant count and the Cohen--Lenstra heuristics. Next, I will explain a new approach we have introduced, in 2021, to attack nilpotent Malle and the partial results obtained so far with that method. Finally, I will discuss work in progress where, under GRH, we prove Malle's conjecture for all odd nilpotent groups, when extensions are ordered by the product of ramified primes.