Abstract:
The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extension of \$K\$ as \$K\$ varies in a family of global fields. In this talk, we will discuss some new developments for this type of question. We will introduce a cohomological invariant of a Galois extension of \$\mathbb{F}_q\$. We show that by keeping track of this invariant we can generalize the nonabelian Cohen—Lenstra Heuristics given by Liu, Wood, and Zureick-Brown to cover the case when the base field contains extra roots of unity; moreover, we show that the new conjecture is a nonabelian generalization of the work by Lipnowski, Tsimerman, and Sawin. We will prove the conjecture with a large \$q\$ limit, and discuss how to make a similar conjecture for number fields.