Recent Progress in Nonabelian Cohen--Lenstra Program

The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extensions of 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; and 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 large $q$ limit, and discuss how to make a similar conjecture for number fields.

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