Distributions of Class Groups of Global Fields

Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups of number fields, found that the Cohen-Lenstra-Martinet heuristics for the distributions of class groups of extensions of a number field seemed incorrect when the base field contains roots of unity. We describe a new conjecture for the distribution of class groups (at primes not dividing the order of the Galois group) that corrects for this issue.  We explain how large q limit function field theorems, along with new results on the moment problem for random groups, lead to proofs of new reflection principles over number fields and our conjecture.  This talk is on joint work with Will Sawin.