Counting problems: class groups, primes, and number fields

Abstract

Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well studied, yet also still mysterious. A central conjecture of Brumer and Silverman states that for each prime $\ell$, every number field has the property that its class group has very few elements of order $\ell$, where “very few” is measured relative to the absolute discriminant of the field. This paper surveys recent progress toward this conjecture, and outlines its close connections to counting prime numbers, counting number fields of fixed discriminant, and counting number fields of bounded discriminant.