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

Counting problems: class groups, primes, and number fields