Counting problems: class groups, primes, and number fields

TL;DR

This paper surveys recent progress on the Brumer-Silverman conjecture, which predicts that class groups of number fields have very few elements of a given prime order relative to the field's discriminant, and explores its connections to counting primes and number fields.

Contribution

It provides an overview of recent advances related to the conjecture and discusses its implications for counting primes and number fields of fixed or bounded discriminant.

Findings

Progress towards the Brumer-Silverman conjecture is summarized.

Connections between class group properties and counting problems are explored.

Open problems and future directions are discussed.

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.