The Gauss Class Number Problem for Imaginary Quadratic Fields

Summary

1. Introduction

Let D < 0 be a fundamental discriminant for an imaginary quadratic field . Such fundamental discriminants D consist of all negative integers that are either ≡ 1 (mod 4) and square-free, or of the form D = 4m with m ≡ 2 or 3 (mod 4) and square-free. We define

to be the cardinality of the ideal class group of K. In the Disquisitiones Arithmeticae, (1801) [G], Gauss showed (using the language of binary quadratic forms) that h(D) is finite. He conjectured that

a result first proved by Heilbronn [H] in 1934. The Disquisitiones also contains tables of binary quadratic forms with small class numbers (actually tables of imaginary quadratic fields of small class number with even discriminant which is a much easier problem to deal with) and Gauss conjectured that his tables were complete. In modern parlance, we can rewrite Gauss’ tables (we are including both even and odd discriminants) in the following form.

The problem of finding an effective algorithm to determine all imaginary quadratic fields with a given class number h is known as the Gauss class number

h problem. The Gauss class number problem is especially intriguing, because if such an effective algorithm did not exist, then the associated Dirichlet L-function would have to have a real zero, and the generalized Riemann hypothesis would necessarily be false. This problem has a long history (see [Go2]) which we do not replicate here, but the first important milestones were obtained by Heegner [Heg], Stark [St1; St2], and Baker [B], whose work led to the solution of the class number one and two problems. The general Gauss class number problem was finally solved completely by Goldfeld, Gross, and Zagier in 1985 [Go1; Go2; GZ]. The key idea of the proof is based on the following theorem (see [Go1] (1976), for an essentially equivalent result) which reduced the problem to a finite amount of computation.