# Heegner’s solution to the ‘Class Number 1 problem’

I have just completed a short monograph on the so-called ‘Class Number 1 problem‘. It was written to fulfill the EPSRC ‘broadening requirement’, having attended the Part C Modular Forms course this term in Oxford,  and was therefore, by design, a little outside my comfort zone — I hope this disclaimer will temper the disdain of any serious algebraic number theorists who happen across this article. No expert, I set out to try to write the kind of exposition of this topic that I , as an interested mathematician with a slightly different specialism, would have liked to have read myself. Although there already exist some very thorough accounts of this topic in the literature — we make reference to a book by Cox, and essays by Booher, Green and Kezuka — I know of no shorter survey which nonetheless gives a detailed description of the entire argument and sketch proofs of most of the important results.

Introduction

Gauss found nine imaginary quadratic fields with class number 1, and conjectured that he had found them all. In 1952 Heegner published a purported proof, based heavily on the work of Weber from the third volume of his landmark, but fearsome, treatise Lehrbuch der Algebra. Heegner was unknown to the mathematical community at the time, and it was felt that his proof contained a serious gap. Stark and Baker independently published the first accepted proofs in 1966, but then Stark examined the argument of Heegner and discovered it to be very similar to his own. Indeed, he went on to show that the ‘gap’ in Heegner’s proof was virtually non-existent. Furthermore, he noticed that enough technical machinery could be avoided to have enabled Weber to prove this result some 60 years earlier.

A detailed historical overview of progress on the problem has been written by Goldfeld. It is worth noting that Heilbronn and Linfoot knew in the 1930s that there were at most 10 imaginary quadratic fields with class number 1.

The aim of this short essay is to outline Heegner’s argument, prove a few of the important constituent lemmas, and to relate some of the theory to that covered in the Part C Modular Forms Course. With this latter aim in mind, we shall focus mostly on the modular functions involved in the proof, rather than the input from algebraic number theory — indeed, two particularly technical propositions will be left entirely unproved. However, we will assume familiarity with basic results concerning the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field, and concerning non-maximal orders $\mathcal{O}\subset \mathcal{O}_K$. This theory is well covered in Chapter 7 of Cox’s ‘Primes of the form $x^2 + ny^2$‘. Regarding the input from modular forms, we will make heavy reference to Eisenstien series and the Ramanujan $\Delta$-function, and introduce other modular functions which are invariant under other congruence subgroups $\Gamma\leqslant SL_2(\mathbb{Z})$. There will also be an analogy to a lemma from the theory of Hecke operators.

Serre has an approach to the class number 1 problem which is much more geometric, constructing a particular modular curve and then counting special points on it, which has more of a flavour of the first half of the Part C Modular Forms course. However, the approach is extremely involved and this is only a short project — Booher  gives details, and also discusses the relationship to Heegner’s argument.

Note on references: We have relied heavily on the excellent essay of Booher, the paper by Stark, and the astonishing book ‘Primes of the form $x^2 + ny^2$ by David Cox, which despite its unassuming title provided a wealth of insight into all aspects of the argument. The master’s essay of Kezuka is comprehensive but is rarely more than a recitation of Cox. Green has an essay which covers much of the background regarding complex multiplication, although from a more high-brow viewpoint than we shall pursue here.