# 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.

# A Lemma of Bateman-Katz

Here we present a formulation of a lemma from Bateman-Katz‘s work on the cap-set problem, which may be of independent interest in additive combinatorics at large. The proof is also novel — Ben Green knows of no other instances of this argument — and may yet find other applications. Indeed, Thomas Bloom recently used this lemma together with other bounds to give his improvement to Roth’s theorem.

[Our main result will be ‘Corollary 3’ below, which will follow from Lemma 1 and Theorem 2. ]

For any abelian group $G$ and finite $A\subseteq G$, we define the ‘2$m$-fold additive energy’  of $A$ by

$E_{2m}(A)=\left\lvert\{(a_{1},\cdots,a_{m},a_{1}^{\prime},\cdots,a_{m}^{\prime})\in A^{2m}:a_{1}+\cdots+a_{m}=a_{1}^{\prime}+\cdots a_{m}^{\prime}\}\right\rvert$

Any $(a_{1},\cdots,a_{m},a_{1}^{\prime},\cdots,a_{m}^{\prime})$ counted on the right-hand-side we call an ‘additive tuple’. The energy is one way of measuring the additive structure of $A$. The main theorems of this note link data concerning the energy to data concerning another notion of additive structure, namely (in the case where $G$ is a finite vector space) dimension.

Lemma 1

If $\lvert S\rvert = d$ and $\text{dim}(\text{span}(S))=d-k$, with $1\leqslant k\leqslant d$, then $E_{2m}(S)\leqslant 2^{2m+1}m^{4m+3}k^{2m}d^{m}$

Remarks on Lemma 1

1. There are about $m^{2m} d^{m}$ diagonal solutions counted in the additive energy, and — thinking of $m$ as much smaller than $d$ — the claim is exactly saying that (up to some tame factors) these dominate when $k$ is very small, i.e. when $S$ is almost completely linearly independent. This matches our intuition.
2. The claim is (worse than) trivial if $k$ is the same order as $d$.
3. We shall see that information about dimension meshes most naturally with equations in the elements of $S$, and not additive tuples. The count for the former is roughly equivalent to Bloom’s notion of ‘restricted energy’, and we will, like him, have to undergo a messy calculation to convert information about this restricted energy  into information about  $E_{2m} (S)$. The issue that $E_{2m} (S)$ cannot be calculated directly does genuinely seem to have been missed by B-K. Broadly speaking this means that their definition giving what it means for the large spectrum to be ‘additive smoothing’ is overly optimistic. I am currently trying to establish whether this has any major effect on their work; this will probably be the subject of a future blog post.
4. My $m$ dependence is extremely wasteful, and no doubt one can do much better. However, don’t be too scared of extra $m^m$ factors; in Bloom’s argument they roughly correspond to $\log\log N$ factors in the final bound. Also remember that there are $m^m$ factors in the trivial lower bound, so we cannot eschew them completely.
5. The reason we go to the trouble of proving these theorems for $E_{2m} (S)$ rather than the restricted energy is that  $E_{2m} (S)$ has a particularly pleasant expression in terms of the Fourier transform, which allows for many useful manipulations (in particular Holder’s inequality).

Viewing Lemma 1 in the contrapositive, it is saying that sets with larger-than-trivial energy have lower-than-trivial dimension. Theorem 2 is a more refined version, namely that sets with large energy have subsets with very low dimension.

Theorem 2

Let integers $d\geqslant 2m\geqslant\text{max}(2,2Ce)$ be arbitrary parameters and $C>0$ be independent of $d$.  Let V be finite vector space, and $A\subseteq V$ satisfy $E_{2m}(A)>\lvert A \rvert ^{2m}d^{-m}\left(2^{4m+3}m^{6m+4}C^{2m}+2m^{2m+1}\right)$. Then $\exists S\subseteq A$ with $\text{dim}(\text{span}(S))\leqslant d$ and $\lvert S\rvert \geqslant\dfrac{C}{d} \lvert A\rvert$

Remarks on Theorem 2

1. This theorem is good in the case where $d$ is a smallish power of $\lvert A\rvert$; B-K have $d\approx \lvert A\rvert ^{\frac{1}{3}}$. It gives some structural information even when the 2$m$-fold additive energy of $A$ is quite a long way below the maximum possible.
2. This is the first main ‘new’ result in Bateman-Katz’s paper, and is the main topic of this post.
3. $2m\geqslant 2Ce$ is just a minor technical condition that streamlines a particular estimation step. It could certainly be removed with a little more effort, and no effort at all if one didn’t care about how the constants in Theorem 2 depended on $m$.
4. One needs Lemma 1 as an ingredient in the proof of Theorem 2, applied to a suitably chosen random subset $S\subseteq A$.
5. For certain choices of the parameters we can use Theorem 2  to bootstrap an improved theorem regarding $E_{8}(A)$, say, which would be better than applying Theorem 2 directly with $m=4$.

Expanding on this last remark, we note that by Holder’s inequality we have

$E_{2m}(A)\geqslant \dfrac{E_{8}(A)^{\frac{m-1}{3}}}{\lvert A \rvert ^{\frac{m-4}{3}}}$

[This follows from the fact that $E_{2m}(A)=\sum\lvert \hat{A}(r)\rvert^{2m}$ and taking Holder in the form $\sum\lvert f\rvert ^{2k}\leqslant\left(\sum\lvert f \rvert^{2m}\right)^{\frac{k-1}{m-1}}\left(\sum\lvert f \rvert ^{2}\right)^{\frac{m-k}{m-1}}$ with $k=4$, followed by Parseval applied to the second bracket].

Therefore we can make the conclusion of Theorem 2 provided that, for some $m$,

$E_{8}(A)>\left(\lvert A\rvert ^{2m+\frac{m-4}{3}}\right)^{\frac{3}{m-1}}d^{\frac{-3m}{m-1}}\left(2^{4m+3}m^{6m+4}C^{2m}+2m^{2m+3}\right)^{\frac{3}{m-1}}$   (1)

Suppose we wanted to be able to take $d=\lvert A \rvert ^{\epsilon}$ for some fixed $\epsilon$. Applying Theorem 2 directly would only allow us to do so when $E_{8}(A)\gg\lvert A\rvert ^{8-4\epsilon}$ (so for example we would be forced to have $\epsilon>\frac{1}{4}$ to get a non-trivial result). However, equation (1) allows us to conclude whenever $E_{8}(A)\gg_{m}\lvert A\rvert ^{7+\frac{3}{m-1}-\frac{3\epsilon m}{m-1}}$. Taking $m$ large enough, depending on $\epsilon$, we win if $E_{8}(A)\gg_{\epsilon}\lvert A\rvert ^{7-2\epsilon}$, which is better for $\epsilon<\frac{1}{2}$. We state this as a corollary:

Corollary 3

Let $\epsilon\in (0,\frac{1}{2})$, and let V be finite vector space, and $A\subseteq V$. Then there exist a constant $K=K(\epsilon)$ such that if $E_{8}(A)\geqslant K(\epsilon)\lvert A\rvert ^{7-2\epsilon}$ then $\exists S\subseteq A$ with $\text{dim}(\text{span}(S))\leqslant \lvert A\rvert^{\epsilon}$ and $\lvert S\rvert\geqslant\lvert A\rvert^{1-\epsilon}$

Proof: Above discussion.

Proofs of Lemma 1 and Theorem 2 are below.

# Erratic bias at the BBC

For many years I have vigorously avoided joining the babbling throng of internet commentators, perceiving their musings to be well-intentioned but unhelpful. Yet, I have been drawn up to such heights of fury by the recent events in Israel — and the BBC coverage of these events — that my vow of silence demands to be broken, if only to release my anger. I assert that, against the morass of partisan bickering, I have something genuinely new to add. But I probably haven’t. Perhaps I’m just another opinionated individual, shouting into the dark.

The source of my madness is not the actions of the Israeli military, or of Hamas, but is the outstandingly poor quality of journalism offered on the conflict by the BBC, particularly on its flagship platform, the 6 o’clock news. Comically, it has been lambasted by supporters from both sides for being biased in favour of the other.

http://www.thecommentator.com/article/5097/is_the_bbc_really_pro_israel

Surely this shows that the BBC is being unbiased? I disagree. Rather, the BBC coverage is erratically preferential, switching its slant depending on the issue but always exhibiting favouritism nonetheless: this is not impartiality. As a general rule, the BBC is pro-Palestinian when describing events on the ground — “everyone loves an underdog,” http://www.theguardian.com/world/2014/jul/21/gaza-crisis-hamas-killed-friend-need-kill — but pro-Israeli when framing the conflict as a whole.

The actions of Hamas are indeed downplayed, such as the number (over 100) of their rockets which have fallen on Gaza itself; Howard Jacobson persuasively argues (http://www.independent.co.uk/voices/commentators/howard-jacobson/howard-jacobson-letrsquos-see-the-criticism-of-israel-for-what-it-really-is-1624827.html) that firing rockets without knowing where they will land shows a greater disregard for civilian life than even the vast civilian collateral incurred by a military campaign. [I don’t entirely agree, but I think that it is certainly at least as bad]. On the other hand, the BBC allows us to forget that Israel took control of its current borders by force, in the wars of 1967 and 1948 (expanding on the already supremely generous UN Partition Plan of 1947), and to forget about its 65 years of provocative bellicosity. The inconvenient reality that Hamas, however badly they are treating the Palestinians, are their elected representatives is also blatantly ignored.

This is the worst possible state of journalistic affairs. It’s even worse than an outrageously one-sided vision, which can at least be kept at arms length and viewed in a detached manner as ‘one version of the story’. As it is, in five minutes on BBC1 we are tossed between extremes of emotion, the linguistic register chosen to shock, amidst an intellectual desert devoid of cogent discussion — all under the pretence of a fair unbiased analysis. The world’s premier news organisation my foot.

Reporters on the 6 o’clock news are also painfully poor at revealing their sources. Where is he getting his detailed knowledge of the military maneuvers from that day’s fighting, which, even if he is on the ground in the Gaza strip, he couldn’t witness all at first hand? I hope not from Israeli government press releases, which have a history of being at best economical with the truth, and at worst outright falsifications: a string of such cases in the first 20 years of Israel’s history were discovered by revisionist Jewish historians, when previously secret documents became available later in the 20th century. Even if all the information given in the report were collected firsthand by the reporter himself, a pipe-dream, he would still exercise editorial judgement. How is he choosing which people to interview, and which interviews to use in his final report?

[A few years ago there was an entertaining miniseries on the BBC, one episode of which involved Sir Ranulph Fiennes and Sir Robin Knox-Johnston following John Simpson around Afghanistan as he compiled a report. He had a thesis about the current situation for the civilian population, and found interview subjects who corroborated it. Those who espoused the opposing view were cut from the report. Simpson argued that, although these views existed, they didn’t represent the general view of most Afghans; he may have been right, or he may have been wrong, but nonetheless his personal judgement defined the colour of the report.]

These questions might have perfectly satisfactory answers, but I don’t know that they do. I fear that they don’t. Indeed, the BBC’s aim seems to be for sensationalism and manufactured shock, in lieu of level-headed analytic reporting, eschewing any debate of the underlying issues.

I can’t help calling to mind the Melian dialogue, from Thucydides’ account of the Peloponnesian War, which is archetypal for the manner of discussion that is so conspicuously absent from the BBC’s discourse. The strong Athenian naval force threatened to destroy the people of Melos, a small island wishing to remain neutral despite Spartan connections. The destruction could only be avoided if the islanders paid homage and tributes to Athens. Understandably they refused, and what followed — at least in Thucydides’ stylised account — was a long subtle philosophical debate between the Athenian envoys and the islanders, pondering whether Athens had the moral right to follow through on its threat. The Athenian argument — that, “the strong do what they can, and the weak suffer what they must,” and that justice only exists between equally matched powers — proved victorious, unsurprisingly. But at least there was a debate: the BBC, had they been there, would have just reported the subsequent bloodshed. The military mismatch which gave birth to the Melian dialogue bears an eerie similarity to the mismatch in Palestine, and Thucydides’ translated words should be being invoked and re-examined everywhere, not least by the BBC. But of course, they are not.

There are comparisons too with our own military history, both recent and past, which beg to be investigated in the mainstream media. How different is Israel’s campaign in Gaza to Britain’s campaign in Afghanistan? Both vast military actions against a less well-armed resistance/terrorist movement, intending to neuter the threat of terrorism, causing vast civilian casualties in the process. Very similar. But in Israel and Palestine the extreme factions of both nations genuinely want to see the total annihilation of the other, civilians and all — this was not the case in Afghanistan. There are other issues too: the blockade, territory, race. So perhaps they are different. Discuss.

Maybe Israel is deliberately targeting civilians. Casting an introspective eye over our previous dealings with our own local insurgents, the Scots, how different is the massacre at Glen Coe — a civilian hit intended to deter the militants — to analogous Israeli actions? Has the international community decided that these things are no longer acceptable? Isn’t that a bit rich from long established countries such as England, who profited from such tactics back when they were ‘allowed’? Discuss.

And then there are the issues of territory and race. Does Israel have a right to exist? If so, should it exist? Given that it exists, what can be done to ensure peace? Is the very notion of a Jewish state a racist one? Is the very notion of a Palestinian state a racist one? Discuss, discuss, discuss!

I find it difficult to envisage the philosophical gymnastics required to interpret the phrase ‘chosen people’ in a way that is not in some way racist. But such an argument no doubt exists, and maybe it’s very convincing. Dear BBC, show me an Israeli explaining this argument; show me a Palestinian disagreeing. I know people are dying, and I wish they weren’t, but filling the airwaves with daily reports of the continuing bloodshed does not make the population better informed. They know people are dying; tell them something they don’t know.

As a national institution of great influence, both here and abroad, the BBC has a responsibility to engender the virtues of critical thinking and analytic debate in its viewers. By renouncing these virtues itself, in favour of sensationalism and shock, it has let us all down. There are legions of anti-Israeli protesters voicing their rage at the inappropriateness of the scale of the Israeli military response; I’m inclined to agree with them, but we must accept that in holding this view we commit ourselves to disagreeing with the Athenian argument wherever it presents itself, lest we be woefully inconsistent. I hope everyone marching against Israel this week also marched against the Allied forces’ actions in Iraq and Afghanistan, or at least expressed a vague disquiet. No doubt many did, but I suspect that many would not see any contradiction in labeling the Israeli war machine evil and ours just, Palestinian deaths murders and Afghan deaths unfortunate. The BBC’s malady is infectious.