## Where can I find a modern write-up of Heegner’s solution of Gauss' class number 1 problem?

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In a recent MO question someone mentioned Heegner's solution of the Gauss "class number 1" problem which takes the following form:

• When the class number of an imaginary quadratic form is 1 an elliptic curve is defined over $\mathbb{Q}$ and a modular function takes on integer values at certain quadratic irrationalities which leads to a collection of Diophantine equations: The solution of which finishes the theorem.

I sadly can't read Heegner's original work (since I cannot read German) but also I don't think it's necessarily the best thing to read for this proof due to an alleged gap. So if anyone recognizes this proof sketch sketch and knows where I could read this in detail that would be wonderful! Thanks.

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The "gap" in Heegner's proof is filled in/shown to be non-existent by the article of Birch I mentioned in my answer to the version of this question that you asked on Math.SE. – Emerton Mar 21 2011 at 1:33
@quanta: Years ago I lectured on a more algebraic version of Heegner's proof, with elliptic curve arguments replacing the use of the Weber functions. I still have handwritten notes from my talk; if I can decipher them, and if you send me a mailing address, I'll put something in the mail to you. – paul Monsky Mar 21 2011 at 19:23

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In her recent paper "Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem," Burcu Baran discusses Heegner's technique in clean modern language. The key observation is that (by the theory of CM), each quadratic number field of class number 1 yields integral points on many modular curves. Generally those curves should have only finitely many integral points, and when the genus is small, one can compute them all. In her paper, after explaining the technique, Baran analyzes (the last) three modular curves where this approach will work. – james-parson Mar 21 2011 at 21:21
Though slightly off-topic, I want to remark that one can also find a proof of the Stark-Heegner theorem in Elkie's exposition of the Klein quartic (the proof originates with Kenku): library.msri.org/books/Book35/files/elkies.pdf – Lennart Meier Feb 22 at 1:18

## 1 Answer

In his article On the "gap'' in a theorem of Heegner, Stark does a pretty thorough job of explaining where people thought the purported gap came from, to what extent it actually was a gap, and what you would need to fix such a thing if it existed. I'm paraphrasing, but he basically argues that the confusion stemmed from some errors (typos?) in some analytic results of Weber that Heegner had heavily used. So in a literal sense, Heegner had not proved it because he had cited faulty results, but Stark shows that he deserved credit for the theorem since using Heegner's argument with the correct versions of Weber results (which were indeed known to Weber), the job gets done.

Here's the mathscinet review of the article:

http://www.ams.org/mathscinet-getitem?mr=241384

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