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

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

2011-03-21T00:40:54Z

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.

Answer by Cam McLeman for Where can I find a modern write-up of Heegner's solution of Gauss' class number 1 problem?

2011-03-21T02:40:08Z

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