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Where can I read a corrected up to date version of Heegner's solution of the class 1 problem of Gauss?

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you can check the references at – yoyo Mar 20 '11 at 1:18
It is a good idea if you also add tags corresponding to the topic that you're looking for references because for example some people subscribe to receive email notifications when a question with certain tag is asked, so you may get more help. – Adrián Barquero Mar 20 '11 at 2:08

The wikipedia entry for Kurt Heegner contains a reference to an article by Harold Stark where he adresses the gap on the original proof and presents an outline of Heegner's argument.

Heegner's original article (cited in the wikipedia entry) seems to be available in SpringerLink if you have access to it.

Another nice reference which you would like to consult is David Cox's excellent book Primes of the form $x^2 + ny^2$. In section 12 of that book you can find a proof under the entry E. Imaginary Quadratic Fields of Class Number 1. I hope this helps.

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+1 for mention of Cox's book Primes of the form $x^2 + ny^2$, a favorite of mine. – hardmath Mar 20 '11 at 1:33
Stark said "It is frequently stated that my proof and Heegner’s proof are the same. The two papers end up with the same Diophantine equations, but I invite anybody to read both papers and then say they give the same proof!" – quanta Mar 20 '11 at 1:39
@quanta I find Stark's comment at the end of the article to be quite amazing, where he says that the class number 1 problem could have been solved just at the start of the 20th century if Weber had made one more observation in his book. – Adrián Barquero Mar 20 '11 at 1:47

There is an article of Birch from 1969, titled Weber's class invariants, in which he proves, using basic class field theory and CM theory, various facts originally due to Weber, but whose proof was in doubt. These are facts that Heegner cited in his proof, and Birch observes that one reason Heegner's proof was doubted. Given this, it seems that Birch's article will be a useful companion when reading Heegner's argument.

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