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The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of Dicks or the proof of Mckay and Wang.

What can be said about the group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$?

Of course, every $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ yields a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$, but there are more $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$, for example, $x \mapsto x^{-1}, y \mapsto y$.

Can one find all of them and characterize that group?

EDIT: After letting me know that I am looking for the Cremona group, I wish to quote from wikipedia: "In two dimensions, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with PGL(3, k), though there was some controversy about whether their proofs were correct, and Gizatullin (1983) gave a complete set of relations for these generators".

What was the problem in their proof (if at all), and is there another published proof?

Thank you very much!

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    $\begingroup$ That's the Cremona group. $\endgroup$
    – Felipe Voloch
    Commented Oct 30, 2016 at 22:00
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    $\begingroup$ From the home page of Serge Cantat: perso.univ-rennes1.fr/serge.cantat/Articles/… $\endgroup$
    – Uri Bader
    Commented Oct 30, 2016 at 22:24
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    $\begingroup$ Knowing a generating subset for a group does not mean that the group is "known". $\endgroup$
    – YCor
    Commented Oct 31, 2016 at 0:16
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    $\begingroup$ Yes there's a big recent literature including modern proofs. You can browse in webusers.imj-prg.fr/~julie.deserti/cremona.html $\endgroup$
    – YCor
    Commented Oct 31, 2016 at 2:10
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    $\begingroup$ I'll be more precise: knowing generators is some piece of information on a group; which can be useless (e.g. if I give you the whole group as set of generators) or not and can be described in many non-equivalent ways. Next you can have a presentation, which can be more or less useful (e.g. an amalgam has interesting special features, which doesn't mean it's always fully understood). But in Cremona "describe the group" can have a totally different meaning. E.g., it can consist in describing the set of pairs of rational functions that indeed define a element of the Cremona group... $\endgroup$
    – YCor
    Commented Oct 31, 2016 at 3:16

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The proof of Noether that the automorphisms of $\mathbb{C}(x,y)$ are generated by $\mathrm{PGL}(3,\mathbb{C})$ and the standard quadratic transformation had troubles because of infinitely near points. The proof of Castelnuovo (1907 is completely valid and is the first one which is accurate. This is why the theorem is Called "Noether-Castelnuovo theorem". You can find a lot of proofs of this result. See for instance the book "Geometry of the Plane Cremona Maps" of Maria Alberich Carramiñana, which gives a detailed proof and explains the history of thre problem.

As YCor said, knowing the generators does not directly give a good understanding of the group. For instance, it is useless here to find conjugacy classes of elements of finite order (but the classification is known using birational geometry). There are many other results on the group that you can find in the literature.

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