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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?

EDIT 2: Let $f$ be an automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$.

Is it possible to find a general form of $(u,v)$, in the special case where $u,v \in \mathbb{C}[x,y]$?

For example: $(u,v)=(x,xy)$; in this example $(u,v)$ is not a Jacobian pair.

Please also see this question.

Thank you very much!

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!

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?

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?

EDIT 2: Let $f$ be an automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$.

Is it possible to find a general form of $(u,v)$, in the special case where $u,v \in \mathbb{C}[x,y]$?

For example: $(u,v)=(x,xy)$; in this example $(u,v)$ is not a Jacobian pair.

Please also see this question.

Thank you very much!

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?

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?