By Fulton W.

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**Example text**

These conjectures have generated wide interest and stimulated intense research. , k which are contained in a finitely generated extension of ޑp , by Mochizuki [1999]; • proof of the birational Section-conjecture for local fields of characteristic zero, by Königsmann [2005]. Here is an incomplete list of other significant result in this area: [Nakamura 1990; Voevodsky 1991a; 1991b; Tamagawa 1997]. In all cases, the proofs relied on nonabelian properties in the structure of the Galois group G K , respectively, the relative Galois group.

Several ingredients of the proof of Theorem 2 sketched above appeared already in Grothendieck’s anabelian geometry, relating the full absolute Galois group of function fields to the geometry of projective models. Specifically, even before Grothendieck’s insight, it was understood by Uchida and Neukirch (in the context of number fields and function fields of curves over finite fields) that the identification of decomposition groups of valuations can be obtained in purely group-theoretic terms as, roughly speaking, subgroups with nontrivial center.

Interesting examples of function fields arise from faithful representations of finite groups G → Aut(V ), where V = ށkn is the standard affine space over k. The corresponding variety X = V /G is clearly unirational. When n ≤ 2 and k is algebraically closed the quotient is rational (even though there exist unirational but nonrational surfaces in positive characteristic). The quotient is also rational when G is abelian and k algebraically closed. Noether’s problem (inspired by invariant theory and the inverse problem in Galois theory) asks whether or not X = V /G is rational for nonabelian groups.

### Algebraic curves by Fulton W.

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