Field Automorphism Groups

The aim of the first part is to explain, how some geometric questions can be translated to the language of representation theory. There will be three categories of representations of G: Sm G ⊃ I G ⊃ Adm, roughly corresponding to birational geometry, to birational motivic questions (like structure of Chow groups of 0-cycles) and to " finite-dimensional " birational motivic questions (in particular, description of " classical " motivic categories). This part is rather motivational, so there will be many conjectures, and just a few particular results. In the second part it is explained how appropriate representation theory could be developed by means of semi-linear representations. Main results suggest an explicit description of the category of admissible semi-linear representations, which is conjecturally sufficient for the geometric applications. 0.1. Notation. Let k be an algebraically closed field of characteristic zero, F be a universal domain over k, i.e., an algebraically closed extension of k of countable transcen-dence degree, and G = G F/k = Aut(F/k) be the field automorphism group of F over k. We consider G as a topological group with the base of open subgroups generated by {G F/k(x) = Aut(F/k(x)) | x ∈ F }. 1. How to translate geometric questions to the language of representation theory? We are interested in representations of G. To specify the type of these representations, we have to ask a geometric question. There will be three categories of representations of G. 1.1. Sm G. In general, geometry deals with varieties. To any variety X over k one can associate the G-module Q[X(F)], i.e. the Q-vector space of 0-cycles on X × k F. This representation is huge, but this is just a starting point. Note that it is smooth, i.e. its stabilizers are open, so all representations we are going to consider will be smooth. Denote by Sm G the category of smooth representations of G over Q. The first question to ask is: what are the finite-dimensional smooth representations of G? Theorem 1.1 ([R1]). Any finite-dimensional smooth representation of G is trivial. This follows from the (topological) simplicity of G: Theorem 1.2. (1) [R1] The subgroup of the automorphism group of an algebraically closed extension of an agebraically closed field of finite transcendence degree, generated by the compact subgroups, is simple. (2) [La, R1] Any closed normal proper subgroup of G is trivial. Remarks.