Amnon Yekutieli, BGU

An averaging process for unipotent group actions

Abstract:
Let G be a unipotent algebraic group over a field K of characteristic 0, and let Z be a torsor under G(K). By this I mean, naively, that Z is a set endowed with a G(K)-action that's transitive and has trivial stabilizers. Suppose we are given a sequence z_0, ..., z_q of points in Z, and a sequence w_0, ..., w_q of scalars in K whose sum is 1 (these are the weights). I will explain how to define the weighted average of z_0, ..., z_q in Z. This averaging process is very well-behaved (it's functorial and simplicial).

Here is an application. Suppose X is an algebraic variety over K and Z is a G-torsor on X, in the usual sense, which is locally trivial for the Zariski topology. Suppose s_0, ..., s_q are local sections of the torsor Z, defined on an open covering of X. The averaging process allows us to construct a global simplicial section s of Z. This situation occurs in the theory of deformation quantization.

There is also an arithmetic application. Suppose Z is a G-torsor over K (in the true meaning of the word). I will show how the averaging process implies that Z has a K-rational point. This (old fact) is usually proved using Galois cohomology. Note that when G is not unipotent, or the characteristic of the field K is positive, such an assertion is false ‒ it is easy to find torsors without rational points.