Groups in pseudofinite fields

Summary

These notes should be read with those of Zoé Chatzidakis. We report some results from Hrushovski and Pillay. The main items in this paper are

• an analogue for pseudofinite fields of ZiPber's Irreducibility Theorem (Theorem 23);

• lemmas relating simplicity properties of an algebraic group G to its restriction G(F) to a pseudofinite field F (§6);

• a fast though non-effective model-theoretic proof of a result of Matthews, Vaserstein and Weispfeiler on reduction at primes (Theorem 33; see also the similar argument in the last section of).

The main difference from is that I avoid the local stability arguments of Hrushovski and Pillay. In fact the proof of the Irreducibility Theorem removes all the stability arguments beyond the ‘(S1) property’, without adding anything in their place. (Later I quote the Theorem of §3, whose proof—at least in its present guise—uses forking, canonical bases and the definability of types.) It took several shoves to remove the parts of the argument that rely on stability; Frank Wagner and John Wilson delivered the final push during the Blaubeuren meeting. I think a fair comment would be that stability theory has powerful methods for showing that things are first-order definable, and this was the role that it played in the original argument. But sometimes, after the event, one sees that other devices may do the job faster.