Bad fields, Mersenne primes and a generalised Lang-Weil estimate

A conjecture by Cherlin and Zilber states that one can characterize simple algebraic groups as abstract simple groups which allow a rudimentary dimension theory. Early attempts to prove this conjecture encountered a pathological obstacle (which incidentally had also figured in the proof of the Feit-Thompson theorem on the solubility of groups of odd order), namely a field K with a distinguished proper infinite divisible multiplicative subgroup T, such that the structure (KT) allows a dimension theory. These are called "bad fields".

After a quick introduction into the model-theoretic background, I shall sketch a proof of the non-existence of bad fields of characteristic p > 0, provided there are infinitely many primes of the form (pn - 1)/(p - 1) (p-Mersenne primes). I shall then present recent results which indicate that in a bad field of positive characteristic, T behaves like a dim(T)/dim(K)-dimensional variety, as far as the number of rational points over finite subfields is concerned.

Frank Wagner

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