**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
(*K*, *T*) 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
(*p*^{n} - 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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