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Algebraic Curves over super-simple fields

Final Activity Report Summary - SUPERSIMPLE FIELDS (Algebraic Curves over Supersimple Fields)

This project was concerned with the study of algebraic curves over supersimple fields, in particular with elliptic curves. The main result was that any elliptic curve defined over a supersimple field had a generic point in case the supersimple field had a unique quadratic extension up to isomorphism.

In addition, exhaustive treatment of Hrushovski's amalgamation procedure in order to obtain omega stable fields equipped with a definable additive, or multiplicative, subgroup according to a predimension was suggested by B. Poizat. In particular, a bad field was constructed in characteristic zero, i.e. a field of finite Morley rank (in this case two) with a divisible torsion-free multiplicative subgroup of rank one. The existence of such fields was originally a major obstacle in proving the algebraicity conjecture, stating that a simple group omega_1-categorical could be seen as an algebraic group over an algebraically closed field. It should also be noted that the existence of a bad field in characteristic p would imply the existence of only finitely many p-Mersenne primes.