The analysis of fields is one of the most active branches of Model Theory, which has found its most spectacular applications in Hrushovski's proofs of the Mordell-Lang and Manin-Mumford conjectures.
There are three principal aspects:
- ways of interpreting a field,
- studying the general properties of fields thus obtained, and
- determining the properties of particular theories of fields (with additional algebraic structure like a derivation or an automorphism).
All three aspects are closely interrelated. This proposal concerns mainly part (b). A theorem of Macintyre, Cherlin and Shelah states that a super-stable field is algebraically closed; this theorem is at the basis of many applications. Recently Kim and Pillay have extended the apparatus of stability theory to a wider class: simple theories; Pillay has conjectured that super-simple fields are perfect, bounded and pseudo-algebraically closed (the converse was shown by Hrushovski).
A positive answer to this conjecture should play a role similar to Macintyre's theorem. Since Pillay and Poizat have shown super-simple fields to be perfect and bounded, only the PAC condition that every absolutely irreducible variety has a rational point needs to be checked; this can be reduced to the consideration of plane curves.
The case of elliptic and hyperelliptic curves with generic modulus has already been dealt with; however, attempts to treat the non-generic case have met with considerable difficulty. We propose to prove triviality of the first cohomology group in order to treat the non-elliptic genus 1 case. We shall also consider isogenies between elliptic curves defined over our field, in order to treat the case of non-generic j-invariant.
Finally, we want to study the question whether super-simple fields are C_1 (related to a question of Ax). A natural approach here will be to study cubic surfaces over a super-simple field. This programme interrelates Algebraic Geometry, Field Theory and Model Theory.
Fields of science
Call for proposal
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