SUPERSIMPLE FIELDS · Algebraic Curves over super-simple fields
6РП — Действия „Мария Кюри“
- Период
- 2005-09-01 → 2007-08-31
- Финансиране от ЕС
- 143 433 €
- Участници
- 1
- Схема
- EIF
Линиите свързват координатора с партньорите. За проекти отпреди 2014 г. CORDIS не винаги дава точни координати. Тези точки са на ниво град или държава.
Накратко на български
Алгебричните криви, като например елиптичните криви, се анализират в рамките на специфични математически структури, наречени суперпрости полета. Работата помага за разбирането на алгебричността на определени групи и връзката им с Мерсенните прости числа.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
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.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
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.
Оригинален текст от CORDIS (на английски).
Участници
- CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION RHONE ALPES - SITE VALLEE DU RHONE · VILLEURBANNEКоординаторНиво градФранция
Връзки
Данни: CORDIS, © Европейски съюз