Cours spécialisé (TN, Log, GA)

Valuations in algebra, model theory, and geometry

Sylvy Anscombe

Contact : sylvy.anscombe à imj-prg.fr

Notes de cours : https://www.sylvyanscombe.com/teaching.html

Langue du cours : Anglais

Présentation

Valuations appear as fundamental objects in algebraic number theory and geometry, as non-archimedean places on global fields or as a measure of the order of a zero for a meromorphic function. Zariski's work on resolution of singularities for surfaces marks the most important geometric use of valuations.

The aim of this course is to study valuations in their modern context, as tools in algebra and geometry, and as objects of study in their own right, with a deep model theory and rich geometry of definable sets.
Key points in this course will include Ax, Kochen, and Ershov's asymptotic resolution of Artin's conjecture; the geometry of definable sets in algebraically closed valued fields; relationships between local uniformization and decidability problems in valued fields; and recent work on Hilbert's Tenth Problem in the context of global fields.

Contenu

• Valuations, places, orderings, the product formula in global fields, Riemann--Zariski space
• Local fields, local-global principles, Hensel's Lemma, Ax--Kochen and Ershov: letting $p\rightarrow\infty$. The property $C_{2}$ and Artin's conjecture for $\mathbb{Q}_{p}$. The method of ultraproducts.
• The geometry of algebraically closed valued fields, $C$-relations, quantifier elimination, elimination of imaginaries.
• Large fields, Hilbert's Tenth Problem in large fields, and relationship to local uniformization. Problems around local uniformization.
• The Generalized Stability Theorem of Kuhlmann, the recent approach of Ducros, Hrushovski, Loeser, and Ye
• Towards Hilbert's Tenth Problem for global fields: Diophantine sets in global fields. Robinson, Poonen, and Koenigsmann's work towards H10 for $\mathbb{Q}$. Pop's conjecture: Dittmann--Pop theorem.

Prérequis

Bibliographie

• J. Ax and S. Kochen. Diophantine problems over local fields. I.. Amer. J. Math., 87 (1965), 605–630.remplacer
• A. Engler and A. Prestel. Valued fields. Springer Monographs in Mathematics. Springer, 2005
• J.-P. Serre. Local Fields.. Springer, 1979.
• L. Bary-Soroker and A. Fehm. Open problems in the theory of ample fields. In Geometric and differential Galois theories, volume 27 of Sémin. Congr., pages 1–11. Soc. Math. France, Paris, 2013.
• F.-V. Kuhlmann. Valuation theoretic and model theoretic aspects of local uniformization. In Hauser, H., Lipman, J., Oort, F., Quirós, A., Resolution of Singularities: A research textbook in tribute to Oscar Zariski, 381–456. Birkhäuser, 2000.
• Yu. L. Ershov. Multi-Valued Fields. Springer, 2001.