Introduction to the Model Theory of p-adic numbers

Bogotá, Colombia, March 23 to 27, 2009

The p-adic numbers were intensively studied during the last century, on account both of their importance in number theory and of their interest as a complete field with norms differing from traditional ones.

In this workshop we intend to provide a basic vision of the Model Theory of p-adic fields, as well as tools to start on a deeper and more detailed investigation of the nature of the interaction between the Model Theory and the Algebra and Analysis of p-adic numbers - an endeavour that promises to be rather fruitful in coming years.

NEW: the Lecture Notes of the Workshop

Activities

The workshop will be held on March 24th, 25th and 26th at Universidad Nacional de Colombia in Bogotá, and will be taught by professors Thomas Scanlon (University of California at Berkeley), Dugald Macpherson (University of Leeds) and Sonat Süer (Universidad Nacional de Colombia).

On Monday, March 23rd, we will offer preparatory lectures for the workshop: an Introduction to Model Theory (aimed at Algebraists and Number Theorists) in the morning and an introductory lecture to p-adic numbers and the Hensel Lemma for Model Theorists and other Mathematicians who are not very familiar with p-adic numbers but may be interested in the workshop.

Time	Tue 3/24	Wed 3/25	Thu 3/26
9:00-9:50	Süer	Süer	Macpherson
10:30-11:20	Scanlon	Scanlon	Macpherson
11:30-12:20	Macpherson	Süer	Scanlon

• Lecture 1: (Süer) Introduction to valued fields [basic definitions (valuation, valuation ring, value group, etc.), topology, examples (order of vanishing, power series rings, p-adic valuation, divisoral valuations), completions, maximal-completeness, Hensel's Lemma; alternate characterizations of henselian fields in terms of extensions of valuations, weakene or strengthened hypotheses for Hensel's Lemma].
• Lecture 2: (Scanlon) Languages for valued fields [language of rings, Denef-van den Dries D-function, valuation as relation, predicate for valuation ring, *multi-sorted* structure --- (K,k,\Gamma), RV sorts, (partial)-cross sections, Macintyre power predicates] and their interpretations (eg definability of valuation in ring language under appropriate hypotheses on nondivisibility of value group); aleph_1-compactness and cross-sections; coarsening construction.
• Lecture 3: (Macpherson) QE for ACVF; Holly's theorem.
• Lecture 4: (Süer) Direct Cohen/Macintyre QE proof for p-adics.
• Lecture 5: (Scanlon) General AKE theorems [we may want to give multiple proofs --- general relative completeness & QE via back-and-forth in the RV-language or in the three-sorted language with cross section].
• Lecture 6: (Süer) Artin conjecture [the algebra required to prove Lang's theorem is not all that difficult, we should present the full proof].
• Lecture 7: (Macpherson) p-adic analytic functions; Denef-van den Dries Theroem.
• Lecture 8: (Macpherson) Rationality of Poincaré series; p-adic exponentials; p-adic power functions.
• Lecture 9: (Scanlon) Relative Frobenius, connection to Teichmüller character; possibly QE proof, at least, QE statement