Pete
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Draft (pdf) (303 pages)

These notes, mostly written after I attended the 2003 Arizona Winter School on model theory and arithmetic, give a sort of introduction to the model theory of fields (assuming, unfortunately, that you know some model theory and some arithmetic geometry and have somehow never managed to combine them!). The point of departure is the search for "invariants" of elementarily equivalent fields. Among other things, I point out that a standard conjecture relating period and index in the Brauer group would play the same role as the Milnor Conjecture did in defining the transcendence degree of an absolutely finitely generated field. But I picked a bad time to try an exposition on the model theory of fields: since this paper has been written, Scanlon has proven the equivalence of elementary equivalence and isomorphism for finitely generated fields of characteristic zero, and Poonen and Pop have presented much stronger definability results than the ones I discuss in Section 2. Perhaps someday I will update the exposition to include these exciting results.

The notes which follow aim to be a "serious" account of all aspects of general field theory. At the moment they cover about half of the material that they should, unfortunately for the most part the better known half. (Update: now about 62%!) Moreover what is present is quite rough: please consider it only a first draft.

Early Draft (pdf) (107 pages)

Chapter I: Basics (16 pages)

Section 1: From Metric Spaces to Topological Spaces (pdf)

Section 2: The Notion of a Topological Space (pdf)

Section 3: Alternative Characterizations of Topological Spaces (pdf)

Section 4: The Lattice of Topologies on a Given Set (pdf)

Section 5: (Neighborhood) Sub/bases (pdf)

Chapter II: Convergence (pdf) (29 pages)

Total: 83 pages

Most of the time, most of us don't need to know more about set theory than the distinction between finite, countably infinite, and uncountable sets. But once in a while it's nice to know a little bit more: e.g. the least uncountable ordinal comes up in topology, or your colleague asks you for a counterexample to the variant of Zorn's Lemma with "chain" replaced by "countable chain." But it is hard to find a treatment of set theory that goes a little beyond Halmos'

Chapter 1: Finite, countable and uncountable sets. (pdf) (11 pages)

Chapter 2: Order and Arithmetic of Cardinalities. (pdf) (8 pages)

Chapter 3: Ordinalities and their arithmetic; von Neumann's ordinals and cardinals. (pdf) (18 pages)

Chapter 4: Cardinality Questions. (pdf) (3 pages)

These are all the notes I typed up for my 2005 ISM course on Shimura varieties. In the lectures, I presented more material on Hilbert and Siegel modular varieties, adelic double coset constructions, and strong approximation than has survived in the lecture notes. Most of the omitted material is of a rather standard sort -- it appears in many places -- which is not to say that it shouldn't appear here as well. The reader will notice that the notes are significantly more polished at the beginning and the end than in the middle. I am quite pleased with the very last lecture, which seems to put some of the pieces of the theory together in a new way. I would like to see more detail on arithmetic groups and lots more detail on quaternion orders and trace formulas. Inevitably for notes of this length, the most important results -- like the existence of rational and integral canonical models -- get stated and kicked around a bit but not proved. To remedy this will require significantly more work.

- Lecture 0: Modular curves.
(pdf) (6 pages)

- Lecture 1: Endomorphisms of elliptic curves.
(pdf) (13 pages)

- Lecture 2: Fuchsian groups.
(pdf) (18 pages)

- Lecture 3: More Fuchsian groups.
(pdf) (6 pages)

- Lecture 4: Arithmetic Fuchsian groups.
(pdf) (6 pages)

- Lecture 4.5: A Crash Course on Linear Algebraic Groups.
(pdf) (7 pages)

- Lecture 5: The Adelic Perspective.
(pdf) (4 pages)

- Lecture 6: Special points and canonical models.
(pdf) (10 pages)

- Lecture 7: Real points.
(pdf) (6 pages)

- Lecture 8: Quaternion orders.
(pdf) (10 pages)

- Lecture 9: Quaternionic moduli.
(pdf) (4 pages)

- Lecture 10: Integral structures, genera and class numbers. (pdf) (16 pages)

These are notes on the basics of uniform distribution of sequences, taken on occasion of the Dover republication of the very nice book on this topic by Kuipers and Niederreiter. The notes are incomplete, not including a full-blown treatment of uniform distribution in a compact (or locally compact) group.

Total as of March, 2013: 2021 pages