On some elementary invariants of fields.
(pdf) (15 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.
There is, I believe, a finite amount of "general field theory". Most graduate level algebra courses concentrate on the structure theory of algebraic field extensions, which is of course
both beautiful and useful. However, many algebraists need to know more than this. Speaking
as an arithmetic algebraic geometer, the structure theory of transcendental extensions --
and especially, the notion of a separable transcendental extension -- inevitably comes up,
as do certain other constructions which don't seem to make it into the standard course:
e.g. linear disjointness, composita, tensor products of fields, a systematic treatment of
norms and traces.
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 70%!) Moreover what is present is quite rough: please consider it only a first draft.
Quadratic Forms: Structure of the Witt Ring.
(pdf) (20 pages)
Quadratic Forms Over Discrete Valuation Fields
(pdf) (13 pages)
Quadratic Forms Over Global Fields
(pdf) (16 pages)
Sequences and Series
Sequences and Series: a Sourcebook.
(pdf) (129 pages)
All the set theory I have ever needed to know. (40 pages total) 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' Naive Set Theory or Kaplansky's Set Theory and Metric Spaces
(both excellent texts) but that isn't off-puttingly foundational and/or
axiomatic (i.e., that treats set theory as mathematics, rather than a
confusing amalgamation of mathematics and meta-mathematics). For
instance, the standard axioms only allow the elements of sets to
themselves be sets (so that, e.g., mathematicians do not form a set)
and forbid a set from containing itself, although neither of these
options seem logically contradictory. But assuming that we are only
interested in sets up to equivalence (i.e., bijection), it doesn't
matter -- a trick of von Neumann allows us to put any set containing
"individuals" and perhaps containing itself into bijection with a
"pure" set that does not have either of these properties. I wish
someone had told me this a long time ago! So, you guessed it, I am
writing up my own notes.
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)
Notes from my 2005 ISM course: (106 pages total)
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 10: Integral structures, genera and class numbers.
(pdf) (16 pages)
(I do not have any good explanation for the bizarre numbering. In
actuality there were many more than 12 lectures, and there was nothing
exceptional about the lecture I gave on linear algebraic groups, except
that when I defined unipotent groups one of the attendees had the guts
and honesty to ask, "What is the point of all this?" The point is that
you need to know about a whole lot of different things to understand
the definition of a Shimura variety!)
Some (unpolished) notes on uniform distribution.
(pdf) (20 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.