Endomorphisms of abelian varieties defined over the field of algebraic numbers.
The classification of endomorphism algebras of g-dimensional abelian varieties over the
complex numbers was initiated by Albert and completed by Shimura. The abelian varieties whose endomorphism algebra contains a given algebra form a subvariety of the Siegel moduli space (a PEL-type Shimura variety) defined over Q-bar; like any algebraic variety
over Q-bar, it has Q-bar valued points iff it has C-valued points. This means however that the varieties whose endomorphism algebra is precisely equal to a given algebra form an algebraic stratum in the Siegel space -- namely, that which remains upon removing from an irreducible algebraic variety a countable union of proper closed subvarieties. Standard topological arguments (e.g. measure zero considerations) show that an algebraic stratum has many C-valued points. However, it is obviously false in general that an algebraic stratum defined over Q-bar must have Q-bar valued points: since Q-bar is countable we could just
remove the points one by one! We show that nevertheless the PEL-type Shimura strata have
Q-bar valued points, or in other words that the classification of endomorphism algebras over Q-bar is the same as the classification over C. Although plausible, this seems to require
some machinery to establish: our proof uses a beautiful local-global principle for
varieties over Q-bar due to (my UGA colleague) R. Rumely, together with the density of CM points in PEL-type Shimura varieties and the Honda-Tate classification of endomorphism algebras of abelian varieties over finite fields.
Ramanujan graphs and Shimura curves.
We make explicit the connection between all known infinite
families of (q+1)-regular Ramanujan graphs and modular (Shimura and Drinfeld-Rapoport-Stuhler) curves. By working systematically in the context of Hecke operators
on Shimura curves defined over suitable totally real number fields -- as opposed to classical modular curves or Drinfeld modular curves -- we construct, for all positive integers q > 1,
infinite families of (q+1)-regular finite graphs all of whose nontrivial eigenvalues have
absolute value at most 2^(omega(q+1)) sqrt(q), where omega(n) is the number of distinct
prime divisors of n. This is a simultaneous generalization of theorems of Ihara, Lubotzsky-Phillips-Sarnak, Chiu, Morgenstern and Pizer.
Algebraic curves uniformized by congruence subgroups of triangle groups (with John Voight).
Stay tuned...
Period-index problems in WC-groups III: biconic curves.
I have conjectured that for any elliptic curve E defined over a global field K and any integers P, I > 1 such that P | I | P^2, there exist infinitely many elements of the Weil-Chatelet group of E with period P and index I. The paper just below with S. Sharif handles the case I = P^2 (and P indivisible by the charateristic of K). This preprint treats the case of P = 2 (and K of characteristic not 2). The approach exploits a "neo-classical 2-descent" found in a 1998 paper of Cassels. The general theory is also applied to the study of WC-groups of elliptic curves over R (a toy case) and R((T)) (building upon work of Scheiderer and Ducros).
Realistically, it seems that not all of these projects will be completed by the end of the 20XY-20X(Y+1) academic year. If you are hoping that a particular one will be finished ASAP, please do let me know...
Period-index problems in WC-groups II: abelian varieties.
(pdf)
Accepted and/or Published:
[10] An "Anti-Hasse Principle'' for prime twists, to appear,
International Journal of Number Theory. (pdf )
[9] On the indices of curves over local fields, to appear, Manuscripta
Math. (pdf)
[8] Abelian points on algebraic varieties, to
appear, Math. Research Letters. (pdf)
[7] Local bounds for torsion points on abelian varieties (with Xavier Xarles), to appear, Canadian J. Math. (pdf)
[6] On the Hasse principle for Shimura curves, to appear, Israel J. Math.
(pdf)
[5] Galois groups via Atkin-Lehner twists, Proc. AMS. (pdf)
[4] There are genus one curves of every index over every number field. J. Reine Angew. Math. 594 (2006), 201-206.
(pdf)
[3] On elementary equivalence, isogeny and isomorphism, to appear in J. Théor. Nombres Bourdeaux.
(pdf)
[2] On the Number of Representations of an Integer by a Linear Form (with G. Alon). Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.2
(free online publication)
[1] Period-index problems in WC-groups I: elliptic curves. J Number Theory 114 (2005), 193-208. (pdf)
The versions of the papers listed here are isomorphic, but not necessarily identical, to
the final published versions. In particular they incorporate all responses to referees'
comments, so in many cases contain substantial expository and mathematical improvements
over the corresponding arXived versions.
Thesis:
[0] Rational points on Atkin-Lehner quotients of Shimura curves. Ph.D. thesis, 2003. (pdf)
Caveat emptor: You are advised not to read either Chapter 1 or Chapter 2. There is a substantial error: Proposition 90 is false, which invalidates Main Theorem 1. (Indeed, later work of Rotger and Murabayashi showed that this theorem is also false as stated. The truth is significantly more complicated than I had expected at the time!) To the best of my
knowledge, the other results are essentially correct. Also, Main Theorem 3 was derived independently by Ogg in the Mauvaise reduction paper cited in the references, although the proof I gave is different and, perhaps, of some interest. In short, it is a bit of a mess, and I would advise you not to use this thesis as a reference in the formal sense:
if you wish to use a result in the thesis, cite an earlier or (see below) later paper, if
possible. If you cannot find another reference, ask me for one!
Published thesis work: Although I have not written any papers which solely contain
material from my thesis, several papers are strongly influenced by my thesis work and
contain results which are improvments / simplifications / mild corrections of certain results
of my thesis. Specifically, [6] began as a writeup of Main Theorem 5
of [0], but when I looked back I found that I was able to exploit new techniques
to prove a much more attractive result. (On the other hand, the results of [6] do not literally include Main Theorem 5.) Main Theorem 6 and its application to XD1(N) appears in [7]. Main Theorem 2 is applied in [10]; as discussed there, in the meantime Rotger, Skorobogatov and Yafaev proved a more general result (by methods which
include and generalize those that I used) and graciously acknowledged my priority
on the special case, so in writing up [10] I felt that there would be no value added to
include my proof.
Still unpublished thesis work: Main Theorem 4 -- a QM analogue of
Mazur's theorem on rational isogenies of prime degree, which I thought of as the
"main" Main Theorem of [0] -- has never been published in any form. So far as I know
the proof is correct (up to some minor changes necessary to circumvent the erroneous result
in Chapter 1) and seems to be of some interest. However, I have always viewed this theorem
as a rough first step to a more satisfactory and complete analysis of rational points on XD0(N). Since many experts in the field have read my thesis and are
hence aware of Main Theorem 4, there doesn't seem to be much advantage to publishing an
"unripe" theorem. Finally, the more precise bounds on torsion on QM abelian surfaces have
never been published. In this regard, it seems to me that the results would be much more
interesting in conjunction with some nontrivial examples, and that this is a reasonable
future project for someone (but maybe not for me).
An Unpublished (Mostly) Expository Paper
A site-theoretic characterization of points in a topological space
(with Andrew Archibald and David Savitt). (pdf)
Motivated by a question of Eyal Goren, Andrew Archibald, David Savitt and I rediscovered a theorem comparing points of a topological space X in the usual sense to the "site theoretic points" -- i.e., the morphisms of sites from the topological site of the one-point space to the topological site of X, a result which was originally found by Grothendieck and Hakim, and stated (but not proved!) in SGA4. It turns out that this result bears approximately the same relationship to a certain school of lattice theory / category theory that the fundamental theorem of arithmetic bears to number theory: it is, to say the least, well known. But what if you are interested in the question -- and its generalizations to the etale site, the flat site, and so on -- from the perspective of broadening your understanding of Grothendieck topologies on schemes, and do not want to learn the lattice-theoretic or topos-theoretic languages? Perhaps then this is the paper for you. I do believe that this 75% expository paper should be published somewhere, but neither I nor my coauthors really know where. Would you like to publish it?
Notes from Seminar Talks
For several years now I have typed up notes accompanying most seminar talks I have given. Nowadays, when I give a talk I prepare some notes in advance, give the talk based on (usually some proper subset of) these notes, and then go back and revise/lengthen/correct the notes based upon the feedback I have received. Finally, I make the edited notes available to the audience members. (Or at least, I claimed that I would; in many cases that claim is being realized here for the first time.) I have found this to be a good system. First, I have a chance to present people with a more durable memory of my talk (and, sometimes, I am more than happy to have that refine or replace their memory of the actual talk!) and conversely they are much more likely to read the notes having attended the talk. And second, typing things up slows me down enough to keep the material in my head for a comparatively long time, and very often this leads me to things I would otherwise not have thought of.
Here are some of these typed up lecture notes. If you heard me give a talk and promise you lecture notes that you don't see here, please let me know. More than likely I have a 75% finished file lying around somewhere. It would be a mistake to assume that I am an expert on all topics that are covered in these notes or that they are in some sense definitive: you should know that they have not been refereed or systematically checked by anyone, except possibly by me.
Ramanujan Graphs Part II.
(pdf)
From a 3/21/07 talk given in the UGA number theory seminar.
Recent thoughts on abelian points.
(pdf)
From a 2/7/2007 talk
given in the UGA number theory seminar. The talk describes the work from my above "abelian
points" paper. Details of proofs -- and especially, Galois cohomological calculations --
were often suppressed in favor of providing additional (and a posteriori) motivation
for these results in terms of Kodaira dimension and field arithmetic. I have recklessly
propounded a conjecture about metabelian points on algebraic varieties and explained a connection with the inverse Galois problem.
Mumford-Tate groups and abelian varieties.
(pdf) From a 11/9/2006 talk given in Elham Izadi's VIGRE seminar on the Hodge Conjecture.
Selections from the arithmetic geometry of modular and Shimura curves.
(pdf) From two talks given at the UGA number theory seminar in August and September 2006. Fielding a request from my new colleagues, I took a crack at ``going from zero to Shimura curves in two talks.''
Rational points on Atkin-Lehner twists of modular curves.
(pdf)
From a talk given at the University of Pennsylvania in July of 2006. If you think you might be interested in twisted modular curves and applications to Galois theory, perhaps you will enjoy reading this document more than the two research papers (Anti-Hasse Principle... and Galois groups...) of mine which cover the same material.
On period-index problems.
(pdf)
These are the "long form" lecture notes of a two hour talk I gave on April 24, 2006 at MSRI. This is the most ambitious talk I have ever given: I described all known period-index problems, their interrelationships, and I pointed out how the period-index problem in
higher-dimensional WC-groups is related to Brauer groups, Shafarevich-Tate groups, and the so-called "elementary obstruction." The notes need to be polished (posting them here makes it much more likely that this polishing will take place). It was while preparing this talk that I began to appreciate how fruitfully the thesis work of Shahed Sharif and my own work could be combined, and portions of this paper give a preview of our forthcoming joint work.
(click here for the short version)
An introductory lecture on WC-groups.
(pdf)
These are notes inspired by a (too) short talk I gave on WC-groups at MSRI on March 17, 2006. After the talk the task became to record everything I wanted to say about WC-groups, so of course the document got longer and longer as I continued to work on it. At present I seem to have approximately half (the easier half) of a fairly substantial survey of the structure of WC-groups in general. Perhaps later I will develop this into a formal paper or a monograph.
Acquisition of rational points on algebraic curves.
(pdf)
These are the slides for my UGA job talk. It turned out that I had at least 50% too much material (so that the talk was in fact not that good; I am thankful to UGA for hiring me anyway!), some of which was not really what I should have been discussing in a job talk but was nevertheless sort of interesting (and does not appear in any of my other writings), so here it is.
Introduction to the real spectrum.
(pdf)
Notes from the second talk I gave at Colorado College in early December 2004 (the first was on the discrete geometry of Chicken McNuggets). The purpose of these notes is to convince a general mathematical audience that the collection of all orderings on a field is an interesting geometric object. Moreover, it develops some of the background material that one needs to appreciate "real arithmetic geometry," a topic of some very interesting recent papers of Colliot-Thelene, Parimala, and Scheiderer.
Notes on the DeRham cohomology of varieties.
(pdf)
Notes from a series of talks given in Eyal Goren's 2003 cohomology theories seminar at McGill University (click here for the seminar webpage). These notes give a discussion of classical Hodge theory and DeRham cohomology first in the analytic category and then algebraically, as a preparation for crystalline cohomology. Not exactly light reading. Chapter 3 still seems to be a reasonably good short introduction to the Hodge Theorem, although in its free use of sheaves and cohomology, it is not maximally elementary.
On beyond line bundles: the Brauer group of almost anything.
(pdf)
Notes from the last talk I gave in Harvard's Trivial Notions seminar (the grad student seminar: organized, lectured and attended by grad students -- and the odd wunderkind undergrad -- no faculty allowed). Here I push the idea that, although "everybody" (at least, everybody at Harvard, where algebraic geometry is tested on the first-year qualifying exams) is familiar with line bundles and the Picard group in a wide variety of geometric contexts, the idea that one can study the projective bundles modulo projectivizations of vector bundles (on almost anything) is much less familiar. All material is taken from Grothendieck's three papers on the subject.
Fundamental groups in characteristic p.
(pdf)
Notes from a guest lecture in Romyar Sharifi's Galois actions on fundamental groups seminar, at Harvard in 2002.