Notes for my MSRI lectures on 3/13/18 pdf
My mathematical work is supported by NSF grant DMS-1100784.
I am an editor for Journal of the AMS, Algebra and Number Theory, and IMRN. Please consider submitting appropriate papers to these journals. All such submissions must go through the journal website; papers cannot be submitted through editors, and any submissions on famous open problems are subject to the strict rules as described at the lower half of this page. [JAMS has acceptance standards on par with Annals of Math and accepts around 30 papers per year across all areas of math, ANT is the top journal for specialized papers in algebra and number theory, and IMRN is a general-interest journal with an acceptance standard roughly at the level just below that of Duke Math Journal.]
If you are taking a class from me then you should find a functioning link to it below (with course information, homeworks, and so on).
If several days go by without a response to email, I am probably away from home.
Number theory and representation theory seminar Analytic number theory, algebraic number theory, arithmetic geometry, automorphic forms, and even some things not beginning with the letter "a". It's a big subject. Schedule and notes for the 2017-18 Seminaire Godement Schedule and notes for the 2016-17 Seminaire Deligne-Laumon Schedule and notes for the 2015-16 Seminaire BSD/Bloch-Kato Schedule and notes for the 2014-15 Seminaire Scholze Schedule and notes for the 2013-14 Seminaire Jacquet-Langlands Schedule and notes for the 2012-13 Seminaire Shimura Schedule and notes for the 2011-12 Seminaire Darmon Schedule and notes for the 2010-11 Mordell seminar Schedule & notes for the 2009-10 modularity lifting seminar Links to current courses: Graduate Algebraic Geometry (fall) ,
Here are handouts and homeworks from some past undergraduate courses:
Undergraduate algebraic geometry , Galois theory, Undergraduate algebraic number theory Here are handouts and homeworks from some past graduate courses:
Linear algebraic groups I , Linear algebraic groups II , Abelian varieties, Alterations , Graduate Algebraic Number Theory, Class field theory, Modular curves, Intro Graduate Algebra (winter)
Some differential geometry I once taught an introductory differential geometry course and was rather disappointed with the course text, so I went overboard (or crazy?) and wrote several hundred pages of stuff to supplement the book. If you are learning elementary differential geometry, maybe you'll find some of these handouts to be interesting. Most likely I will never again teach such a course. Ross Program, PROMYS Program, and Epsilon Fund Some mathematicians are entirely self-created; for the rest of us, assistance and encouragement early on is helpful. These programs do an excellent job in that direction.
Research
First, some caveats.
0. Links to files undergoing revision may be temporarily disabled.
1. If you want to know where something below was published (if it has appeared in print), then please look on MathSciNet.
2. Someday I should join the 21st century and post papers on the arxiv, at least after I can no longer make changes to the version to be published. In particular, I should post my old papers that have already appeared. Unfortunately (?), I tend to rewrite things too many times and don't wish to keep posting revision on top of revision on the arxiv. Posting things here (even in far-from-final state) partially compensates for my pedantry, I hope.
Are you looking for how to get a copy of the pseudo-reductive book with Gabber and Prasad? Or a draft copy of the CM book with Chai and Oort? If so, scroll down to the "Book" section below.
Classification of pseudo-reductive groups pdf
Reductive group schemes (notes for "SGA3 summer school"). pdf This proves main results from SGA3 using Artin's work and the dynamic method from "Pseudo-reductive groups" to simplify proofs. Here is a link to some other notes from the summer school, inspired by a lecture given there by B. Gross on non-split groups over the integers, and a related file by J.-K. Yu providing computer code used in some of the computations therein.
Algebraic independence of periods and logarithms of Drinfeld modules (by C-Y. Chang, M. Papanikolas). pdf
I provided the appendix (a pseudo-application of pseudo-reductive groups).
Lifting global representations with local properties pdf
Universal property of non-archimedean analytification. pdf
Descent for non-archimedean analytic spaces (with M. Temkin). pdf
Finiteness theorems for algebraic groups over function fields. pdf
Moishezon spaces in rigid geometry. pdf
Nagata compactification for algebraic spaces (with M. Lieblich, M. Olsson). pdf
Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence (by K. Prasanna). pdf
I provided the appendix.
Non-archimedean analytification of algebraic spaces (with M. Temkin). pdf
Chow's K/k-image and K/k-trace, and the Lang-Neron theorem (via schemes). pdf
This largely expository note improves the non-effective classical version of the Chow regularity theorem, and generally uses infinitesimal methods and flat descent to replace Weil-style proofs that I could not understand. It is also cited in "Root numbers and ranks in positive characteristic" below.
Prime specialization in higher genus II (with K. Conrad and R. Gross). pdf
Prime specialization in higher genus I (with K. Conrad). pdf
Higher-level canonical subgroups in abelian varieties. pdf
Modular curves and rigid-analytic spaces. pdf
Relative ampleness in rigid-analytic geometry. pdf
Root numbers and ranks in positive characteristic (with K. Conrad and H. Helfgott). pdf
Arithmetic moduli of generalized elliptic curves. pdf
Edixhoven has a different approach to these matters when the moduli stacks are Deligne-Mumford.
Prime specialization in genus 0 (with K. Conrad and R. Gross). pdf
Modular curves and Ramanujan's continued fraction (with B. Cais). pdf
A short erratum to this paper. pdf
"On quasi-reductive group schemes" (by G. Prasad and J-K. Yu). pdf
I provided the appendix.
The Möbius function and the residue theorem (with K. Conrad). pdf
This is a companion to "Prime specialization in genus 0" above.
J1(p) has connected fibers (with S. Edixhoven and W. Stein). pdf
Finite-order automorphisms of a certain torus. pdf
Gross-Zagier revisited (with appendix by W. R. Mann). pdf
Power laws for monkeys typing randomly: the case of unequal probabilities (with M. Mitzenmacher). pdf
Approximation of versal deformations (with A. J. de Jong). pdf
A modern proof of Chevalley's theorem on algebraic groups. pdf
Component groups of purely toric quotients (with W. Stein). pdf
On the modularity of elliptic curves over Q (with C. Breuil, F. Diamond, R. Taylor). pdf
Inertia groups and fibers. pdf Correction to "Inertia groups and fibers" pdf
Irreducible components of rigid spaces. pdf
Modularity of certain potentially Barsotti-Tate representations (with F. Diamond and R. Taylor). pdf
Remarks on mod-ln representations with l = 3, 5 (with S. Wong). pdf
Ramified deformation problems. pdf
Finite group schemes over bases with low ramification. pdf
Books
Here are some books, pre-books, etc.
Complex multiplication and lifting problems (with C-L. Chai, F. Oort) order while supplies last! (this link might be behind a firewall; the ISBN number is ISBN-10: 1-4704-1014-1)
Pseudo-reductive groups (with O. Gabber, G. Prasad) 2nd edition! Some exercises on group schemes and p-divisible groups. | Homework 1: pdf. |
| Homework 2: pdf. |
| Homework 3: pdf. |
| Homework 4: pdf. |
These are the "homework" exercises for a week-long instructional workshop for graduate students co-organized with Andreatta and Schoof in May, 2005. These were too many exercises for the amount of time given. But if you have more than a week to spend on them then perhaps some of the exercises will be helpful or interesting if you are taking your first steps in this direction. Since the lectures that naturally accompany these exercises are not recorded here, a recommended substitute is some of the written lecture notes from the ``Notes on complex multiplication'' (see above) and a lot of asparagus.
What do these 5 people have in common?
Draft of Andreatta's notes for course at 2009 CMI summer school pdf 
