Brian Conrad

Department of Mathematics
Stanford University
Building 380, Sloan Hall
Stanford, CA 94305, USA

office: 383CC, Sloan Hall

Photo Credit: Rod Searcey

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.


    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


    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!

    Grothendieck duality and base change   tar
    Clarifications and corrections to "Grothendieck duality and base change"   pdf
    An addendum to Chapter 5 of "Grothendieck duality and base change"   pdf

    My book on Galois representations and modular forms is still undergoing revisions. (It is now shorter than it was before, with much better proofs; if you have an earlier version, please burn it.) The following link has been disabled.

    Modular forms and the Ramanujan conjecture   pdf

    Here is a very rough draft of a book with M. Lieblich (more to be included, such as exercises, applications to the theory of the monodromy pairing and the p-adic good reduction criterion, etc., and much to be revised). Whoops, the link is now disabled (due to some strange errors that need to be fixed).

    Galois representations arising from p-divisible groups   pdf


    Here are some expository things I have written. These may be updated without warning. Links to files undergoing revision may be temporarily disabled.

    The basic guiding principles for deciding what to write up and post here are two-fold.

    1. Interesting (to me?) alternative proofs of known results, or explanations of important topics that can be difficult for beginners to learn are fair game. (As the literature improves, I may augment postings accordingly.)

    2. With all due respect to the role of Andre Weil in the development of algebraic geometry, nobody should ever again have to read Weil's "Foundations of algebraic geometry": EGA must be an adequate logical starting point for the subject. Hence, if there is an important, interesting, or useful theorem whose published proofs use pre-Grothendieck methods in such an essential way so as to render them impenetrable to later generations (or to me?), and if I have a need to understand why the theorem is true and consequently I figure out a scheme-theoretic proof, then I'll try to write it up.

    The structure of solvable groups over general fields. pdf
    These are notes that mildly revise Appendix B of the "pseudo-reductive" book, to accompany some lectures at the 2011 "SGA3 summer school" in Lumimy.

    Some notes on topologizing the adelic points of schemes, unifying the viewpoints of Grothendieck and Weil.   pdf

    Some lecture notes on p-adic Hodge theory, from a course I taught with Olivier Brinon at the 2009 CMI summer school on Galois representations. It is undergoing regular revision; not yet in final form (so corrections welcome).   pdf

    2007 Arizona Winter School lectures on rigid geometry   pdf
    This appeared as a chapter in a book produced by the Winter School.

    Formal GAGA for Artin stacks.   pdf
    The paper "On proper coverings of Artin stacks" by M. Olsson gives a different point of view on this topic.

    Rosenlicht's unit theorem   pdf
    Rosenlicht proved an extremely interesting theorem on the structure of units on a product of varieties. Since Rosenlicht's proof doesn't address the generality cited without proof in SGA7, and it was written in archaic terminology, this short note gives a "modern" treatment (and eliminates the assumption on existence of rational points).

    Integration in elementary terms   pdf
    Have you told your calculus class that the Gaussian integral cannot be computed in elementary terms, or likewise for elliptic integrals when teaching a course on Riemann surfaces, but you personally have no idea how such a (useless but pretty) result is proved? If so, then you should definitely read Rosenlicht's article "Integration in finite terms" in volume 79 of the American Mathematical Monthly (in 1972). But if you have to explain the proof to talented high school students, this short note (based on a talk I gave to such students at CMI) may be helpful.

    Keel-Mori theorem via stacks.   pdf
    In this note, we use stacks instead of groupoids to give a streamlined proof of the Keel-Mori existence theorem for coarse moduli spaces (useless bonus: noetherian hypotheses eliminated).

    Minimal models for elliptic curves.   pdf
    The book "Algebraic geometry and arithmetic curves" by Q. Liu treats much (but not all) of this material.

    Cohomological descent.   pdf
    WARNING: these notes are essentially written in the context of topology and etale sites; this is adequate for arithmetic applications (potential semistability of l-adic cohomology via alterations) but is woefully inadequate for geometric applications (quasi-coherent cohomology on Artin stacks). The level of generality (and aspects of the writing style) will be much-improved in the next version so that the notes suffice to treat geometric applications too.

    Descent for coherent sheaves on rigid-analytic spaces.   pdf

    The paper "Coherent modules and their descent on relative rigid spaces" by S. Bosch and U. Görtz gives a different approach to this topic (pre-dating mine, as I found out later).

    Nagata's compactification theorem (via schemes).   pdf
    This is a detailed exposition of some private notes of Deligne. The paper "On compactification of schemes" by W. Lütkebohmert gives a different approach to this topic in the noetherian case, but Deligne's approach gives some striking results of general interest for rational maps. If you can understand what is going on in Nagata's original paper then you are either very old or very smart (or both!). These notes were eventually published in Journal of the Ramanujan Math Society (vol. 22, 2007). In the published version there was a tiny mistake in one definition in the middle of a long proof, but this error was localized and easy to fix. There was also a tiny error in the appendix (never used in the main text). Here is the official erratum (but the above link is to a file in which the errors are corrected).   pdf

    The classical Riemann-Hilbert correspondence.   pdf

    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