Deligne’s proof of the Weil conjectures

Today, I will jump on the coat-tails of Terry Tao’s link to Brian Osserman’s article on the Weil Conjectures for the (upcoming) Princeton Companion to Mathematics (edited by Tim Gowers and June Barrow-Green), and use this as an occasion to mention an old text of mine which was meant to be an introduction to Deligne’s (first) proof of the Riemann Hypothesis over finite fields. [The date of 2008 on the PDF file simply reflects that I recompiled it recently to incorporate a PDF version of the single figure in it, which I may as well include here to break the textual monotony…

vanishing.png

this is supposed to illustrate the famous “cycle évanescent”, a terminology which I find somewhat more poetic in French, where “évanescent” is rather old-fashioned and formal, than in the English “vanishing cycle”, although the etymology is in fact the same].

I say “text” because it’s a bit hard to define in what category this could/should be put. I wrote it as a Graduate Student at Rutgers in 1997, as the text for two lectures I gave at the end of a graduate course by my advisor, Henryk Iwaniec, on the general topic of solutions of equations over finite fields. His lectures had ended with a detailed proof of the Riemann Hypothesis for curves (and their all-important consequence, for analytic number theorists, the Weil bound for Kloosterman sums), following Bombieri’s adaptation of Stepanov’s method.

I took this background as starting point to give as complete a sketch as I could (given my understanding of the algebraic geometry at the time…). This means the text is not really self-contained: the basic theory of algebraic curves is assumed to be known, at least informally. However, the somewhat original feature (which is why I am mentioning this text here) was that I tried to explain the origin, the formal definition and some of the properties of étale cohomology “from scratch”. This culminates, at the end of the first part, with an almost complete and rigorous computation of the étale cohomology of an elliptic curve over an algebraically closed field (based on the basic theory of isogenies of elliptic curves). This is in fact not difficult at all, but I had not seen it done elsewhere in a self-contained way, and it certainly seemed enlightening to me.

As implied in the previous paragraph, there are two parts to the paper. The first is the preparation (i.e., supposedly leading to where things stood before Deligne’s breakthrough), and it still seems to have some value for people interested in starting to learn this important theory. The second tries to follow Deligne’s proof very closely, and is in fact probably hard to read, and therefore of less interest. (Moreover, Deligne’s first proof was largely subsumed in his second proof, which leads to much more general statements, and in particular to his Equidistribution Theorem; this second proof is really quite a bit more involved than the first one, in terms of algebraic geometry).

Averages of singular series, or: when Poisson is everywhere

Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text…

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes p for which the gap q-p between p and the next prime q is smaller than ε times the average gap, for arbitrary small ε>0.

This result refers to the behavior, on average over h=(h_1,…,h_k), of the constant S(h) which is supposed to be the leading coefficient in the conjecture

|{n<X | n+h_i is prime for i=1,…,k}|~S(h) X(log X)-k

Gallagher showed that the average value of S(h) is equal to 1, and I’ve extended this in two ways…

[LaTeX 1]

[LaTex 2]

Schlomo Cohen

Quite a few years back, when I was finishing the (in)famous Classes préparatoires, I started writing a series of stories entitled Les fabuleuses aventures de Schlomo Cohen le Mathématicien détective (“The Wonderlous Adventures of Schlomo Cohen, Detective-Mathematician”); after finishing four texts ranging in length from a short story to a modestly-sized novel, this ended around 1994 when I really didn’t have time anymore for the type of imaginative concentration needed for even my level of fiction-writing.

As the title already suggests, there was a strong influence of British-style super-detective crime fiction when I started, based on reading too much Sherlock Holmes and Agatha Christie when I was a few years younger. Indeed, Schlomo Cohen is said to be “the best non-fictional private detective in London”. However, the third, and longest, story, concludes in Los Angeles with a clear debt to R. Chandler, since P. Marlowe makes his appearance, showing at least some improvement in stylistic debt over the years, and the last one is full of direct or indirect quotations of “The Waste Land”…

The other two obvious characteristics are that the hero, the superior detective S. Cohen, is (1) a mathematician; (2) Jewish. The second part may seem somewhat strange (I am not Jewish myself), and is due mostly to the twin influences of I. Bashevis Singer and W. Allen when I started writing the stories (in fact, S. Cohen is theoretically quite orthodox, quoting the Talmud and other sacred texts rather freely, and his mother, Masha Cohen, plays a big part in the last three stories).

The first characteristic (mathematics) is of course more understandable, and was for me the source of much of the fun in writing the stories. The first idea, as far as I remember, was to use sophisticated plots carefully designed so that insights from great theorems and their proofs would be genuinely useful to solve the mysteries S. Cohen was confronted with. This was quickly replaced with essentially completely random associations d’idées, which (quite obviously) make no sense whatsoever, but which nevertheless lead S. Cohen to the guily partie(s).

Here are the main mathematical results Schlomo Cohen claims led him to solve the outstanding problems of the age:

  • Some theorems of Church on ultrapowers (which I don’t remember at all!) in Le vol du traité secret (“The theft of the secret treaty”);
  • The Hahn-Banach theorem in De la banane dans le Bourgogne (“Banana in Burgundy”);
  • The theory of Lefschetz pencils, van der Waerden’s theorem on finite colorings of the integers, Riemannian geometry, in Schlomo Cohen contre les maîtres criminels (“Schlomo Cohen against the master criminals”);
  • Non-euclidean geometry, in Le traducteur subreptice (“The surreptitious translator”).

Quite a few other results are discussed, and they mostly show what type of mathematics I was learning (and finding striking!) at the time.

I am mentioning this today mostly because I have just discovered that the second story (“Banana in Burgundy”) has become part of a (semi?) official bibliography of works about Bourbaki (look at “Anonymes” as author). The point is that, in this particular story, I envision Schlomo Cohen as a great friend of André Weil, so much so that he is invited to attend the first Bourbaki Congress, in Burgundy. There, terrible events occur, involving a conspiration against the wines of Burgundy, the French constabulary, and the transformation of all but one of the Bourbakists into amateur detectives…

People more knowledgeable about the history of Bourbaki than I was at the time will of course realize that the date, place and much else doesn’t make any sense at all, but still, even today, I find pleasure in re-reading the exchanges I wrote between mathematicians, and I think they are quite realistic in a way. Indeed, I feel flattered that the bibliography above says that the story is a Récit plaisant et fantaisiste qui décrit tout de même le groupe Bourbaki de façon réaliste (“A pleasant whimsical story which nevertheless describes realistically the Bourbaki group”).

The story, if you want to read it (it’s in French), is online here (search for “Humour”). Note that most of it does not involve Bourbaki at all; for the best parts, you can look around page 26, up to 44 or 45.

I should say that I had put the stories anonymously on the internet a few years ago (except the first one, which was really too embarrassing to my mind), and this is where the authors of the bibliography must have found it. The third story is therefore also available on this original site (if you look for it a bit)… But from a literary point of view, and indeed also plot-wise, the fourth and last one, Le traducteur subreptice is by far the best. There one finds all ingredients for a smashing hit: Masha Cohen’s theory of “Golems of the second kind”; Schlomo’s monograph on “The babylonian Talmud considered as a formal system”; the beautiful daughter of a mysterious Jesuit Father and her sulfurous thesis, “The judicial arsenal of the Spanish Inquisition”; the Vermont senator Philip P. Mark and his erstwhile English revolutionary friends; and the mysterious criminal of the title, whose devious deed is to replace the original manuscript of “The Waste Land” with a French translation… (One also learns in passing that S. Cohen wrote his thesis under the direction of J.E. Littlewood, on “A refinement of the circle method with applications to Waring’s problem”; in “Banana in Burgundy”, on the other hand, he is said to have interesting results concerning torsion points of elliptic curves).

Version control

When writing my papers (and for many other things), I have been using some kind of version control software since about 1999. Among other things, what version control does is to allow you to preserve as many intermediate versions of your work as you want; any time I reach a point that I may want to preserve, a suitable command indicates that the current state of things must be remembered. Later on, it is also easy to recover those intermediate stages, and/or to compare one of them to the current state of the work (or indeed to compare two of the previous stages directly). In particular, it is useful to tag a paper as soon as it is submitted for publication. Then, even if I make changes or corrections while waiting for the refereeing process to slowly snail itself to send a report back, it remains very easy to look back (one, two, three or four years later) to check what was the exact text I had sent to the journal.

Version control is of course also particularly useful when collaborating because no amount of mistyping can lead to a situation where local changes by me erase a brilliant ten page development by a co-author (well, not really, since it is a theorem that for any computer there is a finite combination of letters which, typed properly, will erase everything on it…). It was particularly useful when I was translating into English a friend’s book on methods of mathematics for physicists. We both had the software configured so that any changes could be communicated to the other by asking the software for what had changed between our local copy and the reference version it controls. Since there were around a hundred source files for this book, including copies of the French original and of the (sometimes partially translated) English versions, plus figures, control files, etc, this was really convenient.

It is unfortunately not so trivial to set up version control, which is more commonly used by computer scientists for their projects (source code replacing LaTeX files). It should be fairly easy for a university to offer this as a service to its faculty (ETH, for instance, has something like this for software — anyone can register a software project at origo.ethz.ch and this platform provides all that is needed, including version control, bug tracker, forums for interaction with users, access control for closed-source projects, wiki,…) but it doesn’t seem to be common for more academic work. (It could of course be a good project to adapt the system to this type of situation).

After moving to ETH, the setup I had been using in Bordeaux for about 8 years was not easily reproducible, and it took me a while here in Zürich to arrive at a smooth operation. (It’s still not quite as nice as before, but it’s getting there). I use the svn software for version control (with Linux), mostly for historical reasons (I know there are many other options). For local work on a single computer, the setup would be quite simple, but part of the point of this exercise is that I can keep multiple copies of my (past and present) work on different computers, and continue writing any paper on any of them, submitting the changes to a central server from which they can be recovered on all the others (this is useful also for backup, and during travels; if I make changes on some important files on my laptop during a conference or a visit to wonders of the earth, I can usually expect to be able to save it to the central server when I have internet access, and thus avoid losing too much if the laptop is destroyed or stolen — which has happened to me once…).

The main configuration issue is to have this server accessible from other computers in a semi-permanent manner. My setup in Bordeaux did not translate well to Zürich (partly for reasons of complexity and having forgotten part of the operating procedure…) However, the following works fine for the moment: I use a VPN connection to ETH to connect to the internet, which provides an IP address visible from the outside, even from behind a router/firewall. Then dyndns allows me to keep a permanent URL to access the server (which runs on an Apache web server with the required SVN module to permit access, and fortunately I just had to copy the old configuration file to make this work…)

An exercise with orthonormal basis

While writing the general case of the large sieve, one question of minor interest arose, which provides a nice exercise (or exam problem) for a course on finite-dimensional Hilbert spaces.

Since it’s not yet possible (as far as I can see, but I will try to investigate the issue) to include either LaTeX formulas (in the style used in a number of WordPress math blogs) or MathML formulas in the ETH blogs, I’ve resorted to the rather embarassing option of using dvipng to produce an image with the LaTeX content of this post…

[LaTeX text]

Coming back to regular HTML, where one can make links, here’s one to the short note I wrote on this, with the proof of the result indicated. Note that I would be surprised is this were really at all new. There’s one lingering question that I haven’t answered at the moment: does there exist a proof by pure thought that, for the uniform density, there is an orthonormal basis of functions with constant modulus 1?