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…

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).