Exponential sums over finite fields, I and II (ETHZ FS 2010, HS 2010)

This page contains links and documents relevant to the first part of the course, which ran in the Spring Semester 2010, as well as the second part, which is running in Fall Semester 2010.

18.11.2010: A slightly updated and corrected version of the lecture notes for Part I are available. Some parts still need work...

Moreover, the first ongoing partial versions for Part II are also available now. These are very incomplete! I haven't had time yet to include much of the material covered in class up to now. It is likely that, from now on, they will be up-to-date starting from Chapter IV ("The Riemann Hypothesis over finite fields"); the missing parts from Chapters I, II, III will be added as time allows.

Here is the abstract in the catalogue for the first part:

This course will introduce exponential sums over finite fields. It will first discuss basic examples (Gauss sums, Kloosterman sums, etc) and motivations for their study, coming from various problems of number theory, such as counting integral solutions of certain diophantine equations using the circle method. We will then develop some of the existing techniques that can be used to obtain interesting bounds for such sums, concentrating on elementary methods which do not involve deep algebraic geometry, such as the Stepanov method (leading to the Riemann Hypothesis for one-variable exponential sums, proved first by Weil) and the more recent methods based on additive combinatorics, introduced by Bourgain. It is expected that the course will be followed by another one in Winter Semester 2010 which will present the methods and results coming from more advanced algebraic geometry (work of Grothendieck, Deligne, Katz and others) and their applications.

And here is the abstract for the second part:

This course will present the modern techniques, based on algebraic geometry, that are used to study exponential sums over finite fields. Because of the large amount of material involved, a number of important facts will be taken as "black boxes", including Deligne's statement of the general form of the Riemann Hypothesis over finite fields. However, the way to use this formalism will be explained in detail, with a particular emphasis on Deligne's Equidistribution Theorem. Various applications will be given, including bounds for multi-variable character sums, families of exponential sums, and certain sieve problems.

Although this course continues the one given in the Spring Semester, the material will be quite independent, and new students should be able to follow by referring to the notes from that course.

Exponential sums over finite fields, I: elementary methods
These are the lecture notes that I wrote for the course as it went along. As of June 22, 2010, the version available is essentially complete; there are a few sections missing, but they correspond to survey sections, appendices, and parts for which I did not give details in class. Some of them will be expanded later on, but the current text is complete in some sense (modulo possible errors...)

Application de la formules des traces aux sommes trigonométriques by P. Deligne. This is the chapter in SGA 4 1/2 explaining the basic formalism of exponential sums arising from Deligne's second proof of the Riemann Hypothesis over finite fields.

Deligne's proof of the Weil conjectures for varieties over finite fields
These are notes (from 1998/1999) of three lectures trying to explain Deligne's first proof of the Riemann Hypothesis over finite fields, and in particular trying to explain the basic ideas behind the \'etale cohomology theory that is fundamental in the Grothendieck-Deligne-Katz approach to exponential sums.

Exponential sums over finite fields and applications
In November 2010, there will be an international conference on exponential sums over finite fields, which I am co-organizing in Zürich with N. Katz, P. Michel and R. Pink, and with the support of the FIM.

Exponential sums over finite fields
This is Chapter 11 from my book Analytic Number Theory with H. Iwaniec, with the bibliography attached.

Some aspects and applications of the Riemann Hypothesis over finite fields
This survey paper was written for the Proceedings of the Verbania Conference celebrating the 150th anniversary of the Riemann Hypothesis. Compared with the next item, it has much more emphasis on the topic of Deligne's Equidistribution Theorem and its applications.

A survey of algebraic exponential sums and some of their applications
This is another survey paper, which was written for the Proceedings of the ICMS Workshop on Motivic Integration held in May 2008. There is some overlap with the previous item, but some discussions are not found there (in particular concerning exponential sums over definable sets of finite fields).

Equations over finite fields: an elementary approach by W.M. Schmidt
This is volume 536 of the Lecture Notes in Mathematics, which gives complete details of Stepanov's method for the Riemann hypothesis in one variable over finite fields (with improvements due to Schmidt himself). The chapters can be downloaded from Springer's web site for those with access to a subscription (e.g. within ETH).

Counting points on curves over finite fields by E. Bombieri
This is Bombieri's Bourbaki lecture concerning Stepanov's method with his own adaptation to prove the Riemann Hypothesis for all (smooth, projective) curves over a finite field.

Last update 18.11.2010 by E. Kowalski