This semester, I am teaching (besides a course on Integration and Measure theory, about which I’ll write later) a course on elementary methods in the study of exponential sums over finite fields. The intent is to describe first the proof of the Riemann Hypothesis of A. Weil for one-variable exponential sums, based on Stepanov’s method (possibly in the version of Bombieri, possibly not), then go to more recent results where the “elementary” methods put to shame the cohomological formalism, e.g. Heilbronn sums or Mordell-type exponential sums involving polynomials with large degree (as in the work of Bourgain, though I haven’t yet quite settled on the detailed programme for that part of the course).
I’m hoping to type my lecture notes as I go along. In fact, the goal of the course is partly to prepare things both for a follow-up in the next semester on the cohomological approach and for a book I’ve been thinking about for quite a while on this topic. I don’t know what will come of this idea (for one thing, I’m starting slowly and as elementarily as I can, which is not really the style of the final book I have in mind, which would be a user guide for already fairly experienced analytic number theorists and other mathematicians interested in applying exponential sum methods to their own problems), and I doubt that even two semesters will be enough to lecture on what I wish to include, but the notes will be available on this page (together with links to various other documents of interest).