Here’s an update on the front of exponential sums…
(1) The conference I co-organized at FIM on exponential sums over finite fields
ended about two weeks ago. It was quite nice (at least from my point of view…). As usual here, the excellent organization made it possible to enjoy the mathematics with no extra issues to take care of. Most talks were on blackboard but, with authorization, here are the files for those that were on beamer (I think I may miss one or two, which I will add later):
- P. Kurlberg, Point count statistics for families of curves over finite fields
- Z. Rudnick, Sums of three squares: beyond equidistribution
- L. Pierce, Class numbers and exponential sums
(2) The second part of my class on exponential sums (cohomological methods) continues. I only barely started the lecture notes I was planning to write, the first few weeks of the semester having been simply too busy. As a result, although they are available and will be updated regularly from now on, they are quite incomplete. Most damagingly, the beginning material (introduction and motivation for going towards describing algebraic exponential sums using traces of Frobenius on suitable “Galois” representations of fundamental groups) is mostly missing for the moment. Still, the end parts of Chapter III are written and the contents, from Chapter IV onwards, will hopefully be kept in sync with the class, and no further gaps will appear. Since Chapter IV will start introducing the Grothendieck-Lefschetz trace formula, the étale cohomology groups, and then go on towards the formalism of weights and Deligne’s general version of the Riemann Hypothesis, this might still be of interest even before I find the time (next semester, probably) for filling up the gaps.
My plan next year, when (and if) this second part is done nicely enough, will be to put it together with the first part of the class (on elementary methods) to produce a proper book on exponential sums over finite fields. I’ll be happy to receive even before any feedback on the way the text shapes up, especially from the point of view of people in analytic number theory or in other fields where exponential sums are applied.