Math 206A - Topics in Algebraic Geometry: Weil cohomology in practice

Course description: Counting the number of points on an algebraic variety over a finite field is one of the oldest topics in algebraic geometry, dating back to the work of Gauss. A profound synthesis of this topic was made by Weil in the mid-20th century, who defined the zeta function of a variety over a finite field and formulated a series of far-reaching conjectures about these zeta functions. The resolution of these conjectures was one of the key driving forces behind the revolution in algebraic geometry led by Grothendieck, and was completed by Deligne's proof of Weil's analogue of the Riemann hypothesis. The resulting theorems form a backbone of modern algebraic geometry, and have seen applications far outside of mathematics (including coding theory and cryptography).

The goal of this course is to give a "practical" introduction to the Weil conjectures and to the primary tool used to study them, the notion of a Weil cohomology theory. There are essentially two different constructions of Weil cohomology: étale cohomology, which is modeled on the interpretion of fundamental groups in terms of covering spaces; and rigid cohomology, which is modeled on the cohomology of differential forms (de Rham cohomology).

While I have some topics in mind (see below), I do not have a set plan for the course. Accordingly, I am open to suggestions about topics from the participants.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours: by appointment.

Lectures: MWF 12:30-1:45, in APM B402A (effective Monday, October 7). If you registered for the course at its old meeting time, you will need to reregister for the new time.

Textbook: No required text. I will suggest various readings as we go along. (Note: some downloads require use of the UCSD VPN.) Also, the class will be assembling typed lecture notes, which will be posted here afterward. For access to them in the meantime, email Peter Wear for access to the course project on CoCalc.

Prerequisites: Math 203C or equivalent. I will be quite generous about what counts as "equivalent", and I will make the lectures as intelligible as possible to a broad audience. If you are interested in this course but have not taken the Math 203 series, please contact me to discuss your situation.

Homework: There will be some problem sets; these will focus on concrete aspects of the subject.

Final exam: None.

Grading: To get a maximal grade for this course, you must do one of the following.

Announcements:

Topics by date:

Problem sets:

Planned topics (more suggestions welcome):