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

Added March 2020: the revised lecture notes from this course will be posted here.

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.

• Contribute complete typed lecture notes for at least one lecture. Peter Wear will be coordinating the compilation of lecture notes.
• Complete and submit at least four problem sets.
• Write a short (8-10 pages) expository paper on a topic not covered in the course.

Announcements:

• First lecture: Monday, September 30. (Week 1 lectures run 1-1:50 in APM 2402.)
• No lecture on Wednesday, November 6. Scheduling for makeup lectures will be decided later.
• University holiday: Monday, November 11.
• Makeup lectures: Tuesday, November 19, 11-11:50am, APM 7321; Tuesday, November 26, 10-10:50am, APM 6402; Tuesday, December 3, 10-10:50am, APM 5829; Monday, December 9, 12-2:50pm, APM B402A.
• Last lecture: Wednesday, December 4 (except for possible makeup lectures).

Topics by date:

• Sep 30: logistics (course grading, lecture notes, meeting time); prehistory of the Weil conjectures (from Gauss to Artin to Weil). READING: Weil, Number of solutions of equations in finite fields (pdf).
• Oct 2: statement of the Weil conjectures; history of their resolution (work of Dwork, Grothendieck, Deligne, et al.) READING: Hartshorne, Algebraic Geometry, appendix C (pdf).
• Oct 7: Weil's cohomology formalism; the obstruction to rational coefficients; the Tate module of an elliptic curve as an example. READING: Silverman, The Arithmetic of Elliptic Curves, 2nd edition, chapters III and V (pdf).
• Oct 9: approaches to rationality, functional equation for curves (Bombieri-Stepanov's, Weil's proof by intersection theory, Jacobians). READING: Lorenzini, An Invitation to Arithmetic Geometry, chapter X (pdf, author's home page).
• Oct 14: approaches to the Riemann hypothesis for curves. READINGS: Lorenzini (see Oct 9).
• Oct 16: abelian varieties over finite fields. READING: J.S. Milne, The Riemann hypothesis over finite fields, from Weil to the present day (pdf).
• Oct 21: Honda-Tate theorem. READING: Waterhouse and Milne, Abelian varieties over finite fields (see CoCalc).
• Oct 23: Lang-Weil theorem; why one needs Weil cohomology to do better. READING: S. Lang and A. Weil, Number of points of varieties in finite fields (pdf).
• Oct 28: etale cohomology and the Faltings-Serre method. READING: R. Livné, Cubic exponential sums and Galois representations, section 4 (see CoCalc).
• Oct 30: continuation.
• Nov 4: p-adic Weil cohomology. READING: B. Dwork, On the rationality of the zeta function of an algebraic variety (pdf).
• Nov 6: NO MEETING.
• Nov 11: NO MEETING (university holiday).
• Nov 13: Bertini theorems over finite fields and zeta functions (lecture by Alina Bucur). READING: B. Poonen, Bertini theorems over finite fields (pdf).
• Nov 18: p-adic Weil cohomology continued. READING: K. Kedlaya, p-adic cohomology from theory to practice (Arizona Winter School lecture notes; pdf).
• Nov 19 (11am-12pm, APM 7321): Trace formula in p-adic Weil cohomology. READING: P. Monsky, Formal cohomology III: Fixed point theorems (pdf).
• Nov 20: continuation.
• Nov 25: etale local systems and "Weil II". READING: N. Katz, L-Functions and monodromy: four lectures on Weil II (pdf).
• Nov 26: continuation.
• Nov 27: NO MEETING.
• Dec 2: p-adic local systems. READING: K.S. Kedlaya, Notes on isocrystals (pdf).
• Dec 3: Continuation.
• Dec 4: Weil 2 in the l-adic and p-adic cases; companions. READING: K.S. Kedlaya, Fourier transforms and p-adic "Weil II" (pdf); K.S. Kedlaya, Etale and crystalline companions, I (arXiv).
• Dec 9: The Tate conjecture; distribution of zeta functions.

Problem sets:

• PS 1: pdf (posted Oct 1).
• PS 2: pdf (posted Oct 8).
• PS 3: pdf (posted Oct 20).
• PS 4: pdf (posted Nov 5).
• PS 5: pdf (posted Nov 24).

Topics we didn't get to:

• Improvements on the Weil bounds for curves (Serre, Igusa, Drinfeld-Vladut), applications to coding theory (Goppa codes).
• Refinements of the Lang-Weil estimate for higher-dimensional varieties.
• Approaches to computing zeta functions (Schoof-Pila, various p-adic methods).
• Newton above Hodge (Mazur's theorem).
• Geometric applications of the Langlands correspondence (e.g., bootstrapping techniques of Deligne and Drinfeld).
• Special value information (Tate conjecture, Artin-Tate formula, etc.).
• Tabulation of Weil polynomials.
• Derived equivalence, Orlov's conjecture.
• Compatible systems (companion relation), independence of l results.