Étale Cohomology and the Weil Conjectures

This is the webpage for the weekly seminar on étale cohomology and the Weil conjectures, a learning seminar with the goal of understanding the proofs due to Grothendieck and Deligne of the Weil conjectures. The (updated) goal for Fall 2026 is to introduce the Weil conjectures, cover historically important "elementary" proofs in special cases, and then understand the proof of the axioms of a Weil cohomology theory for étale cohomology. In the spring, the seminar will continue with the proof of the Riemann hypothesis following Deligne's La conjecture de Weil: I, and possibly Weil II as well.

Logistics

1. Time: 1730-1830 on Mondays
2. Place: Fine Hall 401
3. Contacts: Dhruv Goel (dhruvgoel@princeton.edu) and Patrick Borse (pborse@princeton.edu)
4. Mailing List: There is a mailing list for this seminar. If you are not on it and would like to be on it, or you are on it and would like to be taken off it, please reach out to me at dhruvgoel@princeton.edu.
5. Prerequisites Assumed: Familiarity with algebraic geometry at the level of Vakil or Hartshorne.
6. Notes: The shared Overleaf document can be found here.

Talk Titles

1. Dhruv: Organizational Meeting and Overview of the Weil Conjectures

   • Logistics: 1730-1830 on Monday 260907 in Fine 401
   • Abstract: We will finalize logistics and then review the story of the Weil conjectures. We will start with the Riemann and Dedekind \(\zeta\)-functions, discuss local zeta functions of finite-type \(\mathbb{F}_q\)-schemes, do some easy computations (such as for projective spaces, Grassmannians, and elliptic curves) by hand, and state the Weil conjectures. Then we will overview the proof strategy using Weil cohomology theories and the Lefschetz Trace Formula, lay out the plan for the rest of the seminar, and end by distributing the first few talks.

2. Dhruv: Proof of the Weil Conjectures for Curves via Riemann-Roch, Serre Duality, and the Hodge Index Theorem

   • Logistics: 1730-1830 on Monday 260914 in Fine 401
   • Abstract: We will present the proof of the Weil conjectures for nice curves. The proof of rationality and the functional equation uses only Riemann-Roch with Serre duality; the proof of the Riemann hypothesis uses intersection theory on surfaces, or more specifically, the Hodge Index Theorem. As a bonus, we will discuss a geometric analog of the analytic class number formula for number fields, and illustrate it in examples such as the Fermat quartic and a certain cyclic cover of the line.

3. Dhruv: Proof of the Weil Conjectures for Diagonal Hypersurfaces

   • Logistics: 1730-1830 on Wednesday 260923 in Fine 601
   • Abstract: Following Weil's 1949 paper containing the eponymous conjectures, we will quickly review the basic properties of Gauss and Jacobi sums for finite fields, and use these to prove the Weil conjectures for diagonal hypersurfaces, the key ingredient being the Hasse-Davenport relation. Combined with difficult results of Katz and Scholl, this remains the shortest path to the proof of conjectures to date. In particular, we will encounter our first variety with a negative sign in the functional equation: a smooth quadric surface with a non-split Fano scheme of lines. If time permits, we will explain how this computation relates to the cohomology of complex hypersurfaces via the weak Lefschetz theorem and the Chern-Gauss-Bonnet computation of Euler characteristics.

4. Kenta: Weil Cohomology Theories and the Lefschetz Trace Formula

   • Logistics: 1730-1830 on Monday 260928 in Fine 401
   • Abstract: We will cover the axioms of a Weil cohomology theory, in analogy with the properties of singular cohomology. We will then prove the Lefschetz fixed point theorem for a Weil cohomology theory. Amusingly, the Lefschetz formula already shows that the Betti numbers of a curve \(X\) are \(1, 2g(X),1\) with respect to any Weil cohomology theory. Assuming the existence of a Weil cohomology theory for varieties in positive characteristic, we apply the Lefschetz formula to the Frobenius automorphism and deduce most of the Weil conjectures.

5. Danila: The Étale Topos and Étale Cohomology

   • Logistics: 1730-1830 on Monday 261005 in Fine 401
   • Abstract: Étale cohomology is an important instrument involved in the proof of Weil conjectures. In order to construct étale cohomology, we will discuss a generalization of the standard definition of sheaves on a topological space—sheaves on a site. Then, we will define the étale site and cohomology theory for sheaves on it. We will cover some basic examples and properties of étale sheaves and their cohomology.

6. Paweł: More on Étale Sheaves

   • Logistics: 1730-1830 on Monday 261012 in Fine 401
   • Abstract: We continue our study of étale sheaves. We present useful tools for computations in étale cohomology and some important classes of étale sheaves (e.g., torsion, locally constant, constructible, etc.). We perform some standard cohomology computations, such as using the Kummer and Artin-Schreier sequences, explain the connection to Hilbert 90, and discuss recollement of étale sheaves.

7. Paweł and Dhruv: The Étale Fundamental Group; Recap and Planning

   • Logistics: 1730-1830 on Monday 261026 in Fine 401
   • Abstract: In the first half of the talk, Paweł will introduce the notion of the étale fundamental group \(\pi_1^\mathrm{ét}\) using the fiber functor and study some of its properties. In the second half of the talk, Dhruv will summarize the topics covered in the seminar so far, outline the remaining topics and goals, and discuss the plan for the rest of the semester (and the spring).

8. Michael: Poincaré Duality in Étale Cohomology, Part I

   • Logistics: 1730-1830 on Monday 261102 in Fine 401
   • Abstract: We give a version of Poincaré duality in étale cohomology. The strategy is to try copying a proof of Poincaré duality for singular cohomology and hoping it will still go through in the étale setting. In this vein, we steal the proof from Hatcher's book on algebraic topology. Hatcher's proof has two steps: first prove Poincaré duality for \(\mathbb{R}^d\), then prove Poincaré duality can be checked locally. This week's talk is all on step 2: why is Poincaré duality something that can be checked locally? The key is to use 'derived categories.'
   • Handout: Can be found here.

9. Sam: The Dimension Axiom

   • Logistics: 1730-1830 on Monday 261109 in Fine 401
   • Abstract: We continue verifying that étale cohomology satisfies the axioms for a Weil cohomology theory by proving the dimension axiom: that for \(X\) a variety of dimension \(n\) over an algebraically closed field has cohomological dimension \(\leq 2n\). We prove this by taking geometric generic points and induction to reduce to étale cohomology over points \(\operatorname{Spec}(k)\) which we identify with Galois cohomology for \(G=\operatorname{Gal}(k^\text{sep}/k)\) and then study this problem using Galois cohomology results such as Tsen’s theorem.

Resources

Generalities on Sites

1. Notes on Grothendieck Topologies by Michael Artin
2. Notes on Grothendieck topologies, fibered categories and descent theory by Angelo Vistoli

Étale Cohomology

1. Grothendieck topologies and étale cohomology by Pieter Belmans
2. Lectures on Étale Cohomology by J. S. Milne
3. Lecture Notes for Étale Cohomology by Stefan Patrikis

Weil Conjectures

1. Notes on Deligne's "La Conjecture de Weil. I" by Tony Feng
2. La conjecture de Weil: I and La conjecture de Weil: II by P. Deligne; an English translation of the fomer due to Evgeny Goncharov can be found here
3. Weil Conjectures Exposition by Evgeny Goncharov
4. Appendix C of Algebraic Geometry by R. Hartshorne
5. An Overview of Deligne's Proof of the Riemann Hypothesis for Varieties over Finite Fields by Nick Katz
6. The Riemann Hypothesis over Finite Fields: From Weil to the Present Day by J. S. Milne
7. Chapter II of Abelian Varieties by J. S. Milne
8. Zeta functions in algebraic geometry by Mircea Mustață.
9. Chapter 2. Points over Finite Fields and the Weil Conjectures by Mihnea Popa
10. A course on the Weil Conjectures by Tamás Szamuely (notes by Davide Lombardo)
11. Numbers of Solutions of Equations in Finite Fields by André Weil, the 1949 paper in which Weil first poses the eponymous conjectures. An expanded account of the computation done by Weil for "diagonal hypersurfaces" can also be found in Chapters 10 and 11 of Ireland and Rosen.

Textbook References

1. Cohomologie étale, Séminaire de géométrie algébrique du Bois Marie SGA 4½, par P. Deligne avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Volume 569 of Lecture Notes in Mathematics, Springer-Verlag, 1977
2. Étale Cohomology and the Weil Conjecture by E. Freitag and R. Kiehl, Band 13 der Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Springer-Verlag, 1988
3. Étale Cohomology Theory, Revised Edition by Lei Fu, Volume 14 of Nankai Tracts in Mathematics, World Scientific Publishing Co. Pte. Ltd., 2015
4. Weil Conjectures, Perverse Sheaves and \(\ell\)-adic Fourier Transform by R. Kiehl and R. Weissauer, Volume 42 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Springer-Verlag, 2001
5. Étale Cohomology by J. S. Milne, Princeton University Press, 1980
6. Chapter 7 of Rational Points on Varieties by Bjorn Poonen, Volume 186 of Graduate Studies in Mathematics, American Mathematical Society, 2017
7. Introduction to Étale Cohomology by Günter Tamme, Universitext, Springer-Verlag, 1994

Many more references were collected on this website by Bhargav Bhatt (although beware the broken links). See also the MIT STAGE Seminar from Fall 2025, this seminar by Caleb Ji, and notes from the seminar organized by Brian Conrad and the one by Bhargav Bhatt.