Étale Cohomology and the Weil Conjectures

This is the webpage for the weekly seminar on étale cohomology and the Weil conjectures, a learning seminar that intends to cover the basics of étale cohomology theory and go over the proofs by Grothendieck and Deligne of the Weil conjectures over the course of the fall semester.

Logistics

Talk Titles

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

   • Logistics: [TBD] on [TBD] in [TBD]
   • Abstract: We will finalize the meeting time and place, discuss topics to be covered, and distribute the talks. If time permits, we will review the story of the Weil conjectures.

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

Weil Conjectures

1. Numbers of Solutions of Equations in Finite Fields by André Weil, the 1949 paper in which Weil first poses the eponymous conjectures.
2. An Overview of Deligne's Proof of the Riemann Hypothesis for Varieties over Finite Fields by Nick Katz
3. Chapter 2. Points over Finite Fields and the Weil Conjectures by Mihnea Popa
4. A course on the Weil Conjectures by Tamás Szamuely (notes by Davide Lombardo)

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 by J.S. Milne, Princeton University Press, 1980.
3. Introduction to Étale Cohomology by Günter Tamme, Universitext, Springer-Verlag, 1994.
4. Étale Cohomology Theory, Revised Edition by Lei Fu, Volume 14 of Nankai Tracts in Mathematics, World Scientific Publishing Co. Pte. Ltd., 2015.
5. 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.
6. 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.

Many more refrences were collected on this website by Bhargav Bhatt (although beware the broken links). Another source for references, as well as a potential outline of the present seminar, is the MIT STAGE Seminar from Fall 2025.

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