The goal of this seminar is to present Deligne's first proof of the Weil conjectures. We will introduce Grothendieck topologies, and define the cohomology functors on the étale site. We give outlines of the proofs of the main technical theorems (proper and smooth base change). We then proceed to the proofs of each of the Weil conjectures.

The main reference will be SGA 4½, at least for the first half of the semester. When we get to Deligne's proof of the Riemann hypothesis, we will follow Freitag-Kiehl and Deligne's original article (Weil I).

Other useful references are Milne's notes on étale cohomology, Milne's book with the same title (which is organised very differently), and of course the stacks project. I have also written a detailed account of the construction of the étale site and the cohomology functors on it; see Appendix B of my Parisian master's thesis.

A great introduction to the theory of Henselian local rings is given by Raynaud's book.

**Prerequisites:** A basic command of algebraic geometry is assumed, along the lines of Hartshorne's book. We will try to prove most statements that are not covered, but this may prove impossible. Furthermore, we will assume familiarity with Galois cohomology; see for example Serre's book or Gille-Szamuely. Following Prof. de Jong, we will assume all commutative algebra is trivial.

The language of category theory is used freely; in particular terms like 'abelian category', 'adjoint functor', and '(co)limit' are used without further discussion. For a brief review, see Appendix A of my Parisian master's thesis. Finally, at some point we will encounter some spectral sequences, which we will also assume understood.

**Homework:** Participants are strongly encouraged to do weekly reading, to keep up with the vast amount of theory we have to cover.

**Organiser:** Remy van Dobben de Bruyn.

**Time and location:** Tuesday 16:30-18:00 in Math 507. On the days marked with an asterisk, we will be in room 622.

**Notes:** Pak-Hin Lee has kindly agreed to take notes. They will be uploaded here.

Date | Speaker | Title | Ch. |
---|---|---|---|

8 Sep* |
Remy van Dobben de Bruyn | Introduction: the Weil conjectures | |

15 Sep | Remy van Dobben de Bruyn | Étale morphisms; fpqc descent (affine case); Henselisations | 1.1 |

22 Sep | Remy van Dobben de Bruyn | Grothendieck topologies and sites | 1.1 |

29 Sep | Remy van Dobben de Bruyn | Sheaves on the étale site | 1.2 |

6 Oct | Sam Mundy | Cohomology on curves | 1.3 |

13 Oct* |
Shizhang Li | Proper base change; cohomology with proper support | 1.4 |

20 Oct* |
Ashwin Deopurkar | Local acyclicity of smooth morphisms; smooth base change | 1.5 |

27 Oct | Linus Hamann | Constructible sheaves and finiteness theorems | |

3 Nov | University holiday |
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10 Nov | No meeting |
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17 Nov | Qixiao Ma | Poincaré duality | 1.6 |

24 Nov | Qixiao Ma | Lefschetz trace formula | 2.5 |

1 Dec | Shizhang Li | Sketch of Deligne's proof of the Riemann Hypothesis | |

8 Dec | Raju Krishnamoorthy | The Weil conjectures for K3 surfaces |

* On these dates, the seminar will take place in room 622.