# Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" and by Liu - "Algebraic Geometry and Arithmetic Curves".

• What would be a good learning path towards the proof of the Weil Conjectures for algebraic varieties (not just curves)?

• What modern references are available and in which order should be studied?

Besides the original article I and article II by Deligne and the results on rationality by Dwork, there is the book Freitag/Kiehl - "Étale Cohomology and the Weil Conjecture" and the online pdf by Milne - "Lectures on Étale Cohomology". The first title is out of stock and hard to get and the second seems to me too brief and succinct.

• Is it better to master étale cohomology by itself elsewhere and then refer to the original articles? Is any further algebraic/arithmetic background necessary?

Thank you in advance for any hints on how to approach such a study program, and for any related advice towards a self-learning path in arithmetic geometry. (This question is cross posted to math.stackexchange so all kind of students and professionals can provide their advice regardless of their membership to these forums.)

-
As it happens, there is more than one proof. Laumon gave a proof using a Fourier transform method (still using cohomology of $\ell$-adic sheaves), and Kedlaya has described a proof using Berthelot's rigid cohomology. – S. Carnahan Nov 21 '12 at 8:37
Crossposted to MSE: math.stackexchange.com/questions/241863/… – Zhen Lin Nov 21 '12 at 8:54
As a matter of courtesy, you should have mentioned that you were cross-posting, so people who are answering don't duplicate the efforts of others. – S. Carnahan Nov 21 '12 at 10:01
Yes indeed, I had forgotten, thank you for making me notice. I apologize for the inconvenience. – Javier Álvarez Nov 21 '12 at 10:05
You may be interested in contributing to a proposal Spanish language version of math stackexchange; it could use some input from fluent professors and students: area51.stackexchange.com/proposals/64529/… – Brian Rushton Feb 1 '14 at 3:15

I'm not an expert, but here is how I would plan my trip:

There are obviously two parts: rationality + functional equation + comparison with Betti numbers (which follow from the construction of etale cohomology) and the Riemann hypothesis (which is much deeper).

I would therefore follow the following plan:

1. Learn classical algebraic topology, esp. singular cohomology, Poincare duality, Lefschetz fixed point formula, Leray-Serre spectral sequence.
2. Learn an overview of $\ell$-adic cohomology, without technical details. First, understand how a good cohomology theory like in 1. above will prove the first part of the conjectures. Then understand etale topology and definition of $\ell$-adic cohomology groups and how the Frobenius morphism happens to act on them.
3. Technical machinery underlying $\ell$-adic cohomology. I haven't studied this myself very well, but Milne's book seems to be a standard reference.
4. Read Deligne's Weil I article. It's beautifully written and you don't need much more than 1. and 2. above. The main technical tool is the use of Lefschetz pencils, which is there just to make induction on dimension possible. You can just assume Lefschetz pencils exist, or look to SGA if interested. Note that in Deligne's approach it is crucial to work with constructible sheaves, not just the constant sheaf.
5. Read Deligne's Weil II article. It reproves Weil I and adds much more, but is much longer and more difficult.

Note: 3. and 4. above might be mostly independent. A very good reference is Katz's article "L-functions and monodromy: four lectures on Weil II".

-
Thanks a lot Piotr! This is the kind of guidance I was looking for. (Luckily I can start right away from your point 2.) – Javier Álvarez Nov 21 '12 at 19:19

Dear Javier,

I wasn't sure what to make of your question at first, but I think I understand some of your motivations from your other question. If you have interest in this area, then I think it would be useful to study etale cohomology. Regardless of whether or not you get to the Weil conjectures, the etale topology has all kinds of uses, so the effort would not be wasted. Aside from the references that you mention, see the answers and comments here Textbook for Etale Cohomology

-
Dear Donu, thank you very much for your advice as I was hesitant to study étale topology in detail; I had not come across your linked question in MO as I was looking mainly for "Weil conjectures", but now I have downloaded all the interesting links there, including your notes (maybe they will grow up someday as your recent book on complex algebraic geometry, which I have used extensively since its early stages as pdf draft). – Javier Álvarez Nov 21 '12 at 19:18