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 Deligne's article I and article II and Dwork's result on rationality, 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?

  • What other uses and benefits would have studying étale topology for someone not specializing in arithmetic geometry but in complex algebraic geometry?

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 mathoverflow so all kind of students and professionals can provide their advice regardless of their membership to these forums.)

  • 2
    $\begingroup$ Crossposted to MO. $\endgroup$
    – Zhen Lin
    Commented Nov 21, 2012 at 8:54
  • $\begingroup$ @ZhenLin: Is crossposting a bad practice for this type of questions? $\endgroup$ Commented Nov 21, 2012 at 9:11
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    $\begingroup$ @Javier: crossposting is bad for any type of question. It encourages unnecessary duplication of effort. $\endgroup$ Commented Nov 21, 2012 at 10:20
  • $\begingroup$ @QiaochuYuan: should I remove the question from one of the two sites? $\endgroup$ Commented Nov 21, 2012 at 10:21
  • 2
    $\begingroup$ @Javier: well, it's less bad now that you've indicated the crossposting. But please avoid doing this in the future. Post on one site, then wait awhile and see if you get any good answers before posting it on the other if that, and whenever you do so, link them. $\endgroup$ Commented Nov 21, 2012 at 10:29

1 Answer 1


Edit (2020 July 27): Check out the AMS Open Math Notes on the Weil Conjectures here. The PDF can be downloaded without signing in or even having an AMS membership.

Pasting an image of the Contents from the start of the above-mentioned notes:

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Edit (2016 March 15): Another great source to read over is Tao's blog post on the subject, which is (appropriately) entitled Dwork’s proof of rationality of the zeta function over finite fields.

This answer will no doubt be somewhat unsatisfying, but I can't help but put in a plug for Bernard Dwork, who proved the first of the Weil Conjectures (rationality) before Grothendieck and Deligne came along to dispose of the other two.

With regard to a learning path for Dwork's proof, I recommend the following three sources:

  1. B. Dwork, On the Rationality of the Zeta-Function of an Algebraic Variety, American Journal of Mathematics. 82 (1960), 631-648.

  2. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag, New York, 1984.

  3. F.Q. Gouvea, p-adic Numbers, Springer-Verlag, New York, 2003.

I did an undergraduate thesis on Dwork's proof, which mostly consisted of going through Koblitz's book and trying to write something easily understandable at the undergraduate level. Whether or not I succeeded in doing so - or producing anything novel in the process - I cannot say for sure (probably not), but if it'd be helpful here is a copy: On a Theorem of Dwork.

With regard to the latter two conjectures, someone more knowledgeable than I will have to intervene and suggest sources. Edit: The more knowledgeable response can be found here.

  • $\begingroup$ Thanks a lot, I knew about Dwork's article but it is not freely available to get a copy on my own. I would be very grateful if you could email your work to me. How can we do that without disclosing publicly our emails here? $\endgroup$ Commented Nov 21, 2012 at 8:28

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