# Relatively concise English expositions of the proofs of the various Weil conjectures

Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of étale cohomology, etc.) English expositions of the proofs of the various Weil conjectures? The four Weil conjectures, according to Wikipedia, are as follows.

1. (Rationality) $\zeta(X, s)$ is a rational function of $T = q^{-s}$. More precisely, $\zeta(X, s)$ can be written as a finite alternating product$$\prod_{i = 0}^{2n} P_i(q^{-s})^{(-1)^{i + 1}} = {{P_1(T) \ldots P_{2n - 1}(T)}\over{P_0(T) \ldots P_{2n}(T)}},$$where each $P_i(T)$ is an integral polynomial. Furthermore, $P_0(T) = 1 - T$, $P_{2n}(T) = 1 - q^nT$, and for $1 \le i \le 2n - 1$, $P_i(T)$ factors over $\mathbb{C}$ as $\prod_j (1 - \alpha_{ij}T)$ for some numbers $\alpha_{ij}$.
2. (Functional equation and Poincaré duality) The zeta function satisfies$$\zeta(X, n - s) = \pm q^{{{nE}\over2} - Es}\zeta(X, s)$$or equivalently$$\zeta(X, q^{-n} T^{-1}) = \pm q^{{{nE}\over2}}T^E \zeta(X, T)$$where $E$ is the Euler characteristic of $X$. In particular, for each $i$, the numbers $\alpha_{2n - i, 1}$, $\alpha_{2n - i, 2}$, $\ldots$ equal the numbers $q^n/\alpha_{i, 1}$, $q^n/\alpha_{i, 2}$, $\ldots$ in some order.
3. (Riemann hypothesis) $|\alpha_{i, j}| = q^{i/2}$ for all $1 \le i \le 2n - 1$ and all $j$. This implies that all zeros of $P_k(T)$ lie on the "critical line" of complex numbers $s$ with real part $k/2$.
4. (Betti numbers) If $X$ is a (good) "reduction mod $p$" of a non-singular projective variety $Y$ defined over a number field embedded in the field of complex numbers, then the degree of $P_i$ is the $i$th Betti number of the space of complex points of $Y$.

• For 1. Koblitz's book p-adic Numbers, p-adic Analysis, and Zeta-Functions gives Dwork's proof in a fairly straightforward way. For 2-4, unless you are happy using etale cohomology as a black box, it inevitably takes a long time to develop. Maybe these days the proof using crystalline cohomology is shorter, but I doubt it. – Felipe Voloch Aug 22 '16 at 23:35
• @FelipeVoloch I'm perfectly fine with using étale cohomology as a black box. I've removed the "self-contained" stipulation from my original post. – user97565 Aug 22 '16 at 23:58
• I take it that the book of Freitag-Kiehl has been dismissed? – Hoot Aug 23 '16 at 0:26

What about Nick Katz' expose:

Nicholas M. Katz, MR 1831948 $L$-functions and monodromy: four lectures on Weil II, Adv. Math. 160 (2001), no. 1, 81--132.

As well as Kowalski's notes.

See Kleiman's essay "Algebraic Cycles and the Weil Conjectures", in the volume "Dix exposes sur la cohomologie des schemas". (Despite the French volume title, the article is in English.) This article focuses on the relation between the Weil conjectures and Grothendieck's standard conjectures, but contains a complete proof of all four Weil conjectures modulo the existence of a well-behaved cohomology theory.

(In fact, the proof, which occupies the last section of the paper, is only about three pages long and self-contained modulo some formal properties of that good cohomology theory, some of which are established earlier in the paper and some of which are conjectural, but that you might be willing to take for granted.)

• Are those conjectural properties the ones that the Standard Conjectures supply? Or ones that we know étale cohomology possesses (or perhaps crystalline)? – David Roberts Aug 23 '16 at 8:56
• @DavidRoberts: They are "Lefschetz-like" conjectures, primarily this: On the cohomology of an $n$-dimensional variety $X$, let $L$ denote intersection with a hyperplane section, let $\Lambda$ be the cohomology operator defined as zero on the kernel of $L^{n-i+1}$ acting on $H^i(X)$ and as the inverse of $L$ elsewhere. Then $\Lambda$ should be induced by an algebraic cycle with rational coefficients on $X\times X$ (together with strong Lefschetz and Hodge index conjectures). – Steven Landsburg Aug 23 '16 at 17:01

The second part of J.S. Milne's Lectures on Étale Cohomology is devoted to the proofs of the Weil conjectures. The theory of étale cohomology is developed in the first part, but if you're comfortable using that as a black box, you can skip straight to the second part (pages 151–200 in the current version).

I would like to answer this question with a "moral" (conditional) proof.

Motive-ating the Weil conjecture proof "This post concludes a series of posts I’ve been writing on the attempt to prove the Weil Conjectures through the Standard Conjectures. (Parts 1, 2, 3, 4, 5.)" David Speyer.

Sorry for a bit more of self-promotion but I want to share it somewhere (when I was searching for such an article, it was not available). Maybe I will try to publish this as an expository paper (not sure if that is appropriate, should I?). Anyway, I have covered proofs of the Weil Conjectures by Grothendieck and Deligne, explained all of the arguments that might not be obvious right away and also added an overview on Etale cohomology and tried to provide motivation for all of the steps of the proof - https://drive.google.com/file/d/1KQ6k6vtAowCc9WbuqAlEf7aZIQFEv54Z/view?usp=sharing

Contributing here for completeness. I have translated Deligne's original paper into English (it is only 35 pages) - https://drive.google.com/file/d/1wxVbCrm_0D4jInvG1pd1vhnI7YnvSk0S/view?usp=sharing Still, I suggest to first get acquainted with the theory of etale cohomology and the exposition of the proof in Milne and only then turn to the original paper to fill in the details.

Added: I have found certain typos but they are not crucial for the exposition.

• This is a very useful contribution! You should translate Weil II too! – Andy Putman Jul 31 '18 at 19:37

There are also Uwe Jannsen's lecture notes at Regensburg University: http://www.mathematik.uni-regensburg.de/Jannsen/home/Weil-gesamt-eng.pdf