TITLE: Relatively concise English expositions of the proofs of the various Weil conjectures
QUESTION [25 upvotes]: 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.

(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 $\alpha_{ij}$.
(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.
(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 $k/2$.
(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 $Y$.

Thanks in advance!

REPLY [5 votes]: 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