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!
Best Answer
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.