Weil Conjectures - Statement of The Weil Conjectures

Statement of The Weil Conjectures

Suppose that X is a non-singular n-dimensional projective algebraic variety over the field F_q with q elements. The zeta function ζ(X, s) of X is by definition

where N_m is the number of points of X defined over the degree m extension F_qm of F_q.

The Weil conjectures state:

1. (Rationality) ζ(X, s) is a rational function of T = q−s. More precisely, ζ(X, s) can be written as a finite alternating product
   where each P_i(T) is an integral polynomial. Furthermore, P₀(T) = 1 − T, P_2n(T) = 1 − qnT, and for 1 ≤ i ≤ 2n − 1, P_i(T) factors over C as for some numbers α_ij.
2. (Functional equation and Poincaré duality) The zeta function satisfies
   or equivalently
   where E is the Euler characteristic of X. In particular, for each i, the numbers α_2n-i,1, α_2n-i,2, … equal the numbers qn/α_i,1, qn/α_i,2, … in some order.
3. (Riemann hypothesis) |α_i,j| = qi/2 for all 1 ≤ i ≤ 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 ith Betti number of the space of complex points of Y.