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 Fq with q elements. The zeta function ζ(X, s) of X is by definition

where Nm is the number of points of X defined over the degree m extension Fqm of Fq.

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 Pi(T) is an integral polynomial. Furthermore, P0(T) = 1 − T, P2n(T) = 1 − qnT, and for 1 ≤ i ≤ 2n − 1, Pi(T) factors over C as for some numbers αij.
  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 α2n-i,1, α2n-i,2, … equal the numbers qn/αi,1, qn/αi,2, … in some order.
  3. (Riemann hypothesis) |αi,j| = qi/2 for all 1 ≤ i ≤ 2n − 1 and all j. This implies that all zeros of Pk(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 Pi is the ith Betti number of the space of complex points of Y.

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