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.

Read more about this topic:  Weil Conjectures

Famous quotes containing the words statement of the, statement of, statement, weil and/or conjectures:

    “It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
    —John Dewey (1859–1952)

    “Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
    —William Burroughs (b. 1914)

    “If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
    —J.L. (John Langshaw)

    “Imagination is always the fabric of social life and the dynamic of history. The influence of real needs and compulsions, of real interests and materials, is indirect because the crowd is never conscious of it.”
    —Simone Weil (1909–1943)

    “Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)