@MISC{Deligne_theweil, author = {Pierre Deligne}, title = {THE WEIL CONJECTURE FOR K3 SURFACES}, year = {} }

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Abstract

Denote by Fq a field of q elements, ¯Fq an algebraic closure of Fq, ϕ ∈ Gal(¯Fq/Fq) the Frobenius substitution x ↦ → xq and F = ϕ−1 the “geometric Frobenius”. Denote by X a scheme (separated of finite type) over Fq, and denote by ¯X the scheme over ¯Fq obtained by extension of scalars. For all closed points x of X, let deg(x) = [k(x) : Fq] be the degree over Fq of the residue extension. The zeta function Z(X, t) ∈ Z[[x]] is defined by Z(X, t) = (1 − t deg(x) ) −1 (x a closed point of X) x∈X log Z(X, t) = #X(Fqn)tn n. n>0 For any prime number ℓ coprime to q, the ℓ-adic cohomology with compact support Hi c ( ¯X, Qℓ) of ¯X is a finite-dimensional vector space over ¯Qℓ, on which Gal(¯Fq/Fq) acts by transport of structures. Following Grothendieck, one has Z(X, t) =