Rationalité et valeurs de fonctions L en cohomologie cristalline. (Rationality and values of L-functions in crystalline cohomology) (Q1092133)

In Sémin. Bourbaki 1971-72, Exposé 409, Lect. Notes Math. 317, 167- 200 (1973; Zbl 0259.14007), \textit{N. Katz} conjectured the p-adic meromorphy of the function L(X,E,t) attached to a smooth variety X over a finite field \({\mathbb{F}}_ q (q=p^ a)\) and to an F-crystal E on X. If X is proper and smooth over \({\mathbb{F}}_ q\) we prove that L is rational and given by the usual formula using the action of Frobenius on crystalline cohomology with coefficients in E; this result was only known, via ``Weil conjectures'', for particular unit-root F-crystals: those issued of a representation of \(\pi _ 1(X)\) through a finite quotient. The proof of the theorem involves the formalism of a cohomology class associated to a morphism of crystals, extending the fundamental class of an algebraic cycle, and leading to a Lefschetz trace formula. When E is a unit-root F- crystal the link between crystalline cohomology and De Rham-Witt complex with coefficients in E enables us to interpret zeroes and poles of L of the form \(t=q^{-r}\), r an integer. Under certain hypotheses this complex yields also equivalents of the L function in the neighbourhood of the preceding poles: these results extend those of Milne for zeta functions.