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- W2004258321 abstract "In his Bourbaki talk 409, Katz conjectured the p-adic meromorphy of the function L(X,E,t) attached to a smooth variety X over a finite field F q (q=p a ) and to an F-crystal E on X. If X is proper and smooth over 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 π 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." @default.
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- W2004258321 date "1988-01-01" @default.
- W2004258321 modified "2023-10-01" @default.
- W2004258321 title "Rationalité et valeurs de fonctions $L$ en cohomologie cristalline" @default.
- W2004258321 doi "https://doi.org/10.5802/aif.1149" @default.
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