Theory of weights in p-adic cohomology

Abstract

Let k be a finite field of characteristic p > 0. We construct a theory of weights for overholonomic complexes of arithmetic D-modules with Frobenius structure on varieties over k. The notion of weight behave like Deligne's one in the ℓ-adic framework: first, the six operations preserve weights, and secondly, the intermediate extension of an immersion preserves pure complexes and weights.[BBD82] Gabber proved the stability of purity and mixedness under intermediate extensions, with which the theory of weights for ℓ-adic cohomology may be regarded as complete. However, the problem of obtaining similar results within a p-adic cohomological framework remained opened. After Dwork's p-adic proof of the rationality of zeta functions, a part of Weil's conjectures, it seems natural to expect better computability of zeta functions with a p-adic approach. In this paper, we build a p-adic theory of weights.The first attempt to calculate the weights of some p-adic cohomology was made by Katz and Messing in their famous paper [KM74]. Using Deligne's deep results on weights, especially "Le théorème du pgcd", they showed that, for projective smooth varieties, the weight of crystalline cohomology is the same as that of ℓ-adic one. It is reasonable to hope that the coefficient theory of weights parallel to ℓ-adic cohomology should exist in the spirit of the petit camarade conjecture [Del80, 1.2.10], even though there were many obstacles that prevented the construction of such a theory. After the work of Katz-Messing, efforts were made until Kedlaya finally obtained in [Ked06] the expected estimation of weights of rigid cohomology, a p-adic cohomology constructed by Berthelot. We do not go into more details of the history, and recommend the reader to consult the excellent explanation in the introduction of [Ked06].But the context of rigid cohomology was still not completely satisfactory, in the sense that this is not "functorial enough", namely we do not have six operations formalism, specially push-forwards as pointed out in [Ked06, 5.3.3]. In this paper, we use systematically Berthelot's arithmetic D-modules to complete the program of establishing a p-adic theory of weights stable under six operations. In many applications, such a theory should play as important roles as suggested by the classical situations; for example, the theory of intersection cohomology and its purity, the theory of Springer representations, Lafforgue's proof of Langlands correspondence, etc. In the final part of the paper, we show one such application that the Hasse-Weil L-function for a function field defined by means of ℓ-adic methods and p-adic methods coincide. Now let us explain our results in more details. Let V be a complete discretely valued ring of mixed characteristic (0, p), K its field of fractions, and k its residue field which is assumed to be a finite field. We suppose that there exists a lifting of the Frobenius of k to V. In order to obtain a p-adic cohomology on algebraic varieties over k stable under the six operations whose coefficients...

Langlands correspondence for isocrystals and the existence of crystalline companions for curves

“…Moreover, the t-structure is compatible with this equivalence (cf. [AC1,1.2.8]). The heart of the triangulated category is denoted by Hol(X/K).…”

Constructible isocrystals

“…This is the first step towards an overconvergent Deligne-Kashiwara correspondence. Note that this construction is extended to a slightly more general situation by Tomoyuki Abe and Caro in [2] and was already known to Pierre Berthelot in the case X = P k (proposition 4.4.3 of [6]).…”

Around the Nearby Cycle Functor for Arithmetic -Modules

“…* (F V ) remains to be irreducible. The verification is standard (see, for example, [AC,Proposition 1.4.7]).…”

“…Proof. Only if part follows since i * is right exact by [AC,Proposition 1.3.13]. Assume H 0 F is supported on a reduced scheme Z.…”