Higher Rank Case of Dwork

Dedicated to the memory of Bernard Dwork 1. Introduction In this series of two papers, we prove the p-adic meromorphic continuation of the pure slope L-functions arising from the slope decomposition of an overconvergent F-crystal, as conjectured by Dwork 6]. More precisely, we prove a suitable extension of Dwork's conjecture in our more general setting of-modules, see section 2 for precise deenitions of the various notions used in this introduction. Our main result is the following theorem. Theorem 1.1. Let X be a smooth aane variety deened over a nite eld F q of characteristic p > 0. Let (M;) be a nite rank overconvergent-module over X=F q. Then, for each rational number s, the pure slope s L-function L s (; T) attached to is p-adic meromorphic everywhere. The proof of this theorem will be completed in two papers. In the present higher rank paper, we introduce a reduction approach which reduces Theorem 1.1 to the special case when the slope s (s = 0) part of has rank one and the base space X is the simplest aane space A n. This part is essentially algebraic. It depends on Monsky's trace formula, Grothendieck's specialization theorem, the Hodge-Newton decomposition and Katz's isogeny theorem. In our next paper 23], we will handle the rank one case over the aane space A n. The rank one case is very much analytic in nature and forces us to work in a more diicult innnite rank setting, generalizing and improving the limiting approach introduced in 19]. Dwork's conjecture grew out of his attempt to understand the p-adic analytic variation of the pure pieces of the zeta function of a variety when the variety moves through an algebraic family. To give an important geometric example, let us consider the case that f : Y ! X is a smooth and proper morphism over F q with a smooth and proper lifting to characteristic zero. Berthelot's result 1] says that the relative crystalline cohomology R i f crys; Z p modulo torsion is an overconvergent F-crystal M i over X. Applying Theorem 1.1, we conclude that the pure L-functions arising from these geometric overconvergent F-crystals M i are p-adic meromorphic. In particular, this implies the existence of an exact p-adic formula for geometric p-adic character sums and a suitable p-adic equi-distribution theorem for the roots of zeta functions. For more detailed arithmetic motivations and further …