Higher rank case of Dwork's conjecture

By Daqing Wan. 2000

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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0894-0347(XX)0000-0 arXiv:math/0005309v1 [math.NT] 9 May 2000 HIGHER RANK CASE OF DWORK’S CONJECTURE DAQING WAN 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 σ-modules, see section 2 for precise definitions of the various notions used in this introduction. Our main result is the following theorem. Theorem 1.1. Let X be a smooth affine variety defined over a finite field F q of characteristic p > 0. Let (M, φ) be a finite rank overconvergent σ-module over X/F q . Then, for each rational number s, the pure slope s L-function L s (φ, T ) attached to φ 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 φ has rank one and the base space X is the simplest affine 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 affine space A n . The rank one case is very much analytic in nature and forces us to work in a more difficult infinite 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 con- sider 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 open prob- lems, see the expository papers [20][21]. 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, p-adic meromorphic continuation, σ-modules. This work was partially supported by NSF. c American Mathematical Society

Subfields: Geometry and Topology, Mathematical Physics

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