RANK ONE CASE OF DWORK ' S CONJECTURE 3 Theorem 1

1. Introduction In the higher rank paper 17], we reduced Dwork's conjecture from higher rank case over any smooth aane variety X to the rank one case over the simplest aane space A n. In the present paper, we nish our proof by proving the rank one case of Dwork's conjecture over the aane space A n , which is called the key lemma in 17]. The key lemma had already been proved in 16] in the special case when the Frobenius lifting is the simplest q-th power map (x) = x q. Thus, the aim of the present paper is to treat the general Frobenius lifting case. Our method here is an improvement of the limiting method in 16]. It allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root L-function. As in 16], to handle the rank one case, we are forced to work in the more diicult innnite rank setting, see section 2 for precise deenitions of the various basic innnite rank notions. Let F q denote the nite eld of characteristic p > 0. Our main result of this paper is the following theorem. Theorem 1.1. Let be a nuclear overconvergent-module over the aane n-space A n =F q , ordinary at the slope zero side. Let unit be the unit root (slope zero) part of. Assume that unit has rank one. Let be another nuclear overconvergent-module over A n =F q. Then for each integer k, the L-function L(k unit ; T) is p-adic meromorphic. Furthermore, the family L(k unit ; T) of L-functions parametrized by integers k in each residue class modulo (q ? 1) is a strong family of meromorphic functions with respect to the p-adic topology of k. A nite rank-module is automatically nuclear. Thus, Theorem 1.1 includes the key lemma of 17] over A n as a special case. The basic ideas of the present paper are the same as the limiting approach in 16]. The details are, however, quite diierent. In the simplest q-th power Frobenius lifting case, one has the fundamental Dwork trace formula available, which is completely explicit for uniform estimates. This makes it easy to extend the Dwork trace formula to innnite rank setting. It also makes it possible to see the various analytic subtleties involved in a concrete case. As a result, we were able …