Rank One Case of Dwork’s Conjecture

In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A. In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A, 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 σ is the simplest q-th power map σ(x) = x. 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 difficult infinite rank setting, see section 2 for precise definitions of the various basic infinite rank notions. Let Fq denote the finite field of characteristic p > 0. Our main result of this paper is the following theorem.