Rank one 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/0005308v1 [math.NT] 9 May 2000 RANK ONE CASE OF DWORK’S CONJECTURE DAQING WAN 1. Introduction 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 n . In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine 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 σ is the simplest q-th power map σ(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 difficult infinite rank setting, see section 2 for precise definitions of the various basic infinite rank notions. Let F q denote the finite field of characteristic p > 0. Our main result of this paper is the following theorem. Theorem 1.1. Let φ be a nuclear overconvergent σ-module over the affine n-space A n /F q , ordinary at the slope zero side. Let φ unit be the unit root (slope zero) part of φ. Assume that φ unit has rank one. Let ψ be another nuclear overconvergent σ-module over A n /F q . Then for each integer k, the L-function L(ψ ⊗ φ ⊗k unit , T ) is p-adic meromorphic. Furthermore, the family L(ψ ⊗ φ ⊗k T of L-functions unit 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 finite rank σ-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 different. 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 infinite rank setting. It also makes it possible to see the various analytic subtleties involved in a concrete case. As a result, we were able to prove analytically optimal results in [16]. For a general Frobenius lifting (even over the simplest affine n-space A n as we shall work in this paper), one has to use the much more difficult Monsky trace formula which is a generalization of Dwork’s trace formula. Thus, the first task of this paper is to extend the Monsky trace formula to infinite rank setting and to make it sufficiently 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, Fredholm determinants, p-adic meromorphic continua- tion, nuclear σ-modules and Banach modules. This work was partially supported by NSF. c American Mathematical Society
Subfields: Algebra and Number Theory, Artificial Intelligence, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2964defe7453a3d321e945a958252f0a681a8a6e
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- Counting points on varieties over finite fields of small characteristic (2006, methodology): “It is also different from the notion of an “orthonormal basis” in the literature, where one requires that the coefficient A u goes to zero as w ( u ) goes to ∞ (see [21] for a more detailed discussion of these notions).”