Entireness ofL-Functions ofϕ-Sheaves on Affine Complete Intersections
✍ Scribed by Yuichiro Taguchi; Daqing Wan
- Book ID
- 102974241
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 981 KB
- Volume
- 63
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
In a previous article, the authors showed that the L-function attached to a ?-adic .-sheaf is meromorphic in a certain disk depending on the convergence condition of the .-sheaf. That disk is in general best possible. The purpose of the present article is to show that for an affine complete intersection, either the L-function or its reciprocal is actually analytic (i.e., without poles) in the same disk. 1997 Academic Press
1. Introduction
Let X be a complete intersection of equi-dimension n embedded in some smooth affine variety Y defined over a finite field F q of q elements. Let (E, .) be an : log-convergent ?-adic .-sheaf on X. In this paper, we prove that the L-function L(.ÂX, T) (&1) n&1 is analytic (i.e., without poles) on a certain disk of ``radius q : .'' This result was conjectured in [TW].
As an application, we obtain the following result on L-functions of Drinfeld modules: Let A=F q [t] and K be its fraction field F q (t). Let X be a variety over F q and assume we are given an A-scheme structure X Ä Spec A. For a Drinfeld A-module , over X with everywhere good reduction, its L-function L(,ÂX, s) is defined [G]. Here, s=(z, y) is a variable ranging in the Goss complex plane S =C _ _Z p . Recall [TW] that a Drinfeld module gives rise to an algebraic .-sheaf E on X such that L(,ÂX, s)= L(E F y , z &1 ), where F y is a family of overconvergent -adic .-sheaves article no. NT972055