On the Riemann hypothesis for the characteristic \(p\) zeta function
type: Item
instance of: scholarly article
title: On the Riemann hypothesis for the characteristic \(p\) zeta function
zbMATH Open document ID: 0858.11030
author: Daqing Wan
published in: Journal of Number Theory
publication date: 1996-07-09T00:00:00Z
review text: Let \(\mathbb{F}_q\) be the finite field with \(q=p^{n_0}\) elements. Let \({\mathcal C}\) be a smooth, projective, geometrically connected curve over \(\mathbb{F}_q\) and let \(\infty\in {\mathcal C}\) be a fixed closed point. As is standard, we let \({\mathbf A}\) be the affine ring of those functions regular outside of \(\infty\); so \(\text{Spec} ({\mathbf A}) = {\mathcal C} \backslash \infty\), and we let \({\mathbf k}\) be the quotient field of \({\mathbf A}\). If \(a\in {\mathbf A}\) is an invertible function, then \(a\) must also have no poles at \(\infty\) (as the degree of a principal divisor must be 0) and thus \(a\in \mathbb{F}^*_q\). Moreover it is very easy to see that the class group of \({\mathbf A}\) must be finite. It is precisely such \({\mathbf A}\) that are the base (or ``bottom'') rings of the theory of Drinfeld modules. It is also with such rings \({\mathbf A}\) that we may define characteristic \(p\) \(L\)-functions. The procedure is as follows: let \(\pi\) be a uniformizer in the complete field \({\mathbf K} : = {\mathbf k}_\infty\); we call an element \(x\in {\mathbf K}^*\) ``positive'' if and only if it can be written in the form \(x=\pi^d \langle x \rangle\) where \(\langle x \rangle\) is a 1-unit in \({\mathbf K}\). It is trivial to see that there is a constant \(t\) such that \(I^t\) is principal and generated by a positive element \(i(I)\) for all \({\mathbf A}\)-fractional ideals \(I\). Let \(s=(x,y)\in S_\infty: = \overline {\mathbf K}^* \times \mathbb{Z}_p\), then we set \(I^s: =x^{\deg_{\mathbb{F}_q}I} \cdot (\langle i(I) \rangle^y)^{1 /t}\) where we take the unique 1-unit \(t\)-th root. One sees directly that this has the usual good properties of ``\(n^s\)''. Now let \({\mathcal X}\) be a scheme of finite type over \(\text{Spec} ({\mathbf A})\) and let \(x\in{\mathcal X}\) be a closed point. Let \(x\) lie over \(\wp_x\in \text{Spec} ({\mathbf A})\) and let \(d(x)\) be the relative degree of the residue field at \(x\) over \({\math…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 883845
zbMATH Keywords: characteristic \(p\) zeta function
zbMATH Keywords: \(L\)-series
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: Newton polygons
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1996.0074
OpenAlex ID: W2052780466
DOI: 10.1006/JNTH.1996.0074
description: scientific article; zbMATH DE number 883845