Absolute zeta functions and the automorphy
type: Item
instance of: scholarly article
title: Absolute zeta functions and the automorphy
zbMATH Open document ID: 1390.14067
author: Nobushige Kurokawa
author: Hidekazu Tanaka
published in: Kodai Mathematical Journal
publication date: 2018-01-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1509415235
review text: The paper investigates the zeta function \(\zeta_f(s)\) associated to a particular type of ``absolute automorphic form'' \(f: \mathbb R_+\to \mathbb C\) that satisfies a functional equation \[ f(x^{-1}) = Cx^{-D}f(x)\qquad D\in\mathbb Z,\quad C=\pm 1. \] This study takes the inspiration from the results and the techniques developed in [\textit{A. Connes} and \textit{C. Consani}, Compos. Math. 146, No. 6, 1383--1415 (2010; Zbl 1201.14001)] where the equation \[ -\frac{\partial_s\zeta_N(s)}{\zeta_N(s)}=\int_1^\infty N(x)\, x^{-s}\frac{dx}x\tag{1} \] defines, up-to a multiplication by a non-zero scalar, an integral formula for the Hasse-Weil zeta function \(\zeta_N(s)\) associated, in the limiting case \(q\to 1\), to the counting function \(f(x)=N(x)\) defined for \(x\in [1,\infty)\). The integration in \(s\) of both sides of (1), produces a divergence, at \(x=1\), of the term \(\frac{x^{-s-1}}{\log x}\) on the right hand-side, thus one has to choose a principal value in \[ \log(\zeta_N(s)) = \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}\, dx+c \] near \(x=1\) (the term \(\frac{N(x)}{\log(x)}\) is singular). Here, the authors use the technique that implements a new variable \(w\) to define the principal value as \[ \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}dx :=\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}. \] Thus they obtain, after exponentiation, the formula \[ \zeta_N(s) = \exp\left(\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}\right)\tag{2} \] which fully determines the Hasse-Weil zeta function \(\zeta_N(s)\) associated to \(N\). Notice that the equation (1) of [loc. cit.] defines \(\zeta_N(s)\) only up to a multiplication by a non-zero scalar. The point of view of the authors of the paper under review is that equation (2) can be applied to define the so called ``absolute zeta function'' \[ \zeta_f(s):=\exp\left(\frac{\partial}{\partia…
reviewed by: Caterina Consani
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 33E30
zbMATH DE Number: 6827106
zbMATH Keywords: absolute zeta function
zbMATH Keywords: absolute automorphic form
MaRDI profile type: Publication
OpenAlex ID: W2765667380
DOI: 10.2996/KMJ/1509415235
description: scientific article; zbMATH DE number 6827106