On root numbers connected with special values of \(L\)-functions over \(\mathbb{F}_ q(T)\)
type: Item
instance of: scholarly article
title: On root numbers connected with special values of \(L\)-functions over \(\mathbb{F}_ q(T)\)
zbMATH Open document ID: 0874.11043
author: Jianqiang Zhao
published in: Journal of Number Theory
publication date: 1997-11-11T00:00:00Z
review text: In the paper ``Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)'' [J. Number Theory 60, 165-209 (1996; Zbl 0960.52407)], \textit{G. Anderson} introduced a number of very interesting objects of study in the arithmetic of the polynomial algebra \(\mathbb{F}_q[T]\). Among these are ``special points'', and certain ``root numbers'' which play a role not very different from classical root numbers. Now in classical number theory, root numbers are intimately related to Gauss sums and it is natural to wonder if there is a relationship between Anderson's root numbers and Thakur's Gauss sums for function fields. In this important paper such a connection is presented. Moreover, the answer is quite intriguing and leaves one wondering if something remarkable underlies it. More precisely, Thakur's Gauss sums are made up in the ``Carlitzian'' fashion from ``basic Gauss sums''. That is, for each \(j\) one has a basic Gauss sum \(g(j)\), which is a sum very similar to that of classical arithmetic. Then if \(i\) is an integer written \(q\)-adically as \(\sum^t_{j=0} c_jq^j\), \(0\leq c_j\leq q-1\) for all \(j\), one defines \(g(i):= \prod^t_{j=0} g(j)^{c_j}\). What is so remarkable about the author's results, as illustrated by his Example 2, is that the root numbers are expressed as products of the basic Gauss sums, but where the powers may be strictly greater than \(q-1\). This is most peculiar! Motivated by the present paper, Anderson went on to give another, equally intriguing, basis for the special points. We refer the reader to Section 10.6 of ``Basic structures of function field arithmetic'' [\textit{D. Goss}, Springer (1996; Zbl 0874.11004)]. One again wonders what underlying phenomenon explains these remarkable formulas.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11T23
zbMATH DE Number: 992063
zbMATH Keywords: special values of \(L\)-functions
zbMATH Keywords: root numbers
zbMATH Keywords: Gauss sums for function fields
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1997.2056
OpenAlex ID: W2043779373
cites work: On a new type of \(L\)-function for algebraic curves over finite fields
cites work: Fourier series, measures and divided power series in the theory of function fields
cites work: Explicit Class Field Theory for Rational Function Fields
cites work: Q3134952
cites work: Gauss sums for \({\mathbb F}_ q[T]\)
cites work: Gauss sums for function fields
cites work: Certain quantities transcendental over \(\text{GF}(p^n,x)\)
cites work: Q3943047
DOI: 10.1006/JNTH.1997.2056
Recommended article: Gauss sums for function fields
Recommended article: An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions
Recommended article: On certain functions connected with polynomials in a Galois field
Recommended article: Sums of squares over function fields
Recommended article: Representations of arithmetic functions in \(\mathrm{GF}[p^n,x]\). I
Recommended article: An analogue of the von Staudt-Clausen theorem.
Recommended article: Q5488416
Recommended article: Q4210879
Recommended article: Q3348068
Recommended article: The Gauss sums and Jacobi sums over Galois ring \(\mathrm{GR}(p^2,r)\)
description: scientific article; zbMATH DE number 992063