On missing trace values for the eta multipliers
type: Item
instance of: scholarly article
title: On missing trace values for the eta multipliers
zbMATH Open document ID: 0611.10014
DOI: 10.1016/0022-314X(87)90035-7
author: Hiroshi Saito
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(\eta\) (z) be the Dedekind eta-function, and for \(\sigma \in SL_ 2({\mathbb{Z}})\) let \(\epsilon\) (\(\sigma)\) be the 24th root of unity such that \[ \eta (\sigma z)=\epsilon (\sigma)\quad (cz+d)^{1/2} \eta (z),\quad \sigma =\left( \begin{matrix} a\\ c\end{matrix} \begin{matrix} b\\ d\end{matrix} \right)\quad. \] This paper treats the question for which pairs of integers s and roots of unity \(\zeta =e^{\pi ik/12}\) there is a \(\sigma \in SL_ 2({\mathbb{Z}})\) with \(trace(\sigma)=s\) and \(\epsilon (\sigma)=\zeta\). Given k, the (''missing trace values'') \(s>2\) are determined which satisfy a certain set of necessary congruences desribed by \textit{T. Asai} [Number theory and combinatorics, Proc. Conf., Tokyo/Jap., Okayama/Jap. and Kyoto/Jap. 1984, 43-51 (1985)], but do not occur as a trace. The result is not easy to state, and the proof requires much perseverance.
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 3987364
zbMATH Keywords: eta-multiplier
zbMATH Keywords: Dedekind eta-function
zbMATH Keywords: missing trace values
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
cites work: Q4722140
cites work: Explicit formula of the traces of Hecke operators for \(\Gamma_0(N)\)
cites work: Exceptional values of the Dedekind symbol
cites work: Q5666618
full work available at URL: 0022-314x(87)90035-7
OpenAlex ID: W1972076791
description: scientific article; zbMATH DE number 3987364