Explicit formulas for Hecke Gauss sums in quadratic number fields
type: Item
instance of: scholarly article
title: Explicit formulas for Hecke Gauss sums in quadratic number fields
zbMATH Open document ID: 1228.11126
author: Hatice Boylan
author: Nils-Peter Skoruppa
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2010-11-30T00:00:00Z
review text: Let \(K=\mathbb{Q}(\sqrt{D})\), \(G(\omega)=\sum\limits_{\mu\bmod a}e^{2\pi i\,\text{Tr}(\mu^{2}\omega/\sqrt{D})}\). The following result is obtained: Let \(\omega\) be a nonzero element of \(K\), and let \(M\) be the smallest positive rational integer such that \(M\omega\) is integral. Then \[ G(\omega)/\sqrt{\mathbb{N}(a\;\gcd(2,a))}=\frac{1}{\sqrt{u}}\sum\limits_{\Delta_{1}}\left(\frac{\Delta_{1}}{A}\right) \left(\frac{4\mathbb{N}(\beta)/\Delta_{1}}{N/|\Delta_{1}|}\right)\sqrt{\text{sgn}(\Delta_{1})}. \] Some notations and implicit statements used in the theorem are explained in the paper. The proofs use only elementary algebraic manipulations.
reviewed by: Zaizhao Meng
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11A15
zbMATH DE Number: 5821525
zbMATH Keywords: Hecke reciprocity
zbMATH Keywords: Gauss sums
zbMATH Keywords: quadratic reciprocity
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: s12188-010-0041-0
OpenAlex ID: W2007258121
cites work: A quick proof of reciprocity for Hecke Gauss sums
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DOI: 10.1007/S12188-010-0041-0
description: scientific article; zbMATH DE number 5821525