On two of Liouville's quaternary forms
type: Item
instance of: scholarly article
title: On two of Liouville's quaternary forms
zbMATH Open document ID: 0671.10019
DOI: 10.1007/BF01188674
author: Günter Köhler
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: The quaternary forms referred to in the title are \(x^ 2+2y^ 2+3z^ 2+6t^ 2\) and \(x^ 2+y^ 2+6z^ 2+6t^ 2.\) The generating theta series of these forms are decomposed into Eisenstein series and the cusp eigenform \(H(z)=\eta (z)\eta (2z)\eta (3z)\eta (6z),\) where \(\eta\) denotes Dedekind's eta function. This yields Liouville's and Kloosterman's formulas for the numbers L(n) and K(n) of representations of n by these forms and, in particular, \(K(p)=L(p)=p+1+\lambda (p)\) for all primes \(p>3\), where \(\lambda\) (p) is the p-th Fourier coefficient of H(z). An analysis of L(p) and K(p) modulo small powers of 2 and the theory of ternary quadratic forms yield results on class numbers \(h(-n)\) of imaginary quadratic fields \({\mathbb{Q}}(\sqrt{-n}).\) One of the results is \[ h(-3p)\equiv h(-6p)\equiv (p+1+\lambda (p))(mod 16)\quad for\quad primes\quad p\equiv -1(mod 24).\quad This\quad implies\quad a\quad wea\ker \quad assertion\quad on\quad the\quad sum\quad \] \(h(-3p)+h(-6p)\) which is contained as a special case in \textit{A. Pizer}'s work [J. Number Theory 8, 184-192 (1976; Zbl 0329.12003)].
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11D85
zbMATH DE Number: 4097476
zbMATH Keywords: representation of integers
zbMATH Keywords: quaternary forms
zbMATH Keywords: theta series
zbMATH Keywords: Eisenstein series
zbMATH Keywords: cusp eigenform
zbMATH Keywords: Dedekind's eta function
zbMATH Keywords: Liouville's and Kloosterman's formulas
zbMATH Keywords: class numbers
zbMATH Keywords: imaginary quadratic fields
MaRDI profile type: Publication
cites work: Q3250671
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cites work: Q5511020
cites work: Über die Klassenzahl des Körpers \(P(\sqrt{-2p})\) mit einer Primzahl \(p\ne 2\)
cites work: Q5801410
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cites work: On the 2-part of the class number of imaginary quadratic number fields
cites work: Q3909097
full work available at URL: https://doi.org/10.1007/bf01188674
OpenAlex ID: W2021770140
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description: scientific article; zbMATH DE number 4097476