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Representations of Bell-Type Quaternary Quadratic Forms

  • Chang Heon Kim
  • Kyoungmin KimEmail author
  • Soonhak Kwon
  • Yeong-Wook Kwon
Article
  • 14 Downloads

Abstract

Let f be a quaternary quadratic form and let r(nf) be the number of representations of an integer n by f. A quaternary quadratic form f is said to be a Bell-type quaternary quadratic form if f is isometric to \(x^2+2^{\alpha }y^2+2^{\beta }z^2+2^{\gamma }w^2\) for some nonnegative integers \(\alpha , \beta , \gamma \). In this article, we give a closed formula for r(nf) for each Bell-type quaternary quadratic form f having class number less than or equal to 2 by using the Minkowski–Siegel formula and bases for certain spaces of cusp forms consisting of eta-quotients. Furthermore, we also find some closed formulas for the representations of Bell-type quaternary quadratic forms with some congruence conditions.

Keywords

Bell-type quaternary quadratic forms the Minkowski–Siegel formula cusp forms eta-quotients congruence conditions 

Mathematics Subject Classification

Primary 11E25 11E45 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Chang Heon Kim
    • 1
  • Kyoungmin Kim
    • 1
    Email author
  • Soonhak Kwon
    • 1
  • Yeong-Wook Kwon
    • 1
  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonKorea

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