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The Ramanujan Journal

, Volume 29, Issue 1–3, pp 431–445 | Cite as

A proof of the S-genus identities for ternary quadratic forms

  • Alexander Berkovich
  • Jonathan Hanke
  • Will Jagy
Article

Abstract

In this paper we prove the main conjectures of Berkovich and Jagy about weighted averages of representation numbers over an S-genus of ternary lattices (defined below) for any odd squarefree S∈ℕ. We do this by reformulating them in terms of local quantities using the Siegel–Weil and Conway–Sloane formulas, and then proving the necessary local identities. We conclude by conjecturing generalized formulas valid over certain totally real number fields as a direction for future work.

Keywords

Ternary quadratic forms S-Genus θ-Functions Local densities Siegel’s product 

Mathematics Subject Classification

11E12 11E20 11E25 11F27 11F30 

References

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Alexander Berkovich
    • 1
  • Jonathan Hanke
    • 2
  • Will Jagy
    • 3
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA
  3. 3.Math. Sci. Res. Inst.BerkeleyUSA

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