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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 110105, Gainesville, FL, 32611-8105, USA

    Alexander Berkovich

  2. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Jonathan Hanke

  3. Math. Sci. Res. Inst., 17 Gauss Way, Berkeley, CA, 94720-5070, USA

    Will Jagy

Authors
  1. Alexander Berkovich
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  2. Jonathan Hanke
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  3. Will Jagy
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Correspondence to Alexander Berkovich.

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Berkovich, A., Hanke, J. & Jagy, W. A proof of the S-genus identities for ternary quadratic forms. Ramanujan J 29, 431–445 (2012). https://doi.org/10.1007/s11139-012-9429-z

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  • DOI: https://doi.org/10.1007/s11139-012-9429-z

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