A proof of the S-genus identities for ternary quadratic forms
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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.
KeywordsTernary quadratic forms S-Genus θ-Functions Local densities Siegel’s product
Mathematics Subject Classification11E12 11E20 11E25 11F27 11F30