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A proof of the \(S\)-genus identities for ternary quadratic forms. (English) Zbl 1341.11014

For an odd squarefree natural number \(S\), the \(S\)-genus of ternary quadratic forms is defined to be the set of equivalence classes of ternary integral quadratic forms locally equivalent to a quadratic form \(B(x,y)+2Sz^2\), where \(B(x,y)\) is a positive definite integer-valued binary quadratic form of discriminant \(-8S\). If \(S\) is a product of \(r\) distinct prime factors, then this \(S\)-genus is the union of \(2^r\) genera of integral ternary quadratic forms of discriminant \(16S^2\).
In a previous paper [in: The legacy of Alladi Ramakrishnan in the mathematical sciences. New York, NY: Springer. 211–241 (2010; Zbl 1322.11028)], the first and third authors conjectured the validity of several formulas expressing the weighted average over an \(S\)-genus of representation numbers of certain integers in terms of the number of representations of an integer as a sum of three squares.
In the present paper, the authors provide proofs of the conjectured identities. This is accomplished by utilizing the Siegel-Weil formula for definite ternary quadratic forms and an explicit mass formula due to J. H. Conway and N. J. A. Sloane [Proc. R. Soc. Lond., Ser. A 419, No. 1857, 259–286 (1988; Zbl 0655.10023)]. The paper contains detailed computations of the local representation densities and masses of the genera in an \(S\)-genus that are needed to apply these formulas.

MSC:

11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11E25 Sums of squares and representations by other particular quadratic forms
11F27 Theta series; Weil representation; theta correspondences
11F30 Fourier coefficients of automorphic forms

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References:

[1] Berkovich, A., Jagy, W.C.: Ternary quadratic forms, modular equations and certain positivity conjectures. In: Alladi, K., Klauder, J., Rao, C.R. (eds.) The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pp. 211–241. Springer, Berlin (2010) · Zbl 1322.11028
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