On the number of representations of positive integers by a direct sum of quaternary quadratic forms with discriminant \(19^2\) (Q2709541)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the number of representations of positive integers by a direct sum of quaternary quadratic forms with discriminant \(19^2\) |
scientific article |
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17 November 2002
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representations by quadratic forms
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Eisenstein series
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cusp forms
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generalized theta functions
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explicit exact formulas
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number of representations of positive integers
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On the number of representations of positive integers by a direct sum of quaternary quadratic forms with discriminant \(19^2\) (English)
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Let the quadratic forms \(F_2= x_1^2+ x_1x_2+ 5x_2^2+ x_3^2+ x_3x_4+ 5x_4\) and \(\Phi_2= x_1^2+ 2x_2^2+ 3x_3^2+ 6x_4^2+ x_1x_2- x_1x_3+2x_2x_4+ 3x_3x_4\) both be positive quaternary with discriminant \(19^2\). Explicit exact formulas are obtained in Theorem 2 for the number of representations of positive integers by \(F_2\oplus F_2\), \(F_2\oplus \Phi_2\) and \(\Phi_2\oplus \Phi_2\). The form \(\Phi_2\) was first found by \textit{K. Sh. Shavgulidze} [Tr. Tbilis. Univ. 214, Mat. Mekh. Astron. 9, 194-219 (1980; Zbl 0487.10016)] where he also gave the same formula for \(\Phi_2\oplus \Phi_2\). Theorem 1 proves that a certain system of generalized theta functions provides a basis of the space \(S_4(19,1)\) of cusp forms of type \((-4,19,1)\).
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