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Ternary quadratic forms and quaternion algebras. (English) Zbl 0585.10012

A weighted average of the number of primitive representations of a positive integer by a ternary quadratic form is computed using the arithmetic of the associated quaternion algebra.

MSC:

11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11R11 Quadratic extensions
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11R29 Class numbers, class groups, discriminants
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References:

[1] Eichler, M., Uber die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z., 43, 102-109 (1937) · JFM 63.0093.02
[2] Eichler, M., Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math., 195, 127-151 (1956) · Zbl 0068.03303
[3] Eichler, M., Uber die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math., 195, 156-171 (1956) · Zbl 0068.06601
[4] Eichler, M., The Basis Problem for Modular Forms and the Traces of the Hecke Operators, (Lecture Notes in Math., Vol. 320 (1973), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0258.10013
[5] Gauss, C. F., (Disquisitiones Arithmeticae (1965), Yale Univ. Press: Yale Univ. Press New Haven, Conn), translated by Arthur A. Clarke
[6] Hijikata, H., Explicit formula of the traces of the Hecke operators for \(Γ_0(N)\), J. Math. Soc. Japan, 26, 56-82 (1974) · Zbl 0266.12009
[7] Hijikata, H.; Saito, H., On the representability of modular forms by theta series, (Number Theory, Algebraic Geometry, and Commutative Algebra (1973), Kinokuniya: Kinokuniya Tokyo), 13-21, (in honor of Y. Akizuki)
[8] H. Hijikata, A. Pizer, and T. Shemanske; H. Hijikata, A. Pizer, and T. Shemanske · Zbl 0653.12004
[9] Pizer, A., On the arithmetic of quaternion algebras II, J. Math. Soc. Japan, 28, 676-688 (1976) · Zbl 0344.12005
[10] Pizer, A., The representability of modular forms by theta series, J. Math. Soc. Japan, 28, 689-698 (1976) · Zbl 0344.10012
[11] Pizer, A., The action of the canonical involution on modular forms of weight 2 on \(Γ_0(M)\), Math. Ann., 226, 99-116 (1977) · Zbl 0417.10023
[12] Pizer, A., Theta series and modular forms of level \(p^2M\), Compositio Math., 40, 177-241 (1980) · Zbl 0416.10021
[13] Rehm, H. P., On a Theorem of Gauss concerning the Number of Solutions of the Equation \(x^2 + y^2 + z^2 = m\), (Taussky-Todd, O., Lecture Notes in Pure and Applied Mathematics, Vol. 79 (1982), Dekker: Dekker New York) · Zbl 0499.10020
[14] H. P. Rehm; H. P. Rehm
[15] Reiner, I., (Maximal Orders (1975), Academic Press: Academic Press New York)
[16] Roggenkamp, K.; Huber-Dyson, V., Lattices over Orders I, (Lecture Notes in Math., Vol. 115 (1970), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0205.33501
[17] Shemanske, T., Representations of ternary quadratic forms and the class number of imaginary quadratic fields, Pacific J. Math., 122, 223-250 (1986) · Zbl 0585.10013
[18] Shimura, G., (Introduction to the Arithmetic Theory of Automorphic Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton) · Zbl 0221.10029
[19] Venkov, B. A., On the arithmetic of quaternion algebras, Izv. Akad. Nauk, 607-622 (1929) · Zbl 0024.29001
[20] Vigneras, M-F, Arithmetique des Algebras de Quaternions, (Lecture Notes in Mathematics, Vol. 800 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0422.12008
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