Abstract
Let N and M be quadratic ℤ-lattices, and K be a sublattice of N. A representation σ:K→M is said to be extensible to N if there exists a representation ρ:N→M such that ρ | K =σ. We prove in this paper a local–global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite ℤ-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132–141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88–101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.
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Böcherer, S., Raghavan, S.: On Fourier coefficients of Siegel modular forms. J. Reine Angew. Math. 384, 80–101 (1988)
Cassels, J.W.S.: Rational Quadratic Forms. Academic Press, London (1978)
Hsia, J.S.: Arithmetic of indefinite quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 1–15. Am. Math. Soc., Providence (1999)
Hsia, J., Kitaoka, Y., Kneser, M.: Representations of positive definite quadratic forms. J. Reine Angew. Math. 301, 132–141 (1978)
Jagy, W.C., Kaplansky, I., Schiemann, A.: There are 913 regular ternary forms. Mathematika 44, 332–341 (1997)
Jöchner, M.: On the representation theory of positive definite quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 73–86. Am. Math. Soc., Providence (1999)
Jöchner, M., Kitaoka, Y.: Representation of positive definite quadratic forms with congruence and primitive conditions. J. Number Theory 48, 88–101 (1994)
Kim, B.M., Kim, M.-H., Oh, B.-K.: 2-universal positive definite integral quinary quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 51–62. Am. Math. Soc., Providence (1999)
Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993)
Oh, B.-K.: The representation of quadratic forms by almost universal forms of higher rank. Math. Z. 244, 399–413 (2003)
O’Meara, O.T.: Introduction to Quadratic Forms. Grundlehren der mathematischen Wissenschaften, vol. 117. Springer, Berlin (1963)
Schulze-Pillot, R.: Representation by integral quadratic forms—a survey. In: Contemporary Mathematics, vol. 344, pp. 303–321. Am. Math. Soc., Providence (2004)
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Research of the first author was partially supported by the National Science Foundation. The third author was partially supported by KRF Research Fund (2003-070-C00001).
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Chan, W.K., Kim, B.M., Kim, MH. et al. Extensions of representations of integral quadratic forms. Ramanujan J 17, 145–153 (2008). https://doi.org/10.1007/s11139-007-9023-y
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DOI: https://doi.org/10.1007/s11139-007-9023-y