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Representation of Quadratic Forms by Integral Quadratic Forms

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Quadratic and Higher Degree Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 31))

Abstract

The number of representations of a positive definite integral quadratic form of rank n by another positive definite integral quadratic form of rank m ≥ n has been studied by arithmetic, analytic, and ergodic methods. We survey and compare in this article the results obtained by these methods.

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References

  1. A. N. Andrianov, V. G. Zhuravlev: Modular forms and Hecke operators. Translations of Mathematical Monographs, 145. American Mathematical Society, Providence, RI, 1995.

    Google Scholar 

  2. M. Bhargava: On the Conway-Schneeberger fifteen theorem. Quadratic forms and their applications (Dublin, 1999), 27–37, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000.

    Google Scholar 

  3. V. Blomer, V, J. Brüdern, R. Dietmann: Sums of smooth squares. Compos. Math. 145 (2009), 1401–1441.

    Google Scholar 

  4. V. Blomer, G. Harcos: Hybrid bounds for twisted L-functions. J. Reine Angew. Math. 621 (2008), 53–79

    MathSciNet  MATH  Google Scholar 

  5. V. Blomer, G. Harcos: Twisted L-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), 1–52

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Böcherer: On the Fourier-Jacobi-coefficients of Eisenstein series of Klingen type, Preprint

    Google Scholar 

  7. S. Böcherer, W. Kohnen: Estimates for Fourier coefficients of Siegel cusp forms. Math. Ann. 297 (1993), 499–517

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Böcherer, S. Raghavan: On Fourier coefficients of Siegel modular forms. J. Reine Angew. Math. 384 (1988), 80–101.

    MathSciNet  MATH  Google Scholar 

  9. J.W.S. Cassels: Rational quadratic forms. London Mathematical Society Monographs, 13. Academic Press, Inc., London-New York, 1978.

    Google Scholar 

  10. W.K. Chan, M.I. Icaza: Effective results on representations of quadratic forms. Algebraic and arithmetic theory of quadratic forms, 73–83, Contemp. Math., 344, Amer. Math. Soc., Providence, RI, 2004

    Google Scholar 

  11. W.K. Chan, B.M. Kim, M.-H. Kim, B.-K. Oh: Extensions of representations of integral quadratic forms, Ramanujan J. 17 (2008), 145–153

    Article  MathSciNet  MATH  Google Scholar 

  12. W. Duke, R. Schulze-Pillot: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99 (1990), 49–57.

    Article  MathSciNet  MATH  Google Scholar 

  13. A.G. Earnest, J.S. Hsia: Spinor norms of local integral rotations. II. Pacific J. Math. 61 (1975), 71–86.

    Article  MathSciNet  MATH  Google Scholar 

  14. A.G. Earnest, J.S. Hsia, D.C. Hung: Primitive representations by spinor genera of ternary quadratic forms. J. London Math. Soc. (2) 50 (1994), 222–230

    Google Scholar 

  15. M. Eichler: Die Ähnlichkeitsklassen indefiniter Gitter. Math. Z. 55 (1952), 216–252.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Ellenberg, A. Venkatesh: Local-global principles for representations of quadratic forms. Invent. Math. 171 (2008), 257–279.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Hasse: Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen. J. für Math. 152, 129–148 (1923)

    MATH  Google Scholar 

  18. J.S. Hsia: Spinor norms of local integral rotations. I. Pacific J. Math. 57 (1975), 199–206

    Article  MathSciNet  MATH  Google Scholar 

  19. J. S. Hsia: Representations by spinor genera. Pacific J. Math. 63 (1976), 147–152.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. S. Hsia, Y. Kitaoka, M. Kneser: Representations of positive definite quadratic forms. J. Reine Angew. Math. 301 (1978), 132–141.

    MathSciNet  MATH  Google Scholar 

  21. J.S. Hsia, J.P. Prieto-Cox: Representations of positive definite Hermitian forms with approximation and primitive properties. J. Number Theory 47 (1994), 175–189.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Jöchner, Y. Kitaoka: Representations of positive definite quadratic forms with congruence and primitive conditions. J. Number Theory 48 (1994), 88–101.

    Article  MathSciNet  MATH  Google Scholar 

  23. B.M. Kim, M.-H. Kim, B.-K. Oh: A finiteness theorem for representability of quadratic forms by forms. J. Reine Angew. Math. 581 (2005), 23–30

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Kitaoka: Modular forms of degree n and representation by quadratic forms. II. Nagoya Math. J. 87 (1982), 127–146.

    MathSciNet  MATH  Google Scholar 

  25. Y. Kitaoka: Lectures on Siegel modular forms and representation by quadratic forms. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 77. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1986.

    Google Scholar 

  26. Y. Kitaoka: Modular forms of degree n and representation by quadratic forms. V. Nagoya Math. J. 111 (1988), 173–179.

    MathSciNet  MATH  Google Scholar 

  27. Y. Kitaoka: Local densities of quadratic forms. Investigations in number theory, 433–460, Adv. Stud. Pure Math., 13, Academic Press, Boston, MA, 1988.

    Google Scholar 

  28. Y. Kitaoka: Some remarks on representations of positive definite quadratic forms. Nagoya Math. J. 115 (1989), 23–41.

    MathSciNet  MATH  Google Scholar 

  29. Y. Kitaoka: Representations of positive definite quadratic forms and analytic number theory. Translation of Sugaku 43 (1991), no. 2, 115–127

    MathSciNet  MATH  Google Scholar 

  30. Y. Kitaoka: The minimum and the primitive representation of positive definite quadratic forms. Nagoya Math. J. 133 (1994), 127–153.

    MathSciNet  MATH  Google Scholar 

  31. Y. Kitaoka: The minimum and the primitive representation of positive definite quadratic forms. II. Nagoya Math. J. 141 (1996), 1–27.

    MathSciNet  MATH  Google Scholar 

  32. H.D. Kloosterman: On the representation of numbers in the form \(a{x}^{2} + b{y}^{2} + c{z}^{2} + d{t}^{2}.\), Acta Math. 49 (1927), 407–464.

    Google Scholar 

  33. H.D. Kloosterman: Asymptotische Formeln für die Fourierkoeffizienten ganzer Modulformen. (German) Abh. Math. Sem. Hamburg 5 (1927), 337–352.

    Google Scholar 

  34. M. Kneser: Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen. Arch. Math. 7 (1956), 323–332.

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Kneser: Darstellungsmaße indefiniter quadratischer Formen. Math. Z. 77 (1961), 188–194.

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Kohnen: On the fourier coefficients of certain special Siegel cusp forms. Math. Z. 248 (2004), 345–350

    Article  MathSciNet  MATH  Google Scholar 

  37. G.A. Margulis, G.M. Tomanov: Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116 (1994), 347–392.

    Article  MathSciNet  MATH  Google Scholar 

  38. O.T. O’Meara: Introduction to quadratic forms, Springer-Verlag 1973.

    Google Scholar 

  39. S. Raghavan: Modular forms of degree n and representation by quadratic forms. Ann. of Math. (2) 70 (1959) 446–477

    Google Scholar 

  40. M. Ratner: Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups. Duke Math. J. 77 (1995), 275–382

    Article  MathSciNet  MATH  Google Scholar 

  41. H. L. Resnikoff and R. L. Saldaña: Some properties of Fourier coefficients of Eisenstein series of degree two. J. reine angew. Math. 265 (1974) 90–109.

    MathSciNet  MATH  Google Scholar 

  42. R. Schulze-Pillot: Darstellung durch Spinorgeschlechter ternärer quadratischer Formen, J. Number Theory 12 (1980), 529–540

    Article  MathSciNet  MATH  Google Scholar 

  43. R. Schulze-Pillot: Darstellungsmaße von Spinorgeschlechtern ternärer quadratischer Formen, J. Reine Angew. Math. 352 (1984), 114–132

    MathSciNet  MATH  Google Scholar 

  44. R. Schulze-Pillot: Representation by integral quadratic forms—a survey. Algebraic and arithmetic theory of quadratic forms, 303–321, Contemp. Math., 344, Amer. Math. Soc., Providence, RI, 2004

    Google Scholar 

  45. R. Schulze-Pillot: Local conditions for global representations of quadratic forms, Acta Arith. 138 (2009), 289–299

    Article  MathSciNet  MATH  Google Scholar 

  46. C.L. Siegel: Über die analytische Theorie der quadratischen Formen. Ann. of Math. (2) 36 (1935), 527–606

    Google Scholar 

  47. C.L. Siegel: Über die analytische Theorie der quadratischen Formen II, Ann. of Math. (2) 37 (1936), 230–263

    Google Scholar 

  48. C.L. Siegel: Über die analytische Theorie der quadratischen Formen III, Ann. of Math. (2) 38 (1937), 212–291

    Google Scholar 

  49. C. L. Siegel: On the theory of indefinite quadratic forms, Ann. of Math. 45 (1944), 577–622

    Article  MathSciNet  MATH  Google Scholar 

  50. W. Tartakowsky: Die Gesamtheit der Zahlen, die durch eine positive quadratische Form \(F(x_{1},x_{2},\ldots,x_{s})\) (\(s \geqq 4\)) darstellbar sind. I, II. Bull. Ac. Sc. Leningrad 7 2 (1929); 111–122, 165–196 (1929)

    Google Scholar 

  51. A. Weil: Sur la théorie des formes quadratiques. in: Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) pp. 9–22

    Google Scholar 

  52. Fei Xu: Representation masses of spinor genera. Duke Math. J. 110 (2001), 279–307

    Article  MathSciNet  MATH  Google Scholar 

  53. Fei Xu: On representations of spinor genera II. Math. Ann. 332 (2005), 37–53.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    Rainer Schulze-Pillot

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  1. Rainer Schulze-Pillot
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Correspondence to Rainer Schulze-Pillot .

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Editors and Affiliations

  1. Department of Mathematics, University of Florida Department of Mathematics, Gainesville, Florida, USA

    Krishnaswami Alladi

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Manjul Bhargava

  3. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    David Savitt

  4. Department of Mathematics, University of Arizona, Tuscon, Arizona, USA

    Pham Huu Tiep

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Schulze-Pillot, R. (2013). Representation of Quadratic Forms by Integral Quadratic Forms. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_9

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