Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids

This is a preview of subscription content, access via your institution.

References

  1. [Ci]

    Cipra, B.: On the Niwa-Shintani theta-kernel lifting of modular forms. Nagoya Math. J. 91, 49–117 (1983)

    Google Scholar 

  2. [De]

    Deligne, P.: La conjecture de Weil. I. Publ. Math., Inst. Hautes Etud. Sci. 43, 273–308 (1973)

    Google Scholar 

  3. [Du]

    Duke, W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)

    Google Scholar 

  4. [Ea]

    Earnest, A.G.: Representation of spinor exceptional integers by ternary quadratic forms. Nagoya Math. J. 93, 27–38 (1984)

    Google Scholar 

  5. [E]

    Eichler, M.: Quadratische Formen und orthogonale Gruppen. Berlin Göttingen Heidelberg: Springer 1952

    Google Scholar 

  6. [Go-Fol]

    Golubeva, E.P., Fomenko, O.M.: Application of spherical functions to a certain problem in the theory of quadratic forms. J. Sov. Math. 38, 2054–2060 (1987)

    Google Scholar 

  7. [Go-Fo2]

    Golubeva, E.P., Fomenko, O.M.: Asymptotic equidistribution of integral points on the three-dimensional sphere (russian). Zapiski Nauchnykh Seminarov Leningradskogo Otd. Math. Inst. im. V.A. Steklova AN SSSR 160, 54–71 (1987)

    Google Scholar 

  8. [I]

    Iwaniec, H.: Fourier coefficients of modular forms of half integral weight. Invent. Math. 87, 385–401 (1987)

    Google Scholar 

  9. [Jo]

    Jones, B.W.: The arithmetic theory of quadratic forms. Carus Math. Monographs no. 10, New York: Wiley 1950

    Google Scholar 

  10. [Kn]

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

    Google Scholar 

  11. [Ma]

    Malyshev, A.V.: Representation of integers by positive quadratic forms. Trudy Mat. Inst. Akad. Nauk. SSSR 65 (1962)

  12. [Ni]

    Niwa, S.: Modular forms of half integral weight and the integral of certain theta functions. Nagoya Math. J. 56, 147–161 (1974)

    Google Scholar 

  13. [OM]

    O'Meara, O.T.: Introduction to quadratic forms. Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  14. [Pe]

    Peters, M.: Darstellungen durch definite ternäre quadratische Formen, Acta Arith. 34, 57–80 (1977)

    Google Scholar 

  15. [Pf]

    Pfetzer, W.: Die Wirkung der Modulsubstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Variablenzahl. Arch. Math. 4, 448–454 (1953)

    Google Scholar 

  16. [Pod]

    Podsypanin, E.V.: The number of integral points in an elliptic region (a remark on a theorem of A.V. Malyshev). J. Sov. Math. 18, 923–925 (1982)

    Google Scholar 

  17. [Pom]

    Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227–257 (1959)

    Google Scholar 

  18. [Shi]

    Shimura, G.: On modular forms of half integral weight. Ann. Math. 97, 440–481 (1973)

    Google Scholar 

  19. [Si]

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

    Google Scholar 

  20. [Si2]

    Siegel, C.L.: Über die Classenzahl quadratischer Zahlkörper. Acta Arith. 1, 83–86 (1935)

    Google Scholar 

  21. [SPI]

    Schulze-Pillot, R.: Thetareihen positiv definiter quadratischer Formen. Invent. Math. 75, 283–299 (1984)

    Google Scholar 

  22. [SP2]

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

    Google Scholar 

  23. [SP3]

    Schulze-Pillot, R.: Ternary quadratic forms and Brandt matrices. Nagoya Math. J. 102, 117–126 (1986)

    Google Scholar 

  24. [Stu]

    Sturm, J.: Theta series of weight 3/2. J. Number Th. 14, 353–361 (1982)

    Google Scholar 

  25. [Te]

    Teterin, Yu.G.: Representation of integers by positive ternary quadratic forms. J. Sov. Math. 29, 1312–1342 (1985)

    Google Scholar 

  26. [vB]

    Van der Blij, F.: On the theory of quadratic forms. Ann. Math. 50, 875–899 (1949)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Duke, W., Schulze-Pillot, R. Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent Math 99, 49–57 (1990). https://doi.org/10.1007/BF01234411

Download citation

Keywords

  • Quadratic Form
  • Lattice Point
  • Ternary Quadratic Form
  • Positive Ternary Quadratic Form