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Hyperbolic distribution problems and half-integral weight Maass forms

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References

  1. Andrianov, A.N., Maloletkin, G.N.: The behavior of the theta-series of genusn of indefinite quadratic forms under modular substitutions. Trudy Mat. Inst. Steklova148, 5–15, 271 (1978) Eng. trans.: Proc. Steklov Inst.148, 1–12 (1978)

    Google Scholar 

  2. Arenas, A.: An arithmetic problem on the sums of three squares. Acta Arith.51 no. 3 (1987)

    Google Scholar 

  3. Arenstorf, R.F., Johnson, D.: Uniform distribution of integral points on 3-dimensional spheres via modular forms. J. Number Theory11, 218–238 (1979)

    Google Scholar 

  4. Burgess, D.: On character sums andL-series II. Proc. Lond. Math. Soc., III Ser.13, 524–536 (1963)

    Google Scholar 

  5. Bykovskii, V.A.: On a summation formula in the spectral theory of automorphic functions and its applications in analytic number theory. Dokl. Akad. Nauk. SSSR264, 275–277 (1982) Eng. trans.: Sov. Math. Dokl.25, 608–610 (1982)

    Google Scholar 

  6. Friedberg, S.: On theta functions associated to indefinite quadratic forms. J. Number Theory23, 255–267 (1986)

    Google Scholar 

  7. Goldfeld, D., Hoffstein, J.: Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series. Invent. Math.80, 185–208 (1985)

    Google Scholar 

  8. Golubeva, E.P., Fomenko, O.M.: Theta-series associated with indefinite ternary quadratic forms and the related Dirichlet series. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (LOMI)112, 41–50 (1981) Eng. trans.: J. Sov. Math.25, 999–1005 (1984)

    Google Scholar 

  9. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York: Academic Press 1980

    Google Scholar 

  10. Hecke, E.: Lectures on the theory of algebraic numbers. Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  11. Hecke, E.: Werke. Göttingen: Vandenhoek and Ruprecht 1983

  12. Hejhal, D.A.: The Selberg trace formula for PSL(2,R), vol.2. (Lect. Notes Math., vol. 1001) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  13. Hejhal, D.A.: Some Dirichlet series with coefficients related to periods of automorphic eigenforms I, II. Proc. Japan Acad., Ser. A.58, 413–417 (1982),59, 335–338 (1983)

    Google Scholar 

  14. Hejhal, D.A.: Roots of quadratic congruences and eigenvalues of the non-Euclidean laplacian. In: Sarnak, P., Terras, A.A. (eds.) The Selberg trace formula and related topics. Proceedings AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences, Brunswick 1984. (Contemporary Math. vol. 53, pp. 277–339) Providence: AMS 1986

    Google Scholar 

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

    Google Scholar 

  16. Koblitz, N.: Introduction to elliptic curves and modular forms. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  17. Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann.271, 237–268 (1985)

    Google Scholar 

  18. Kudla, S.S.: Relations between automorphic forms produced by theta-functions. In: Serre, J.-P., Zagier, B.D. (eds.) Modular functions of one variable VI. Proceedings, Bonn 1976. (Lect. Notes Math., vol. 627, pp. 277–285) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  19. Kuznetsov, N.V.: Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums. Mat. Sb., Nov. Ser.111 (153) 334–383 (1980) Eng. trans.: Math. USSR Sb.39, 299–342 (1981)

    Google Scholar 

  20. Linnik, Y.V.: Ergodic properties of algebraic fields. Berlin Heidelberg New York: Springer 1968

    Google Scholar 

  21. Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.121, 141–183 (1949)

    Google Scholar 

  22. Maass, H.: Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Ann.138, 287–315 (1959)

    Google Scholar 

  23. Maclachlan, C.: Groups of units of zero ternary quadratic forms. Proc. R. Soc. Edinb. Sect. A.88, 141–157 (1981)

    Google Scholar 

  24. Magnus, W., Oberhettinger, F.: Formulas and theorems for the functions of mathematical physics. Berlin Heidelberg New York: Springer 1966

    Google Scholar 

  25. Malyshev, A.V.: Yu V. Linnik's ergodic method in number theory. Acta Arith.27, 555–598 (1975)

    Google Scholar 

  26. Mennicke, J.: On the groups of units of ternary quadratic forms with rational coefficients. Proc. R. Soc. Edinb. Sect. A67, 309–351 (1967)

    Google Scholar 

  27. Niwa, S.: Modular forms of half-integral weight and the integral of certain theta functions. Nagoya Math. J.56, 183–202 (1974)

    Google Scholar 

  28. Ogg, A.P.: Modular forms and Dirichlet series. New York: W.A. Benjamin, Inc. 1969

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  31. Proskurin, N.V.: General Kloosterman sums. (Russian) Preprint LOMI R-3-80 Akad. Nauk. SSSR, Mat. Inst. Leningr. Otd. (1980) MR 82E 10040

  32. Proskurin, N.V.: Summation formulas for general Kloosterman sums. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (LOMI)82, 103–135, 167 (1979) Eng. trans.: J. Sov. Math.18, 925–957 (1982)

    Google Scholar 

  33. Ramanathan, K.G.: On the analytic theory of quadratic forms. Acta Arith.21, 423–436 (1972)

    Google Scholar 

  34. Roelke, W.: Das Eigenwert Problem der automorphen Formen in der hyperbolischen Ebene I, II. Math. Ann.167, 292–337 (1966),167, 261–324 (1967)

    Google Scholar 

  35. Sarnak, P.: Additive number theory and Maass forms. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds.) Number theory. Proceedings, New York 1982. (Lect. Notes Math. vol. 1052, pp. 286–309) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  36. Sarnak, P.: Class numbers of indefinite binary quadratic forms. J. Number Theory15, 229–247 (1982)

    Google Scholar 

  37. Schoeneberg, B.: Elliptic modular functions. Berlin Heidelberg New York: Springer 1974

    Google Scholar 

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

    Google Scholar 

  39. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc.20, 47–87 (1956)

    Google Scholar 

  40. Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Symposia Pure Math.8, 1–15 (1965)

    Google Scholar 

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

    Google Scholar 

  42. Shintani, T.: On construction of holomorphic cusp forms of half-integral weight. Nagoya Math. J.58, 83–126 (1975)

    Google Scholar 

  43. Siegel, C.L.: The average measure of quadratic forms with given determinant and signature. Ann. Math.45, 667–685 (1944)

    Google Scholar 

  44. Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie I. Math. Ann.124, 17–54 (1951)

    Google Scholar 

  45. Siegel, C.L.: Lectures on advanced analytic number theory. Bombay: Tata Institute 1961

    Google Scholar 

  46. Siegel, C.L.: Lectures on quadratic forms. Bombay: Tata Institute 1967

    Google Scholar 

  47. Stark, H.M.: On the transformation formula for the symplectic theta function and applications. J. Fac. Sci. Univ. Tokyo, Sect. IA.29, 1–12 (1982)

    Google Scholar 

  48. Tunnell, J.: A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.72, 323–334 (1983)

    Google Scholar 

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  1. W. Duke

    Present address: Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

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  1. Department of Mathematics, University of California, 92093, San Diego, La Jolla, CA, USA

    W. Duke

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Duke, W. Hyperbolic distribution problems and half-integral weight Maass forms. Invent Math 92, 73–90 (1988). https://doi.org/10.1007/BF01393993

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