Hyperbolic distribution problems and half-integral weight Maass forms. (English) Zbl 0628.10029

Duke, W.

Invent. Math. 92, No. 1, 73-90 (1988).

A recent estimate of H. Iwaniec [Invent. Math. 87, 385-401 (1987)] for certain Fourier coefficients of a holomorphic cusp form of weight half an odd integer is extended to include a class of Maass forms. Several applications are given, in particular the uniform distribution of Heegner points and cycles on the hyperbolic surface PSL(2,\({\mathbb{Z}})\setminus H\).

MSC:

11F12

Automorphic forms, one variable

Keywords:

Fourier coefficients; Maass forms; uniform distribution of Heegner points; cycles; hyperbolic surface; half-integral weight; Kuznetsov formula; non-holomorphic automorphic forms

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) · Zbl 0443.10023
[2]	Arenas, A.: An arithmetic problem on the sums of three squares. Acta Arith.51 no. 3 (1987)
[3]	Arenstorf, R.F., Johnson, D.: Uniform distribution of integral points on 3-dimensional spheres via modular forms. J. Number Theory11, 218-238 (1979) · Zbl 0413.10037
[4]	Burgess, D.: On character sums andL-series II. Proc. Lond. Math. Soc., III Ser.13, 524-536 (1963) · Zbl 0123.04404
[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) · Zbl 0508.10013
[6]	Friedberg, S.: On theta functions associated to indefinite quadratic forms. J. Number Theory23, 255-267 (1986) · Zbl 0587.10014
[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) · Zbl 0564.10043
[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) · Zbl 0479.10016
[9]	Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York: Academic Press 1980 · Zbl 0521.33001
[10]	Hecke, E.: Lectures on the theory of algebraic numbers. Berlin Heidelberg New York: Springer 1981 · Zbl 0504.12001
[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
[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) · Zbl 0516.10018
[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
[15]	Iwaniec, H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math.87, 385-401 (1987) · Zbl 0606.10017
[16]	Koblitz, N.: Introduction to elliptic curves and modular forms. Berlin Heidelberg New York: Springer 1984 · Zbl 0553.10019
[17]	Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann.271, 237-268 (1985) · Zbl 0553.10020
[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
[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) · Zbl 0427.10016
[20]	Linnik, Y.V.: Ergodic properties of algebraic fields. Berlin Heidelberg New York: Springer 1968 · Zbl 0162.06801
[21]	Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.121, 141-183 (1949) · Zbl 0034.31702
[22]	Maass, H.: Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Ann.138, 287-315 (1959) · Zbl 0089.06102
[23]	Maclachlan, C.: Groups of units of zero ternary quadratic forms. Proc. R. Soc. Edinb. Sect. A.88, 141-157 (1981) · Zbl 0467.10018
[24]	Magnus, W., Oberhettinger, F.: Formulas and theorems for the functions of mathematical physics. Berlin Heidelberg New York: Springer 1966 · Zbl 0143.08502
[25]	Malyshev, A.V.: Yu V. Linnik’s ergodic method in number theory. Acta Arith.27, 555-598 (1975) · Zbl 0303.10020
[26]	Mennicke, J.: On the groups of units of ternary quadratic forms with rational coefficients. Proc. R. Soc. Edinb. Sect. A67, 309-351 (1967) · Zbl 0169.03801
[27]	Niwa, S.: Modular forms of half-integral weight and the integral of certain theta functions. Nagoya Math. J.56, 183-202 (1974)
[28]	Ogg, A.P.: Modular forms and Dirichlet series. New York: W.A. Benjamin, Inc. 1969 · Zbl 0191.38101
[29]	Pfetzer, W.: Die Wirkung der Modulsubstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Variablenzahl. Arch. Math. IV 448-454 (1953) · Zbl 0052.08703
[30]	Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten aufm-dimensionalen Ellipsoiden. Acta Arith.5, 227-257 (1959) · Zbl 0089.26802
[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) · Zbl 0425.10027
[33]	Ramanathan, K.G.: On the analytic theory of quadratic forms. Acta Arith.21, 423-436 (1972) · Zbl 0265.10014
[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) · Zbl 0152.07705
[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
[36]	Sarnak, P.: Class numbers of indefinite binary quadratic forms. J. Number Theory15, 229-247 (1982) · Zbl 0499.10021
[37]	Schoeneberg, B.: Elliptic modular functions. Berlin Heidelberg New York: Springer 1974 · Zbl 0285.10016
[38]	Schulze-Pillot, R.: Thetareihen positiv definiter quadratischer Formen. Invent. Math.75, 283-299 (1984) · Zbl 0533.10021
[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) · Zbl 0072.08201
[40]	Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Symposia Pure Math.8, 1-15 (1965) · Zbl 0142.33903
[41]	Shimura, G.: On modular forms of half-integral weight. Ann. Math., II Ser.97, 440-481 (1973) · Zbl 0266.10022
[42]	Shintani, T.: On construction of holomorphic cusp forms of half-integral weight. Nagoya Math. J.58, 83-126 (1975) · Zbl 0316.10016
[43]	Siegel, C.L.: The average measure of quadratic forms with given determinant and signature. Ann. Math.45, 667-685 (1944) · Zbl 0063.07007
[44]	Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie I. Math. Ann.124, 17-54 (1951) · Zbl 0043.27402
[45]	Siegel, C.L.: Lectures on advanced analytic number theory. Bombay: Tata Institute 1961
[46]	Siegel, C.L.: Lectures on quadratic forms. Bombay: Tata Institute 1967 · Zbl 0248.10019
[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) · Zbl 0497.10022
[48]	Tunnell, J.: A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.72, 323-334 (1983) · Zbl 0515.10013