Kloosterman sums and Fourier coefficients of cusp forms

Springer Science and Business Media LLC - Tập 70 Số 2 - Trang 219-288 - 1982
Jean-Marc Deshouillérs1, Henryk Iwaniec2
1Laborative Associé au CNRS N0 226, Université de Bordeaux I, Mathematiques 351, Cours de la Liberation, F-33405, Talence, France
2Mathematics Institute Polish Academy of Sciences ul, Sniadeckich 8, PL-00-950, Warszawa

Tóm tắt

Từ khóa


Tài liệu tham khảo

[Bru] Bruggeman, R.W.: Fourier coefficients of cusp forms. Invent. Math.45, 1?18 (1978)

[Bus] Buser, P.: On Cheeger's inequality? 1?h 2/4. Proc. Symposia, in Pure Math., XXXVI, pp. 29?77 AMS, Providence 1981

[Che] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis, A symposium in honour of Bochner, Princeton University Press, Princeton, N.J. 1970 pp. 195?199

[Del] Deligne, P.: La conjecture de Weil I. Publ. Math. I.H.E.S.,43, 273?307 (1974)

[Dl 1] Deshouillers, J.-M., Iwaniec, H.: On the greatest prime factor ofn 2+1 Ann. Inst. Fourier. in press (1982)

[DI2] Deshouillers, J.-M., Iwaniec, H.: An additive divisor problem. J. London Math. Soc.26, (2), 1?14 (1982)

[Di3] Deshouillers, J.-M., Iwaniec, H.: Power mean-values for the Riemann zeta-function. Mathematika in press (1982)

[DI4] Deshouillers, J.-M., Iwaniec, H.: On the Brun-Titchmarsh theorem and the greatest prime factor ofp+a. Proc. Janos Bolyai Soc. Conf. in press (1982)

[EMOT] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions II. New York-Toronto-London: Mc Graw-Hill 1953

[Fad] Faddeev, L.D.: Expansions in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Loba?evskii plane. Trudy Moscov, Mat. Obsc.,17, 323?350 (1967)

[G-R] Gradsztejn, I.S., Ry?yk, I.M.: Tablice ca?ek, sum, szeregów i iloczynow. PWN-Warszawa, 1964

[Gun] Gunning, R.C.: Lectures on modular forms. Princeton University Press, 1962

[H.-B.] Heath-Brown, D.R.: The fourth power moment of the Riemann zeta-function. Proceeding London Math. Soc.38(3), 385?422 (1979)

[Hej] Hejhal, D.A.: Some observations concerning eigenvalues of the Laplacian and DirichletL-series in Recent progress in analytic number theory. London: Academic Press 1981

[Hoo] Hooley, C.: On the greatest prime factor of a quadratic polynomial. Acta Math.117, 281?299 (1967)

[Iwa 1] Iwaniec, H.: Fourier coefficients of cusp forms and the Riemann zeta-function. Sém. Th. Nb. Bordeaux (1979?1980), exposé no18,36 pages

[Iwa 2] Iwaniec, H.: Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums. Proceedings from the Journées Arithmétiques at Exeter in press (1982)

[Iwa 3] Iwaniec, H.: On mean values for Dirichlet's polynomials and the Riemann zeta-function. J. London Math. Soc.22, (2), 39?45 (1980)

[Iwa 4] Iwaniec, H.: On the Brun-Titchmarsh theorem. J. Math. Soc. Japan34, 95?123 (1982)

[Kub] Kubota, T.: Elementary Theory of Eisenstein Series. New-York: John Wiley and Sons 1973

[Kuz1] Kuznietsov, N.V.: Petersson hypothesis for forms of weight zero and Linnik hypothesis (in Russian), Preprint no02, Khab. KHII, Khabarovsk 1977

[Kuz2] Kuznietsov, N.V.: Petersson hypothesis for parabolic forms of weight zero and Linnik hypothesis. Sums of Kloostermann sums. Math. Sbornik111, (153), no3, 334?383 (1980)

[Lin] Linnik, Yu.V.: Additive problems and eigenvalues of the modular operators. Proc. Internat. Congr. Math. Stockholm 270?284 (1962)

[Maa1] Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen. Math. Ann.121, (o2) 141?183 (1949)

[Maa2] Maass, H.: On modular functions of one complex variable. Tata Institute, Bombay, 1964

[Mon] Montgomery, H.L.: Topics in multiplicative number theory. Lect. Notes in Math. vol. 227. Berlin-New-York: Springer 1971

[Pet] Petersson, H.: Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math.58, 169?215 (1932)

[Pro1] Proskurin, N.V.: Summation formulas for generalized Kloosterman sums (in Russian). Zap. Nau?n. Sem. Leningrad. Otdel. Mat. Inst. Steklov.82, 103?135 (1979)

[Pro2] Proskurin, N.V.: On a hypothesis of Yu.V. Linnik (in Russian). Zap. Nau?n. Sem. Leningrad Otdel. Mat. Inst. Steklov.91, 94?118 (1979)

[Pro3] Proskurin, N.V.: The estimations of the eigenvalues of Hecke's operators in the space of parabolic forms of weight zero. Studies in Number Theory, vol. 5, 136?143 (1979). Pub. Math. Inst. Steklov

[Ran] Rankin, R.: Modular forms and functions. Cambridge-London-New-York: Cambridge University Press 1977

[Roe1] Roelcke, W.: Über die Wellengleichung bei Grenzkreisgruppen erster Art. Sitz. Ber. Heidl. Akad. der Wiss. (Math. natur. Kl.) 1956, 4. Abh.

[Roe2] Roelcke, W.: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I. II. Math. Ann.167, 292?337 (1966);168, 261?324 (1967)

[Sel1] Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet's series. J. Indian Mat. Soc.20, 47?87 (1956)

[Sel2] Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Symposia in Pure Math. VIII, A.M.S., Providence 1965, pp. 1?15

[Shi] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton: Princeton University Press 1971