Higher Moments of Fourier Coefficients of Cusp Forms

Lü, Guangshi ; Sankaranarayanan, Ayyadurai

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 548-560

Résumé

Let ${{S}_{k}}\left( \Gamma\right)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL\left( 2,\mathbb{Z} \right)$ . Let ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{g}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{h}}\left( n \right)$ be the $n$ -th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f\left( z \right)\in {{S}_{{{k}_{1}}}}\left( \Gamma\right),g\left( z \right)\in {{S}_{{{k}_{2}}}}\left( \Gamma\right)$ , and $h\left( z \right)\in {{S}_{{{k}_{3}}}}\left( \Gamma\right)$ , respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\text{ }{{\lambda }_{f}}{{\left( n \right)}^{4}}{{\lambda }_{g}}{{\left( n \right)}^{2}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{6}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{2}}{{\lambda }_{h}}{{\left( n \right)}^{4}}$ , and ${{\text{ }\!\!\lambda\!\!\text{ }}_{g}}{{\left( {{n}^{3}} \right)}^{2}}$ twisted by the arithmetic function ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$ .

Détail

DOI : 10.4153/CMB-2015-031-1

Mots-clés : 11F30, 11F66, Fourier coefficients of automorphic forms, Dirichlet series, triple product L-function, Perron’s formula

Bibliographie

[1] [1] Davenport, H., On certain exponential sums. J. Reine angew. Math. 169(1932), 158–176. Google Scholar

[2] [2] Deligne, P., La Conjecture de Weil. Inst. Hautes Etudes Sci. Pub. Math. 43(1974), 29–39. Google Scholar

[3] [3] Fomenko, O. M., Fourier coefficients of parabolic forms and automorphic L-functions. J. Math. Sci. (New York). 95(1999), no. 3, 2295–2316. Google Scholar

[4] [4] Gelbart, S. and Jacquet, H., A relation between automorphic representations o/GL(2) and GL(3). Ann. Sci. École Norm. Sup. 11(1978), no. 4, 471–552. Google Scholar

[5] [5] Good, A., The square mean ofDirichlet series associated with cusp forms. Mathematika. 29(1982), no. 2, 278–295. http://dx.doi.Org/10.1112/S0025579300012377 Google Scholar

[6] [6] Hafner, J. L. and Me, A., On sums of Fourier coefficients of cusp forms. Enseign.Math. 35(1989), no. 3-4, 375–382. Google Scholar

[7] [7] Hecke, E., Théorie der Eisensteinsche Reihenhoherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh.Math.Sem. Univ. Hamburg 5(1927), no. 1, 199–224. Google Scholar | DOI

[8] [8] Ivic, A., On zeta-functions associated with Fourier coefficients of cusp forms. In: Proceedings of theAmalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, 1992, pp. 231–246. Google Scholar

[9] [9] Iwaniec, H. and Kowalski, E., Analytic number theory. American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. Google Scholar

[10] [10] Jacquet, H., Piatetski-Shapiro, I.I., and Shalika, J. A., Rankin-Selberg convolutions. Amer. J. Math. 105(1983), no. 2, 367–464. Google Scholar | DOI

[11] [11] Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations. I. Amer. J. Math. 103(1981), no. 3, 499–558. http://dx.doi.Org/10.2307/2374103 Google Scholar

[12] [12] Jacquet, H. and Shalika, J. A. , On Euler products and the classification of automorphic forms. II. Amer. J. Math. 103(1981), no. 4, 777–815. Google Scholar | DOI

[13] [13] Kloosterman, H. D., AsymptotischeFormeln fur die Fourier-koeffizientenganzerModulformen. Abh.Math. Sem. Univ. Hamburg. 5(1927), no. 1, 337–352. http://dx.doi.Org/10.1 OO7/BFO295253O Google Scholar

[14] [14] Kim, H., Functoriality for the exterior square ofGL\ and symmetric fourth ofGL2- With appendix 1 by DinakarRamakrishnan and appendix 2 by Kim and Peter Sarnak, J. Amer. Math. Soc. 16(2003), no. 1, 139–183. Google Scholar | DOI

[15] [15] Kim, H. and Shahidi, E., Functorial products for GL2 x GL3 and the symmetric cube for GL2 With an appendix by Colin J. Bushnell and Guy Henniart, Ann. of Math. 155(2002), no. 3, 837–893. Google Scholar | DOI

[16] [16] Kim, H. and Shahidi, E. , Cuspidality of symmetric power with applications. Duke Math. J. 112(2002), no. 1, 177–197. Google Scholar | DOI

[17] [17] Lau, Y.-K. and Lu, G. S., Sums of Fourier coefficients of cusp forms. Q. J. Math. 62(2011), no. 3, 687–716. http://dx.doi.Org/10.1093/qmath/haqO12 Google Scholar

[18] [18] Lii, G. S., Average behavior of Fourier coefficients of cusp forms. Proc. Amer. Math. Soc. 137(2009), no. 6, 1961–1969. Google Scholar

[19] [19] Lii, G. S., On sixth and eighth moments of Fourier coefficients of cusp forms. J. Number Theory. 129(2009), no. 11, 2790–2880. http://dx.doi.Org/1 0.1 01 6/j.jnt.2009.01 .01 9 Google Scholar

[20] [20] Lii, G. S., On higher moments of Fourier coefficients of holomorphic cusp forms. Canad. J. Math. 63(2011), no. 3, 634–647. Google Scholar | DOI

[21] [21] Lii, G. S., On higher moments of Fourier coefficients of holomorphic cusp forms. II. Montash Math. 169(2013), no. 3-4, 409–422. Google Scholar | DOI

[22] [22] Moreno, C. J. and Shahidi, F., The fourth moment of the Ramanujan r-function. Math. Ann. 266(1983), no. 2, 233–239. Google Scholar | DOI

[23] [23] Rankin, R. A., Contributions to the theory of Ramanujan's function T(«)and similar arithemtical functions. II. The order of the Fourier coefficients of the integral modular forms. Proc. Cambridge Phil. Soc. 35(1939), 357–372. Google Scholar

[24] [24] Rankin, R. A., Sums of cusp form coefficients. In: Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, Montreal, QC, 1990, pp. 115–121. Google Scholar

[25] [25] Rudnick, Z. and Sarnak, P., Zeros of principal L-functions and random matrix theory. Duke Math. J. 81(1996), no. 2, 269–322. Google Scholar | DOI

[26] [26] Ramakrishnan, D. and Wang, S., A cuspidality criterion for the functorial product on GL(2) x GL(3) with a cohomological application. Int. Math. Res. Not. 2004, no. 27,1355–1394. Google Scholar

[27] [27] Salie, H., ZurAbschatzung der FourierkoeffizientenganzerModulformen. Math. Z. 36(1933), no. 1, 263–278. Google Scholar | DOI

[28] [28] Selberg, A., BemerkungeniibereineDirichletscheReihe, die mit der Théorie der Modulformen naheverbundenist. Arch. Math. Naturvid. 43(1940), 47–50. Google Scholar

[29] [29] Selberg, A., On certain L-functions. Amer. J. Math. 103(1981), no. 2, 297–355. Google Scholar | DOI

[30] [30] Selberg, A., Fourier transforms of intertwining operators and Plancherel measures for GL(«). Amer. J. Math. 106(1984), no. 1, 67–111. http://dx.doi.Org/10.2307/2374430 Google Scholar

[31] [31] Selberg, A., Local coefficients as Artin factors for real groups. Duke Math. J. 52(1985), no. 4, 973–1007. Google Scholar | DOI

[32] [32] Selberg, A., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math. 132(1990), no. 2, 273–330. http://dx.doi.Org/10.2307/1 971 524 Google Scholar

[33] [33] Walfisz, A., Ûberdie KoeffizientensummeneinigerModulformen. Math. Ann. 108(1933), no. 1, 75–90. Google Scholar | DOI

[34] [34] Wilton, J. R., A note on Ramanujan's arithmetical function T(«). Proc. Cambridge Philos. Soc. 25(1928), 121–129. Google Scholar

[35] [35] Weil, A., On some exponential sums. Proc. Acad. Sci. U.S.A. 34(1948), 204–207. http://dx.doi.Org/10.1073/pnas.34.5.204 Google Scholar

[36] [36] Wu, J., Power sums of Heche eigenvalues and application. Acta Arith. 137(2009), no. 4, 333–344. Google Scholar | DOI