×

On the higher power moments of cusp form coefficients over sums of two squares. (English) Zbl 07655785

Summary: Let \(f\) be a normalized primitive holomorphic cusp form of even integral weight for the full modular group \(\Gamma =\mathrm{SL}(2,\mathbb{Z})\). Denote by \(\lambda_{f}(n)\) the \(n\)th normalized Fourier coefficient of \(f\). We are interested in the average behaviour of the sum \[\sum_{a^{2}+b^{2}\leq x}\lambda_{f}^{j}(a^{2}+b^{2})\] for \(x\geq 1\), where \(a,b\in\mathbb{Z}\) and \(j\geq 9\) is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power \(L\)-functions and Rankin-Selberg \(L\)-functions.

MSC:

11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Clozel, L.; Thorne, J. A., Level-raising and symmetric power functoriality. I, Compos. Math. 150 (2014), 729-748 · Zbl 1304.11040 · doi:10.1112/S0010437X13007653
[2] Clozel, L.; Thorne, J. A., Level raising and symmetric power functoriality. II, Ann. Math. (2) 181 (2015), 303-359 · Zbl 1339.11060 · doi:10.4007/annals.2015.181.1.5
[3] Clozel, L.; Thorne, J. A., Level-raising and symmetric power functoriality. III, Duke Math. J. 166 (2017), 325-402 · Zbl 1372.11054 · doi:10.1215/00127094-3714971
[4] Deligne, P., La conjecture de Weil. I, Publ. Math., Inst. Hautes Étud. Sci. 43 (1974), 273-307 French · Zbl 0287.14001 · doi:10.1007/BF02684373
[5] Fomenko, O. M., Fourier coefficients of parabolic forms, and automorphic \(L\)-functions, J. Math. Sci., New York 95 (1999), 2295-2316 · Zbl 0993.11023 · doi:10.1007/BF02172473
[6] Fomenko, O. M., Identities involving the coefficients of automorphic \(L\)-functions, J. Math. Sci., New York 133 (2006), 1749-1755 · doi:10.1007/s10958-006-0086-x
[7] Fomenko, O. M., Mean value theorems for automorphic \(L\)-functions, St. Petersbg. Math. J. 19 (2008), 853-866 · Zbl 1206.11061 · doi:10.1090/S1061-0022-08-01024-8
[8] Gelbart, S.; Jacquet, H., A relation between automorphic representations of \(GL(2)\) and \(GL(3)\), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 471-542 · Zbl 0406.10022 · doi:10.24033/asens.1355
[9] Hafner, J. L.; Ivić, A., On sums of Fourier coefficients of cusp forms, Enseign. Math., II. Sér. 35 (1989), 375-382 · Zbl 0696.10020 · doi:10.5169/seals-57381
[10] He, X., Integral power sums of Fourier coefficients of symmetric square \(L\)-functions, Proc. Am. Math. Soc. 147 (2019), 2847-2856 · Zbl 1431.11062 · doi:10.1090/proc/14516
[11] Hecke, E., Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Semin. Univ. Hamb. 5 (1927), 199-224 German \99999JFM99999 53.0345.02 · doi:10.1007/BF02952521
[12] Huang, B., On the Rankin-Selberg problem, Math. Ann. 381 (2021), 1217-1251 · Zbl 1483.11098 · doi:10.1007/s00208-021-02186-7
[13] Ivić, A., On zeta-functions associated with Fourier coefficients of cusp forms, Proceedings of the Amalfi Conference on Analytic Number Theory Universitá di Salerno, Salerno (1992), 231-246 · Zbl 0787.11035
[14] Iwaniec, H.; Kowalski, E., Analytic Number Theory, Colloquium Publications. American Mathematical Society 53. AMS, Providence (2004) · Zbl 1059.11001 · doi:10.1090/coll/053
[15] Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A., Rankin-Selberg convolutions, Am. J. Math. 105 (1983), 367-464 · Zbl 0525.22018 · doi:10.2307/2374264
[16] Jacquet, H.; Shalika, J. A., On Euler products and the classification of automorphic representations. I, Am. J. Math. 103 (1981), 499-558 · Zbl 0473.12008 · doi:10.2307/2374103
[17] Jacquet, H.; Shalika, J. A., On Euler products and the classification of automorphic forms. II, Am. J. Math. 103 (1981), 777-815 · Zbl 0491.10020 · doi:10.2307/2374050
[18] Jiang, Y.; Lü, G., Uniform estimates for sums of coefficients of symmetric square \(L\)-function, J. Number Theory 148 (2015), 220-234 · Zbl 1380.11037 · doi:10.1016/j.jnt.2014.09.008
[19] Kim, H. H., Functoriality for the exterior square of \(GL_4\) and the symmetric fourth of \(GL_2\), J. Am. Math. Soc. 16 (2003), 139-183 · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1
[20] Kim, H. H.; Shahidi, F., Cuspidality of symmetric power with applications, Duke Math. J. 112 (2002), 177-197 · Zbl 1074.11027 · doi:10.1215/S0012-9074-02-11215-0
[21] Kim, H. H.; Shahidi, F., Functorial products for \(GL_2\times GL_3\) and the symmetric cube for \(GL_2\), Ann. Math. (2) 155 (2002), 837-893 · Zbl 1040.11036 · doi:10.2307/3062134
[22] Lao, H.; Luo, S., Sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms, Rocky Mt. J. Math. 51 (2021), 1701-1714 · Zbl 1486.11056 · doi:10.1216/rmj.2021.51.1701
[23] Lau, Y.-K.; Lü, G., Sums of Fourier coefficients of cusp forms, Q. J. Math. 62 (2011), 687-716 · Zbl 1269.11044 · doi:10.1093/qmath/haq012
[24] Lau, Y.-K.; Lü, G.; Wu, J., Integral power sums of Hecke eigenvalues, Acta Arith. 150 (2011), 193-207 · Zbl 1300.11042 · doi:10.4064/aa150-2-7
[25] Lü, G., Average behavior of Fourier coefficients of cusp forms, Proc. Am. Math. Soc. 137 (2009), 1961-1969 · Zbl 1241.11054 · doi:10.1090/S0002-9939-08-09741-4
[26] Lü, G., The sixth and eighth moments of Fourier coefficients of cusp forms, J. Number Theory 129 (2009), 2790-2800 · Zbl 1195.11060 · doi:10.1016/j.jnt.2009.01.019
[27] Lü, G., Uniform estimates for sums of Fourier coefficients of cusp forms, Acta Math. Hung. 124 (2009), 83-97 · Zbl 1200.11031 · doi:10.1007/s10474-009-8153-7
[28] Lü, G., On higher moments of Fourier coefficients of holomorphic cusp forms, Can. J. Math. 63 (2011), 634-647 · Zbl 1250.11046 · doi:10.4153/CJM-2011-010-5
[29] Luo, S.; Lao, H.; Zou, A., Asymptotics for the Dirichlet coefficients of symmetric power \(L\)-functions, Acta Arith. 199 (2021), 253-268 · Zbl 1477.11079 · doi:10.4064/aa191112-24-12
[30] Moreno, C. J.; Shahidi, F., The fourth moment of Ramanujan \(\tau \)-function, Math. Ann. 266 (1983), 233-239 · Zbl 0508.10014 · doi:10.1007/BF01458445
[31] Newton, J.; Thorne, J. A., Symmetric power functoriality for holomorphic modular forms, Publ. Math., Inst. Hautes Étud. Sci. 134 (2021), 1-116 · Zbl 1503.11085 · doi:10.1007/s10240-021-00127-3
[32] Newton, J.; Thorne, J. A., Symmetric power functoriality for holomorphic modular forms. II, Publ. Math., Inst. Hautes Étud. Sci. 134 (2021), 117-152 · Zbl 1503.11086 · doi:10.1007/s10240-021-00126-4
[33] Rankin, R. A., Contributions to the theory of Ramanujan’s function \(\tau(n)\) and similar arithmetical functions. II. The order of the Fourier coefficients of the integral modular forms, Proc. Camb. Philos. Soc. 35 (1939), 357-372 · Zbl 0021.39202 · doi:10.1017/S0305004100021101
[34] Rankin, R. A., Sums of cusp form coefficients, Automorphic Forms and Analytic Number Theory University Montréal, Montréal (1990), 115-121 · Zbl 0735.11023
[35] Rudnick, Z.; Sarnak, P., Zeros of principal \(L\)-functions and random matrix theory, Duke Math. J. 81 (1996), 269-322 · Zbl 0866.11050 · doi:10.1215/S0012-7094-96-08115-6
[36] Sankaranarayanan, A., On a sum involving Fourier coefficients of cusp forms, Lith. Math. J. 46 (2006), 459-474 · Zbl 1162.11337 · doi:10.1007/s10986-006-0042-y
[37] Sankaranarayanan, A.; Singh, S. K.; Srinivas, K., Discrete mean square estimates for coefficients of symmetric power \(L\)-functions, Acta Arith. 190 (2019), 193-208 · Zbl 1465.11109 · doi:10.4064/aa180819-6-10
[38] Selberg, A., Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47-50 German
[39] Shahidi, F., On certain \(L\)-functions, Am. J. Math. 103 (1981), 297-355 · Zbl 0467.12013 · doi:10.2307/2374219
[40] Shahidi, F., Fourier transforms of intertwining operators and Plancherel measure for \(GL(n)\), Am. J. Math. 106 (1984), 67-111 · Zbl 0567.22008 · doi:10.2307/2374430
[41] Shahidi, F., Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973-1007 · Zbl 0674.10027 · doi:10.1215/S0012-7094-85-05252-4
[42] Shahidi, F., Third symmetric power \(L\)-functions for \(GL(2)\), Compos. Math. 70 (1989), 245-273 · Zbl 0684.10026
[43] Shahidi, F., A proof of Langland’s conjecture on Plancherel measures; Complementary series for \(p\)-adic groups, Ann. Math. (2) 132 (1990), 273-330 · Zbl 0780.22005 · doi:10.2307/1971524
[44] Tang, H., Estimates for the Fourier coefficients of symmetric square \(L\)-functions, Arch. Math. 100 (2013), 123-130 · Zbl 1287.11061 · doi:10.1007/s00013-013-0481-8
[45] Tang, H.; Wu, J., Fourier coefficients of symmetric power \(L\)-functions, J. Number Theory 167 (2016), 147-160 · Zbl 1417.11050 · doi:10.1016/j.jnt.2016.03.005
[46] Wu, J., Power sums of Hecke eigenvalues and application, Acta Arith. 137 (2009), 333-344 · Zbl 1232.11054 · doi:10.4064/aa137-4-3
[47] Zhai, S., Average behavior of Fourier coefficients of cusp forms over sum of two squares, J. Number Theory 133 (2013), 3862-3876 · Zbl 1295.11041 · doi:10.1016/j.jnt.2013.05.013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.