Article
| Title: | On the higher power moments of cusp form coefficients over sums of two squares (English) |
| Author: | Hua, Guodong |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1089-1104 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma ={\rm SL} (2,\mathbb {Z})$. 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. (English) |
| Keyword: | Fourier coefficient |
| Keyword: | automorphic $L$-function, Langlands program |
| MSC: | 11F11 |
| MSC: | 11F30 |
| MSC: | 11F66 |
| idZBL: | Zbl 07655785 |
| idMR: | MR4517598 |
| DOI: | 10.21136/CMJ.2022.0358-21 |
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| Date available: | 2022-11-28T11:38:50Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151132 |
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| Reference: | [1] Clozel, L., Thorne, J. A.: Level-raising and symmetric power functoriality. I.Compos. Math. 150 (2014), 729-748. Zbl 1304.11040, MR 3209793, 10.1112/S0010437X13007653 |
| Reference: | [2] Clozel, L., Thorne, J. A.: Level raising and symmetric power functoriality. II.Ann. Math. (2) 181 (2015), 303-359. Zbl 1339.11060, MR 3272927, 10.4007/annals.2015.181.1.5 |
| Reference: | [3] Clozel, L., Thorne, J. A.: Level-raising and symmetric power functoriality. III.Duke Math. J. 166 (2017), 325-402. Zbl 1372.11054, MR 3600753, 10.1215/00127094-3714971 |
| Reference: | [4] Deligne, P.: La conjecture de Weil. I.Publ. Math., Inst. Hautes Étud. Sci. 43 (1974), 273-307 French. Zbl 0287.14001, MR 0340258, 10.1007/BF02684373 |
| Reference: | [5] Fomenko, O. M.: Fourier coefficients of parabolic forms, and automorphic $L$-functions.J. Math. Sci., New York 95 (1999), 2295-2316. MR 1691291, 10.1007/BF02172473 |
| Reference: | [6] Fomenko, O. M.: Identities involving the coefficients of automorphic $L$-functions.J. Math. Sci., New York 133 (2006), 1749-1755. Zbl 1094.11018, MR 2119744, 10.1007/s10958-006-0086-x |
| Reference: | [7] Fomenko, O. M.: Mean value theorems for automorphic $L$-functions.St. Petersbg. Math. J. 19 (2008), 853-866. Zbl 1206.11061, MR 2381948, 10.1090/S1061-0022-08-01024-8 |
| Reference: | [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, MR 0533066, 10.24033/asens.1355 |
| Reference: | [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, MR 1039952, 10.5169/seals-57381 |
| Reference: | [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, MR 3973888, 10.1090/proc/14516 |
| Reference: | [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. MR 3069476, 10.1007/BF02952521 |
| Reference: | [12] Huang, B.: On the Rankin-Selberg problem.Math. Ann. 381 (2021), 1217-1251. Zbl 07498337, MR 4333413, 10.1007/s00208-021-02186-7 |
| Reference: | [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, MR 1220467 |
| Reference: | [14] Iwaniec, H., Kowalski, E.: Analytic Number Theory.Colloquium Publications. American Mathematical Society 53. AMS, Providence (2004). Zbl 1059.11001, MR 2061214, 10.1090/coll/053 |
| Reference: | [15] Jacquet, H., Piatetski-Shapiro, I. I., Shalika, J. A.: Rankin-Selberg convolutions.Am. J. Math. 105 (1983), 367-464. Zbl 0525.22018, MR 0701565, 10.2307/2374264 |
| Reference: | [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, MR 0618323, 10.2307/2374103 |
| Reference: | [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, MR 0623137, 10.2307/2374050 |
| Reference: | [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, MR 3283177, 10.1016/j.jnt.2014.09.008 |
| Reference: | [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, MR 1937203, 10.1090/S0894-0347-02-00410-1 |
| Reference: | [20] Kim, H. H., Shahidi, F.: Cuspidality of symmetric power with applications.Duke Math. J. 112 (2002), 177-197. Zbl 1074.11027, MR 1890650, 10.1215/S0012-9074-02-11215-0 |
| Reference: | [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, MR 1923967, 10.2307/3062134 |
| Reference: | [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, MR 4382993, 10.1216/rmj.2021.51.1701 |
| Reference: | [23] Lau, Y.-K., Lü, G.: Sums of Fourier coefficients of cusp forms.Q. J. Math. 62 (2011), 687-716. Zbl 1269.11044, MR 2825478, 10.1093/qmath/haq012 |
| Reference: | [24] Lau, Y.-K., Lü, G., Wu, J.: Integral power sums of Hecke eigenvalues.Acta Arith. 150 (2011), 193-207. Zbl 1300.11042, MR 2836386, 10.4064/aa150-2-7 |
| Reference: | [25] Lü, G.: Average behavior of Fourier coefficients of cusp forms.Proc. Am. Math. Soc. 137 (2009), 1961-1969. Zbl 1241.11054, MR 2480277, 10.1090/S0002-9939-08-09741-4 |
| Reference: | [26] Lü, G.: The sixth and eighth moments of Fourier coefficients of cusp forms.J. Number Theory 129 (2009), 2790-2800. Zbl 1195.11060, MR 2549533, 10.1016/j.jnt.2009.01.019 |
| Reference: | [27] Lü, G.: Uniform estimates for sums of Fourier coefficients of cusp forms.Acta Math. Hung. 124 (2009), 83-97. Zbl 1200.11031, MR 2520619, 10.1007/s10474-009-8153-7 |
| Reference: | [28] Lü, G.: On higher moments of Fourier coefficients of holomorphic cusp forms.Can. J. Math. 63 (2011), 634-647. Zbl 1250.11046, MR 2828536, 10.4153/CJM-2011-010-5 |
| Reference: | [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, MR 4296723, 10.4064/aa191112-24-12 |
| Reference: | [30] Moreno, C. J., Shahidi, F.: The fourth moment of Ramanujan $\tau$-function.Math. Ann. 266 (1983), 233-239. Zbl 0508.10014, MR 0724740, 10.1007/BF01458445 |
| Reference: | [31] Newton, J., Thorne, J. A.: Symmetric power functoriality for holomorphic modular forms.Publ. Math., Inst. Hautes Étud. Sci. 134 (2021), 1-116. Zbl 07458825, MR 4349240, 10.1007/s10240-021-00127-3 |
| Reference: | [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 07458826, MR 4349241, 10.1007/s10240-021-00126-4 |
| Reference: | [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, MR 0000411, 10.1017/S0305004100021101 |
| Reference: | [34] Rankin, R. A.: Sums of cusp form coefficients.Automorphic Forms and Analytic Number Theory University Montréal, Montréal (1990), 115-121. Zbl 0735.11023, MR 1111014 |
| Reference: | [35] Rudnick, Z., Sarnak, P.: Zeros of principal $L$-functions and random matrix theory.Duke Math. J. 81 (1996), 269-322. Zbl 0866.11050, MR 1395406, 10.1215/S0012-7094-96-08115-6 |
| Reference: | [36] Sankaranarayanan, A.: On a sum involving Fourier coefficients of cusp forms.Lith. Math. J. 46 (2006), 459-474. Zbl 1162.11337, MR 2320364, 10.1007/s10986-006-0042-y |
| Reference: | [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, MR 3984265, 10.4064/aa180819-6-10 |
| Reference: | [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. Zbl 0023.22201, MR 0002626 |
| Reference: | [39] Shahidi, F.: On certain $L$-functions.Am. J. Math. 103 (1981), 297-355. Zbl 0467.12013, MR 0610479, 10.2307/2374219 |
| Reference: | [40] Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measure for $GL(n)$.Am. J. Math. 106 (1984), 67-111. Zbl 0567.22008, MR 0729755, 10.2307/2374430 |
| Reference: | [41] Shahidi, F.: Local coefficients as Artin factors for real groups.Duke Math. J. 52 (1985), 973-1007. Zbl 0674.10027, MR 0816396, 10.1215/S0012-7094-85-05252-4 |
| Reference: | [42] Shahidi, F.: Third symmetric power $L$-functions for $GL(2)$.Compos. Math. 70 (1989), 245-273. Zbl 0684.10026, MR 1002045 |
| Reference: | [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, MR 1070599, 10.2307/1971524 |
| Reference: | [44] Tang, H.: Estimates for the Fourier coefficients of symmetric square $L$-functions.Arch. Math. 100 (2013), 123-130. Zbl 1287.11061, MR 3020126, 10.1007/s00013-013-0481-8 |
| Reference: | [45] Tang, H., Wu, J.: Fourier coefficients of symmetric power $L$-functions.J. Number Theory 167 (2016), 147-160. Zbl 1417.11050, MR 3504040, 10.1016/j.jnt.2016.03.005 |
| Reference: | [46] Wu, J.: Power sums of Hecke eigenvalues and application.Acta Arith. 137 (2009), 333-344. Zbl 1232.11054, MR 2506587, 10.4064/aa137-4-3 |
| Reference: | [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, MR 3084303, 10.1016/j.jnt.2013.05.013 |
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