On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 1089-1104
  • ISSN: 0011-4642

Abstract

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Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL ( 2 , ) . Denote by λ f ( n ) the n th normalized Fourier coefficient of f . We are interested in the average behaviour of the sum a 2 + b 2 x λ f j ( a 2 + b 2 ) for x 1 , where a , b and j 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.

How to cite

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Hua, Guodong. "On the higher power moments of cusp form coefficients over sums of two squares." Czechoslovak Mathematical Journal 72.4 (2022): 1089-1104. <http://eudml.org/doc/298935>.

@article{Hua2022,
abstract = {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 \[ \sum \_\{a^\{2\} + b^\{2\}\le x\} \lambda \_\{f\}^\{j\}(a^\{2\}+b^\{2\}) \] for $x\ge 1$, where $a,b\in \mathbb \{Z\}$ and $j\ge 9$ 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.},
author = {Hua, Guodong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fourier coefficient; automorphic $L$-function; Langlands program},
language = {eng},
number = {4},
pages = {1089-1104},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the higher power moments of cusp form coefficients over sums of two squares},
url = {http://eudml.org/doc/298935},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hua, Guodong
TI - On the higher power moments of cusp form coefficients over sums of two squares
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1089
EP - 1104
AB - 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 \[ \sum _{a^{2} + b^{2}\le x} \lambda _{f}^{j}(a^{2}+b^{2}) \] for $x\ge 1$, where $a,b\in \mathbb {Z}$ and $j\ge 9$ 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.
LA - eng
KW - Fourier coefficient; automorphic $L$-function; Langlands program
UR - http://eudml.org/doc/298935
ER -

References

top
  1. Clozel, L., Thorne, J. A., 10.1112/S0010437X13007653, Compos. Math. 150 (2014), 729-748. (2014) Zbl1304.11040MR3209793DOI10.1112/S0010437X13007653
  2. Clozel, L., Thorne, J. A., 10.4007/annals.2015.181.1.5, Ann. Math. (2) 181 (2015), 303-359. (2015) Zbl1339.11060MR3272927DOI10.4007/annals.2015.181.1.5
  3. Clozel, L., Thorne, J. A., 10.1215/00127094-3714971, Duke Math. J. 166 (2017), 325-402. (2017) Zbl1372.11054MR3600753DOI10.1215/00127094-3714971
  4. Deligne, P., 10.1007/BF02684373, Publ. Math., Inst. Hautes Étud. Sci. 43 (1974), 273-307 French. (1974) Zbl0287.14001MR0340258DOI10.1007/BF02684373
  5. Fomenko, O. M., 10.1007/BF02172473, J. Math. Sci., New York 95 (1999), 2295-2316. (1999) MR1691291DOI10.1007/BF02172473
  6. Fomenko, O. M., 10.1007/s10958-006-0086-x, J. Math. Sci., New York 133 (2006), 1749-1755. (2006) Zbl1094.11018MR2119744DOI10.1007/s10958-006-0086-x
  7. Fomenko, O. M., 10.1090/S1061-0022-08-01024-8, St. Petersbg. Math. J. 19 (2008), 853-866. (2008) Zbl1206.11061MR2381948DOI10.1090/S1061-0022-08-01024-8
  8. Gelbart, S., Jacquet, H., 10.24033/asens.1355, Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 471-542. (1978) Zbl0406.10022MR0533066DOI10.24033/asens.1355
  9. Hafner, J. L., Ivić, A., 10.5169/seals-57381, Enseign. Math., II. Sér. 35 (1989), 375-382. (1989) Zbl0696.10020MR1039952DOI10.5169/seals-57381
  10. He, X., 10.1090/proc/14516, Proc. Am. Math. Soc. 147 (2019), 2847-2856. (2019) Zbl1431.11062MR3973888DOI10.1090/proc/14516
  11. Hecke, E., 10.1007/BF02952521, Abh. Math. Semin. Univ. Hamb. 5 (1927), 199-224 German 9999JFM99999 53.0345.02. (1927) MR3069476DOI10.1007/BF02952521
  12. Huang, B., 10.1007/s00208-021-02186-7, Math. Ann. 381 (2021), 1217-1251. (2021) Zbl07498337MR4333413DOI10.1007/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. (1992) Zbl0787.11035MR1220467
  14. Iwaniec, H., Kowalski, E., 10.1090/coll/053, Colloquium Publications. American Mathematical Society 53. AMS, Providence (2004). (2004) Zbl1059.11001MR2061214DOI10.1090/coll/053
  15. Jacquet, H., Piatetski-Shapiro, I. I., Shalika, J. A., 10.2307/2374264, Am. J. Math. 105 (1983), 367-464. (1983) Zbl0525.22018MR0701565DOI10.2307/2374264
  16. Jacquet, H., Shalika, J. A., 10.2307/2374103, Am. J. Math. 103 (1981), 499-558. (1981) Zbl0473.12008MR0618323DOI10.2307/2374103
  17. Jacquet, H., Shalika, J. A., 10.2307/2374050, Am. J. Math. 103 (1981), 777-815. (1981) Zbl0491.10020MR0623137DOI10.2307/2374050
  18. Jiang, Y., Lü, G., 10.1016/j.jnt.2014.09.008, J. Number Theory 148 (2015), 220-234. (2015) Zbl1380.11037MR3283177DOI10.1016/j.jnt.2014.09.008
  19. Kim, H. H., 10.1090/S0894-0347-02-00410-1, J. Am. Math. Soc. 16 (2003), 139-183. (2003) Zbl1018.11024MR1937203DOI10.1090/S0894-0347-02-00410-1
  20. Kim, H. H., Shahidi, F., 10.1215/S0012-9074-02-11215-0, Duke Math. J. 112 (2002), 177-197. (2002) Zbl1074.11027MR1890650DOI10.1215/S0012-9074-02-11215-0
  21. Kim, H. H., Shahidi, F., 10.2307/3062134, Ann. Math. (2) 155 (2002), 837-893. (2002) Zbl1040.11036MR1923967DOI10.2307/3062134
  22. Lao, H., Luo, S., 10.1216/rmj.2021.51.1701, Rocky Mt. J. Math. 51 (2021), 1701-1714. (2021) Zbl1486.11056MR4382993DOI10.1216/rmj.2021.51.1701
  23. Lau, Y.-K., Lü, G., 10.1093/qmath/haq012, Q. J. Math. 62 (2011), 687-716. (2011) Zbl1269.11044MR2825478DOI10.1093/qmath/haq012
  24. Lau, Y.-K., Lü, G., Wu, J., 10.4064/aa150-2-7, Acta Arith. 150 (2011), 193-207. (2011) Zbl1300.11042MR2836386DOI10.4064/aa150-2-7
  25. Lü, G., 10.1090/S0002-9939-08-09741-4, Proc. Am. Math. Soc. 137 (2009), 1961-1969. (2009) Zbl1241.11054MR2480277DOI10.1090/S0002-9939-08-09741-4
  26. Lü, G., 10.1016/j.jnt.2009.01.019, J. Number Theory 129 (2009), 2790-2800. (2009) Zbl1195.11060MR2549533DOI10.1016/j.jnt.2009.01.019
  27. Lü, G., 10.1007/s10474-009-8153-7, Acta Math. Hung. 124 (2009), 83-97. (2009) Zbl1200.11031MR2520619DOI10.1007/s10474-009-8153-7
  28. Lü, G., 10.4153/CJM-2011-010-5, Can. J. Math. 63 (2011), 634-647. (2011) Zbl1250.11046MR2828536DOI10.4153/CJM-2011-010-5
  29. Luo, S., Lao, H., Zou, A., 10.4064/aa191112-24-12, Acta Arith. 199 (2021), 253-268. (2021) Zbl1477.11079MR4296723DOI10.4064/aa191112-24-12
  30. Moreno, C. J., Shahidi, F., 10.1007/BF01458445, Math. Ann. 266 (1983), 233-239. (1983) Zbl0508.10014MR0724740DOI10.1007/BF01458445
  31. Newton, J., Thorne, J. A., 10.1007/s10240-021-00127-3, Publ. Math., Inst. Hautes Étud. Sci. 134 (2021), 1-116. (2021) Zbl07458825MR4349240DOI10.1007/s10240-021-00127-3
  32. Newton, J., Thorne, J. A., 10.1007/s10240-021-00126-4, Publ. Math., Inst. Hautes Étud. Sci. 134 (2021), 117-152. (2021) Zbl07458826MR4349241DOI10.1007/s10240-021-00126-4
  33. Rankin, R. A., 10.1017/S0305004100021101, Proc. Camb. Philos. Soc. 35 (1939), 357-372. (1939) Zbl0021.39202MR0000411DOI10.1017/S0305004100021101
  34. Rankin, R. A., Sums of cusp form coefficients, Automorphic Forms and Analytic Number Theory University Montréal, Montréal (1990), 115-121. (1990) Zbl0735.11023MR1111014
  35. Rudnick, Z., Sarnak, P., 10.1215/S0012-7094-96-08115-6, Duke Math. J. 81 (1996), 269-322. (1996) Zbl0866.11050MR1395406DOI10.1215/S0012-7094-96-08115-6
  36. Sankaranarayanan, A., 10.1007/s10986-006-0042-y, Lith. Math. J. 46 (2006), 459-474. (2006) Zbl1162.11337MR2320364DOI10.1007/s10986-006-0042-y
  37. Sankaranarayanan, A., Singh, S. K., Srinivas, K., 10.4064/aa180819-6-10, Acta Arith. 190 (2019), 193-208. (2019) Zbl1465.11109MR3984265DOI10.4064/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. (1940) Zbl0023.22201MR0002626
  39. Shahidi, F., 10.2307/2374219, Am. J. Math. 103 (1981), 297-355. (1981) Zbl0467.12013MR0610479DOI10.2307/2374219
  40. Shahidi, F., 10.2307/2374430, Am. J. Math. 106 (1984), 67-111. (1984) Zbl0567.22008MR0729755DOI10.2307/2374430
  41. Shahidi, F., 10.1215/S0012-7094-85-05252-4, Duke Math. J. 52 (1985), 973-1007. (1985) Zbl0674.10027MR0816396DOI10.1215/S0012-7094-85-05252-4
  42. Shahidi, F., Third symmetric power L -functions for G L ( 2 ) , Compos. Math. 70 (1989), 245-273. (1989) Zbl0684.10026MR1002045
  43. Shahidi, F., 10.2307/1971524, Ann. Math. (2) 132 (1990), 273-330. (1990) Zbl0780.22005MR1070599DOI10.2307/1971524
  44. Tang, H., 10.1007/s00013-013-0481-8, Arch. Math. 100 (2013), 123-130. (2013) Zbl1287.11061MR3020126DOI10.1007/s00013-013-0481-8
  45. Tang, H., Wu, J., 10.1016/j.jnt.2016.03.005, J. Number Theory 167 (2016), 147-160. (2016) Zbl1417.11050MR3504040DOI10.1016/j.jnt.2016.03.005
  46. Wu, J., 10.4064/aa137-4-3, Acta Arith. 137 (2009), 333-344. (2009) Zbl1232.11054MR2506587DOI10.4064/aa137-4-3
  47. Zhai, S., 10.1016/j.jnt.2013.05.013, J. Number Theory 133 (2013), 3862-3876. (2013) Zbl1295.11041MR3084303DOI10.1016/j.jnt.2013.05.013

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