Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ESTIMATES FOR CERTAIN SHIFTED CONVOLUTION SUMS INVOLVING HECKE EIGENVALUES

  • Guodong Hua (School of Mathematics and Statistics Weinan Normal University, School of Mathematics Shandong University )
  • Received : 2021.11.04
  • Accepted : 2022.11.15
  • Published : 2023.04.30
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Abstract

In this paper, we obtain certain estimates for averages of shifted convolution sums involving Hecke eigenvalues of classical holomorphic cusp forms. This generalizes some results of Lü and Wang in this direction.

Keywords

Acknowledgement

The author would like to express his sincere gratitude to Professor Guangshi Lü and Professor Bin Chen for their constant encouragement and valuable suggestions. The author is extremely grateful to the anonymous referees for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and readable.

References

  1. L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality, I, Compos. Math. 150 (2014), no. 5, 729-748. https://doi.org/10.1112/S0010437X13007653
  2. L. Clozel and J. A. Thorne, Level raising and symmetric power functoriality, II, Ann. of Math. (2) 181 (2015), no. 1, 303-359. https://doi.org/10.4007/annals.2015.181.1.5
  3. L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality, III, Duke Math. J. 166 (2017), no. 2, 325-402. https://doi.org/10.1215/00127094-3714971
  4. P. Deligne, La conjecture de Weil. I, Inst. Hautes Etudes Sci. Publ. Math. No. 43 (1974), 273-307.
  5. L. Dieulefait, Automorphy of Symm5(GL(2)) and base change, J. Math. Pures Appl. (9) 104 (2015), no. 4, 619-656. https://doi.org/10.1016/j.matpur.2015.03.010
  6. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), no. 4, 471-542. https://doi.org/10.24033/asens.1355
  7. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499-558. https://doi.org/10.2307/2374103
  8. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777-815. https://doi.org/10.2307/2374050
  9. A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function, translated from the Russian by Neal Koblitz, De Gruyter Expositions in Mathematics, 5, Walter de Gruyter & Co., Berlin, 1992. https://doi.org/10.1515/9783110886146
  10. H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. https://doi.org/10.1090/S0894-0347-02-00410-1
  11. H. H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177-197. https://doi.org/10.1215/S0012-9074-02-11215-0
  12. H. H. Kim and F. Shahidi, Functorial products for GL2×GL3 and the symmetric cube for GL2, Ann. of Math. (2) 155 (2002), no. 3, 837-893. https://doi.org/10.2307/3062134
  13. Y.-K. Lau and J. Wu, A density theorem on automorphic L-functions and some applications, Trans. Amer. Math. Soc. 358 (2006), no. 1, 441-472. https://doi.org/10.1090/S0002-9947-05-03774-8
  14. G. Lu, Exponential sums with Fourier coefficients of automorphic forms, Math. Z. 289 (2018), no. 1-2, 267-278. https://doi.org/10.1007/s00209-017-1950-8
  15. G. Lu and D. Wang, Averages of shifted convolutions of general divisor sums involving Hecke eigenvalues, J. Number Theory 199 (2019), 342-351. https://doi.org/10.1016/j.jnt.2018.11.016
  16. R. Munshi, Shifted convolution sums for GL(3) × GL(2), Duke Math. J. 162 (2013), no. 13, 2345-2362. https://doi.org/10.1215/00127094-2371416
  17. J. Newton and J. A. Thorne, Symmetric power functoriality for holomorphic modular forms, Publ. Math. Inst. Hautes Etudes Sci. 134 (2021), 1-116. https://doi.org/10.1007/s10240-021-00127-3
  18. J. Newton and J. A. Thorne, Symmetric power functoriality for holomorphic modular forms, II, Publ. Math. Inst. Hautes Etudes Sci. 134 (2021), 117-152. https://doi.org/10.1007/s10240-021-00126-4
  19. A. Perelli, General L-functions, Ann. Mat. Pura Appl. (4) 130 (1982), 287-306. https://doi.org/10.1007/BF01761499
  20. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269-322. https://doi.org/10.1215/S0012-7094-96-08115-6
  21. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), no. 2, 297-355. https://doi.org/10.2307/2374219
  22. F. Shahidi, Third symmetric power L-functions for GL(2), Compositio Math. 70 (1989), no. 3, 245-273.
  23. Q. Sun, Averages of shifted convolution sums for GL(3) × GL(2), J. Number Theory 182 (2018), 344-362. https://doi.org/10.1016/j.jnt.2017.07.005