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The cubic moment of Hecke–Maass cusp forms and moments of L-functions

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Abstract

In this paper, we prove that the smooth cubic moments dissipate for the Hecke–Maass cusp forms, which gives a new case of the random wave conjecture. In fact, we can prove a polynomial decay for the smooth cubic moments, while for the smooth second moment (i.e. QUE) no rate of decay is known unconditionally for general Hecke–Maass cusp forms. The proof is based on various estimates of moments of central L-values. We prove the Lindelöf on average bound for the first moment of \({\text {GL}}(3)\times {\text {GL}}(2)\) L-functions in short intervals of the subconvexity strength length, and the convexity strength upper bound for the mixed moment of \({\text {GL}}(2)\) and the triple product L-functions. In particular, we prove new subconvexity bounds of certain \({\text {GL}}(3)\times {\text {GL}}(2)\) L-functions.

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References

  1. Berry, M.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083–2091 (1977)

    Article  MathSciNet  Google Scholar 

  2. Blomer, V.: Subconvexity for twisted \(L\)-functions on \(\text{ GL }(3)\). Am. J. Math. 134(5), 1385–1421 (2012)

    Article  MathSciNet  Google Scholar 

  3. Blomer, V., Khan, R., Young, M.: Distribution of mass of holomorphic cusp forms. Duke Math. J. 162(14), 2609–2644 (2013)

    Article  MathSciNet  Google Scholar 

  4. Buttcane, J., Khan, R.: On the fourth moment of Hecke-Maass forms and the random wave conjecture. Compos. Math. 153(7), 1479–1511 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chandee, V., Li, X.: The second moment of \(\text{ GL }(4)\times \text{ GL }(2)\)\(L\)-functions at special points. Adv. Math. 365, 107060 (2020)

    Article  MathSciNet  Google Scholar 

  6. Conrey, J.B., Iwaniec, H.: The cubic moment of central values of automorphic \(L\)-functions. Ann. Math. (2) 151(3), 1175–1216 (2000)

    Article  MathSciNet  Google Scholar 

  7. Djanković, G., Khan, R.: On the random wave conjecture for Eisenstein series. Int. Math. Res. Not. IMRN 23, 9694–9716 (2020)

    Article  MathSciNet  Google Scholar 

  8. Gelbart, S., Jacquet, H.: A relation between automorphic representations of \(\text{ GL }(2)\) and \(\text{ GL }(3)\). Ann. Sci. École Norm. Sup. (4) 11(4), 471–542 (1978)

    Article  MathSciNet  Google Scholar 

  9. Goldfeld, D.: Automorphic forms and \(L\)-functions for the group \(\text{ GL }(n,{\mathbb{R}})\). With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, vol. 99. Cambridge University Press, Cambridge (2006). (xiv+493 pp)

    Google Scholar 

  10. Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. Ann. Math. (2) 140(1), 161–181 (1994)

    Article  MathSciNet  Google Scholar 

  11. Huang, B.: Hybrid subconvexity bounds for twisted \(L\)-functions on \(GL(3)\). Sci. China Math. 64(3), 443–478 (2021)

    Article  MathSciNet  Google Scholar 

  12. Huang, B.: On the Rankin-Selberg problem. Math. Ann. 381(3–4), 1217–1251 (2021)

    Article  MathSciNet  Google Scholar 

  13. Huang, B.: Uniform bounds for \(\text{ GL }(3) \times GL(2)\)\(L\)-functions. ArXiv preprint (2021), arXiv:2104.13025

  14. Humphries, P.: Equidistribution in shrinking sets and \(L^4\)-norm bounds for automorphic forms. Math. Ann. 371(3–4), 1497–1543 (2018)

    Article  MathSciNet  Google Scholar 

  15. Humphries, P., Khan, R.: On the random wave conjecture for dihedral Maaß forms. Geom. Funct. Anal. 30(1), 34–125 (2020)

    Article  MathSciNet  Google Scholar 

  16. Humphries, P., Khan, R.: \(L^p\)-norm bounds for automorphic forms via spectral reciprocity. Preprint

  17. Ivić, A.: On sums of Hecke series in short intervals. J. Théor. Nombres Bordeaux 13(2), 453–468 (2001)

    Article  MathSciNet  Google Scholar 

  18. Iwaniec, H.: Small eigenvalues of Laplacian for \(\Gamma _0(N)\). Acta Arith. 56(1), 65–82 (1990)

    Article  MathSciNet  Google Scholar 

  19. Iwaniec, H.: The spectral growth of automorphic \(L\)-functions. J. Reine Angew. Math. 428, 139–159 (1992)

    MathSciNet  Google Scholar 

  20. Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence, RI (2004)

    Google Scholar 

  21. Jutila, M.: On spectral large sieve inequalities. Dedicated to Włodzimierz Staś on the occasion of his 75th birthday. Funct. Approx. Comment. Math. 28, 7–18 (2000)

    MathSciNet  Google Scholar 

  22. Khan, R., Young, M.: Moments and hybrid subconvexity for symmetric-square \(L\)-functions. To appear in J. Inst. Math. Jussieu (2021)

  23. Kim, H.: Functoriality for the exterior square of \(\text{ GL}_4\) and the symmetric fourth of \(\text{ GL}_2\). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Amer. Math. Soc. 16(1), 139–183 (2003)

    Article  MathSciNet  Google Scholar 

  24. Kıral, E., Petrow, I., Young, M.: Oscillatory integrals with uniformity in parameters. J. Théor. Nombres Bordeaux 31(1), 145–159 (2019)

    Article  MathSciNet  Google Scholar 

  25. Kwan, C.: Spectral Moment Formulae for \(\text{ GL }(3)\times GL(2)\)\(L\)-functions. ArXiv preprint, arXiv:2112.08568

  26. Lapid, E.M.: On the nonnegativity of Rankin-Selberg \(L\)-functions at the center of symmetry. Int. Math. Res. Not. 2, 65–75 (2003)

    Article  MathSciNet  Google Scholar 

  27. Li, X.: Upper bounds on \(L\)-functions at the edge of the critical strip. Int. Math. Res. Not. IMRN 4, 727–755 (2010)

    MathSciNet  Google Scholar 

  28. Li, X.: Bounds for \(\text{ GL }(3)\times GL(2)\)\(L\)-functions and \(\text{ GL }3)\)\(L\)-functions. Ann. Math. (2) 173(1), 301–336 (2011)

    Article  MathSciNet  Google Scholar 

  29. Lin, Y., Nunes, R., Qi, Z.: Strong subconvexity for self-dual \(\text{ GL }(3)\)\(L\)-functions. To appear in Int. Math. Res. Not. IMRN (2022)

  30. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. (2) 163(1), 165–219 (2006)

    Article  MathSciNet  Google Scholar 

  31. Liu, J., Ye, Y.: Subconvexity for Rankin-Selberg \(L\)-functions of Maass forms. Geom. Funct. Anal. 12(6), 1296–1323 (2002)

    Article  MathSciNet  Google Scholar 

  32. Luo, W.: Spectral mean-value of automorphic \(L\)-functions at special points. Analytic number theory, Vol. 2 (Allerton Park, IL, 1995), Progr. Math., vol. 139, pp. 621–632. Birkhäuser Boston, Boston, MA (1996)

  33. Luo, W.: \(L^4\)-norms of the dihedral Maass forms. Int. Math. Res. Not. IMRN 2014, 2294–2304 (2014)

    Article  Google Scholar 

  34. Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on \(\text{ PSL}_2({ Z })\backslash { H}^2\). Inst. Hautes Études Sci. Publ. Math. (81), 207–237 (1995)

  35. McKee, M., Sun, H., Ye, Y.: Improved subconvexity bounds for \(\text{ GL }2)\times GL(3)\) and \(\text{ GL }3)\)\(L\)-functions by weighted stationary phase. Trans. Am. Math. Soc. 370(5), 3745–3769 (2018)

    Article  Google Scholar 

  36. Miller, S.D., Zhou, F.: The balanced Voronoi formulas for \(\text{ GL }n)\). Int. Math. Res. Not. IMRN 11, 3473–3484 (2019)

    Article  Google Scholar 

  37. Nelson, P.: Bounds for standard \(L\)-functions. ArXiv preprint, arXiv:2109.15230

  38. Ramakrishnan, D.: Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\text{ SL }2)\). Ann. Math. (2) 152(1), 45–111 (2000)

    Article  MathSciNet  Google Scholar 

  39. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys. 161(1), 195–213 (1994)

    Article  MathSciNet  Google Scholar 

  40. Sarnak, P.: Estimates for Rankin-Selberg \(L\)-functions and quantum unique ergodicity. J. Funct. Anal. 184(2), 419–453 (2001)

    Article  MathSciNet  Google Scholar 

  41. Sarnak, P.: Spectra of hyperbolic surfaces. Bull. Am. Math. Soc. (N.S.) 40(4), 441–478 (2003)

    Article  MathSciNet  Google Scholar 

  42. Sarnak, P.: Letter to Morawetz (2004). https://publications.ias.edu/node/480

  43. Soundararajan, K.: Quantum unique ergodicity for \(\text{ SL }2({\mathbb{Z} }) \backslash {\mathbb{H} }\). Ann. Math. (2) 172(2), 1529–1538 (2010)

    Article  MathSciNet  Google Scholar 

  44. Spinu, F.: The \(L^4\)-norm of Eisenstein series, Princeton PhD thesis (2003)

  45. Watson, T.: Rankin triple products and quantum chaos. arXiv:0810.0425

  46. Young, M.: Weyl-type hybrid subconvexity bounds for twisted \(L\)-functions and Heegner points on shrinking sets. J. Eur. Math. Soc. (JEMS) 19(5), 1545–1576 (2017)

    Article  MathSciNet  Google Scholar 

  47. Zagier, D.: The Rankin–Selberg method for automorphic functions which are not of rapid decay. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 415–437 (1981) (1982)

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Acknowledgements

The author would like to thank Prof. Jianya Liu and Zeév Rudnick for their valuable discussions and constant encouragement. He also wants to thank Peter Humphries and Yongxiao Lin for their comments. He gratefully thanks to the referees for the constructive comments and recommendations which definitely improve the readability and quality of the paper.

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  1. Data Science Institute and School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Bingrong Huang

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This work was supported by the National Key R &D Program of China (No. 2021YFA1000700) and NSFC (Nos. 12001314 and 12031008).

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Huang, B. The cubic moment of Hecke–Maass cusp forms and moments of L-functions. Math. Ann. 389, 899–945 (2024). https://doi.org/10.1007/s00208-023-02668-w

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