Home Global decomposition of GL(3) Kloosterman sums and the spectral large sieve
Article
Licensed
Unlicensed Requires Authentication

Global decomposition of GL(3) Kloosterman sums and the spectral large sieve

  • Valentin Blomer EMAIL logo and Jack Buttcane
Published/Copyright: September 21, 2017

Abstract

We prove best-possible bounds for bilinear forms in Kloosterman sums for GL(3) associated with the long Weyl element. As an application we derive a best-possible spectral large sieve inequality on GL(3).

Award Identifier / Grant number: 1128155

Funding statement: The first author was supported by the Volkswagen Foundation and NSF grant 1128155 while enjoying the hospitality of the Institute for Advanced Study. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

Acknowledgements

We would like to thank the referee for many useful suggestions.

References

[1] V. Blomer, Applications of the Kuznetsov formula on GL(3), Invent. Math. 194 (2013), no. 3, 673–729. 10.1007/s00222-013-0454-3Search in Google Scholar PubMed PubMed Central

[2] V. Blomer and J. Buttcane, On the subconvexity problem for GL(3), preprint (2015), https://arxiv.org/abs/1504.02667. Search in Google Scholar

[3] V. Blomer and G. Harcos, A hybrid asymptotic formula for the second moment of Rankin–Selberg L-functions, Proc. Lond. Math. Soc. (3) 105 (2012), no. 3, 473–505. 10.1112/plms/pdr069Search in Google Scholar

[4] V. Blomer, R. Khan and M. Young, Distribution of mass of holomorphic cusp forms, Duke Math. J. 162 (2013), no. 14, 2609–2644. 10.1215/00127094-2380967Search in Google Scholar

[5] D. Bump, S. Friedberg and D. Goldfeld, Poincaré series and Kloosterman sums for SL(3,𝐙), Acta Arith. 50 (1988), no. 1, 31–89. 10.4064/aa-50-1-31-89Search in Google Scholar

[6] D. Bump and J. Huntley, Unramified Whittaker functions for GL(3,), J. Anal. Math. 65 (1995), 19–44. 10.1007/BF02788764Search in Google Scholar

[7] J. Buttcane, The spectral Kuznetsov formula on SL(3), Trans. Amer. Math. Soc. 368 (2016), no. 9, 6683–6714. 10.1090/tran/6833Search in Google Scholar

[8] R. Dabrowski and B. Fisher, A stationary phase formula for exponential sums over 𝐙/pm𝐙 and applications to GL(3)-Kloosterman sums, Acta Arith. 80 (1997), no. 1, 1–48. 10.4064/aa-80-1-1-48Search in Google Scholar

[9] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. 10.1007/BF01390728Search in Google Scholar

[10] P. X. Gallagher, A large sieve density estimate near σ=1, Invent. Math. 11 (1970), 329–339. 10.1007/BF01403187Search in Google Scholar

[11] D. Goldfeld, Automorphic forms and L-functions for the group GL(n,𝐑), Cambridge Stud. Adv. Math. 99, Cambridge University Press, Cambridge 2006. 10.1017/CBO9780511542923Search in Google Scholar

[12] H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence 2004. 10.1090/coll/053Search in Google Scholar

[13] H. Iwaniec and X. Li, The orthogonality of Hecke eigenvalues, Compos. Math. 143 (2007), no. 3, 541–565. 10.1112/S0010437X07002679Search in Google Scholar

[14] M. Jutila and Y. Motohashi, Uniform bound for Hecke L-functions, Acta Math. 195 (2005), 61–115. 10.1007/BF02588051Search in Google Scholar

[15] N. V. Kuznetsov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums, Math. USSR Sb. 39 (1981), 299–342. 10.1070/SM1981v039n03ABEH001518Search in Google Scholar

[16] X. Li, Upper bounds on L-functions at the edge of the critical strip, Int. Math. Res. Not. IMRN 2010 (2010), no. 4, 727–755. 10.1093/imrn/rnp148Search in Google Scholar

[17] G. Stevens, Poincaré series on GL(r) and Kloostermann sums, Math. Ann. 277 (1987), no. 1, 25–51. 10.1007/BF01457276Search in Google Scholar

[18] M. P. Young, Bilinear forms with GL3 Kloosterman sums and the spectral large sieve, Int. Math. Res. Not. IMRN 2016 (2016), no. 21, 6453–6492. 10.1093/imrn/rnv347Search in Google Scholar

Received: 2015-12-13
Revised: 2017-08-07
Published Online: 2017-09-21
Published in Print: 2019-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2017-0034/html
Scroll to top button