Abstract
We prove best-possible bounds for bilinear forms in Kloosterman sums for
Funding source: National Science Foundation
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
[2]
V. Blomer and J. Buttcane,
On the subconvexity problem for
[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
[6]
D. Bump and J. Huntley,
Unramified Whittaker functions for
[7]
J. Buttcane,
The spectral Kuznetsov formula on
[8]
R. Dabrowski and B. Fisher,
A stationary phase formula for exponential sums over
[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
[11]
D. Goldfeld,
Automorphic forms and L-functions for the group
[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
[18]
M. P. Young,
Bilinear forms with
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The structure of spaces with Bakry–Émery Ricci curvature bounded below
- Global decomposition of GL(3) Kloosterman sums and the spectral large sieve
- Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs
- The evolution of complete non-compact graphs by powers of Gauss curvature
- Littlewood–Richardson coefficients for Grothendieck polynomials from integrability
- Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups
- Systolic geometry and simplicial complexity for groups
- Identifiability of homogeneous polynomials and Cremona transformations
- Systems of cubic forms in many variables
Articles in the same Issue
- Frontmatter
- The structure of spaces with Bakry–Émery Ricci curvature bounded below
- Global decomposition of GL(3) Kloosterman sums and the spectral large sieve
- Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs
- The evolution of complete non-compact graphs by powers of Gauss curvature
- Littlewood–Richardson coefficients for Grothendieck polynomials from integrability
- Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups
- Systolic geometry and simplicial complexity for groups
- Identifiability of homogeneous polynomials and Cremona transformations
- Systems of cubic forms in many variables