Abstract
We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelöf on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maaß forms violating the Ramanujan conjecture at finite places.
Similar content being viewed by others
References
Arthur, J.: Eisenstein series and the trace formula, In: Automorphic forms, representations and L-functions, Corvallis/Oregon 1977, Proc. Symp. Pure Math. 33, 253–274 (1979)
Balakci, D.: Automorphic forms for congruence subgroups of \({{\rm SL}}(3,{\mathbb{Z}})\), Ph.D. thesis, Göttingen (2015)
Blomer, V.: Applications of the Kuznetsov formula on \({\rm GL}(3)\). Invent. math. 194, 673–729 (2013)
Blomer, V., Buttcane, J., Raulf, N.: A Sato-Tate law for \({\rm GL}(3)\). Comm. Math. Helv. 89, 895–919 (2014)
Blomer, V., Harcos, G., Michel, P.: Bounds for modular L-functions in the level aspect. Ann. Sci. Ecole Norm. Sup. 40, 697–740 (2007)
Blomer, V., Khan, R., Young, M.: Mass distribution of holomorphic cusp forms. Duke Math. J. 162, 2609–2644 (2013)
Bruggeman, R.: Fourier coefficients of cusp forms. Invent. Math. 45, 1–18 (1978)
Bump, D., Friedberg, S., Goldfeld, D.: Poincaré series and Kloosterman sums for \({\rm SL}(3, {\mathbb{Z}})\). Acta Arith. 50, 31–89 (1988)
Buttcane, J.: Sums of \({\rm SL}(3,{\mathbb{Z}})\) Kloosterman sums, Ph.D. thesis, UCLA (2012)
Dabrowski, R., Fisher, B.: A stationary phase formula for exponential sums over \({\mathbb{Z}}/p^m{\mathbb{Z}}\) and applications to \(GL(3)\)-Kloosterman sums. Acta Arith. 80, 1–48 (1997)
Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982/1983)
Duke, W., Friedlander, F., Iwaniec, H.: The subconvexity problem for Artin \(L\)-functions. Invent. Math. 149, 489–577 (2002)
Duke, W., Kowalski, E.: Large sieve inequalities for \({\rm GL}(n)\)-forms in the conductor aspect (with appendix by D. Ramakrishnan). Invent. Math. 139, 1–39 (2000)
Friedberg, S.: Poincaré series for \({\rm GL}(n)\): Fourier expansion, Kloosterman sums, and algebreo-geometric estimates. Math. Z. 196, 165–188 (1987)
Goldfeld, D.: Automorphic forms and \(L\)-functions for the group \({\rm GL}(n, {\mathbb{R}})\), Cambridge studies in advanced mathematics 99. Cambridge University Press, Cambridge (2006)
Iwaniec, H., Kowalski, E.: Analytic number theory, vol. 53. American Mathematical Society, Providence, RI (2004)
Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256, 199–214 (1981)
Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic representations. I. Am. J. Math. 103, 499–558 (1981)
Li, X.: Upper bounds on \(L\)-functions at the edge of the critical strip. Int. Math. Res. Notices 2010(4), 727–755 (2010)
Kuznetsov, N.: The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums. Math. USSR-Sb 39, 299–342 (1981)
Stevens, G.: Poincarè series on \({\rm GL}( r)\) and Kloostermann sums. Math. Ann. 277, 25–51 (1987)
Venkatesh, A.: Large sieve inequalities for \({\rm GL}(n)\)-forms in the conductor aspect. Adv. Math. 200, 336–356 (2006)
Acknowledgements
The authors would like to thank the referee for a careful reading of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
V. Blomer was supported by the Volkswagen Foundation and a Starting Grant of the European Research Council. J.Buttcane was supported by a Starting Grant of the European Research Council. P. Maga was supported by a Starting Grant of the European Research Council and OTKA Grant No. NK104183.
Rights and permissions
About this article
Cite this article
Blomer, V., Buttcane, J. & Maga, P. Applications of the Kuznetsov formula on GL(3) II: the level aspect. Math. Ann. 369, 723–759 (2017). https://doi.org/10.1007/s00208-017-1558-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-017-1558-7