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Applications of the Kuznetsov formula on GL(3) II: the level aspect

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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.

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Notes

  1. This corrects an error [3, (8.7)] where \(X_1^2 X_2\) should be replaced with \(X_1^2X_2^2\).

  2. Notice that in [3, Lemma 3] the indices should be exchanged and read as in (4.1) above as a consequence of Remark 1 after Theorem 6.

References

  1. 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)

  2. Balakci, D.: Automorphic forms for congruence subgroups of \({{\rm SL}}(3,{\mathbb{Z}})\), Ph.D. thesis, Göttingen (2015)

  3. Blomer, V.: Applications of the Kuznetsov formula on \({\rm GL}(3)\). Invent. math. 194, 673–729 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blomer, V., Buttcane, J., Raulf, N.: A Sato-Tate law for \({\rm GL}(3)\). Comm. Math. Helv. 89, 895–919 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blomer, V., Harcos, G., Michel, P.: Bounds for modular L-functions in the level aspect. Ann. Sci. Ecole Norm. Sup. 40, 697–740 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blomer, V., Khan, R., Young, M.: Mass distribution of holomorphic cusp forms. Duke Math. J. 162, 2609–2644 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruggeman, R.: Fourier coefficients of cusp forms. Invent. Math. 45, 1–18 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bump, D., Friedberg, S., Goldfeld, D.: Poincaré series and Kloosterman sums for \({\rm SL}(3, {\mathbb{Z}})\). Acta Arith. 50, 31–89 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Buttcane, J.: Sums of \({\rm SL}(3,{\mathbb{Z}})\) Kloosterman sums, Ph.D. thesis, UCLA (2012)

  10. 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)

    MathSciNet  MATH  Google Scholar 

  11. Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982/1983)

  12. Duke, W., Friedlander, F., Iwaniec, H.: The subconvexity problem for Artin \(L\)-functions. Invent. Math. 149, 489–577 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Friedberg, S.: Poincaré series for \({\rm GL}(n)\): Fourier expansion, Kloosterman sums, and algebreo-geometric estimates. Math. Z. 196, 165–188 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

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

  17. Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256, 199–214 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic representations. I. Am. J. Math. 103, 499–558 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, X.: Upper bounds on \(L\)-functions at the edge of the critical strip. Int. Math. Res. Notices 2010(4), 727–755 (2010)

  20. 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)

    Article  MATH  Google Scholar 

  21. Stevens, G.: Poincarè series on \({\rm GL}( r)\) and Kloostermann sums. Math. Ann. 277, 25–51 (1987)

    Article  MathSciNet  Google Scholar 

  22. Venkatesh, A.: Large sieve inequalities for \({\rm GL}(n)\)-forms in the conductor aspect. Adv. Math. 200, 336–356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for a careful reading of the manuscript.

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Correspondence to Valentin Blomer.

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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.

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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

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  • DOI: https://doi.org/10.1007/s00208-017-1558-7

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