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ON A TWISTED VERSION OF LINNIK AND SELBERG’S CONJECTURE ON SUMS OF KLOOSTERMAN SUMS

Part of: Discontinuous groups and automorphic forms Exponential sums and character sums

Published online by Cambridge University Press:  29 January 2019

Raphael S. Steiner
Raphael S. Steiner*
Affiliation:
FH-317, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, U.S.A. email raphael.steiner.academic@gmail.com
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Abstract

We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations
Secondary: 11L07: Estimates on exponential sums 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 437 - 474
Copyright
Copyright © University College London 2019 

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References

Blomer, V. and Milićević, D., Kloosterman sums in residue classes. J. Eur. Math. Soc. (JEMS) 17(1) 2015, 51–69.Google Scholar
Blomer, V. and Milićević, D., The second moment of twisted modular L-functions. Geom. Funct. Anal. 25(2) 2015, 453–516.Google Scholar
Browning, T. D., Vinay Kumaraswamy, V. and Steiner, R. S., Twisted Linnik implies optimal covering exponent for S 3 . Int. Math. Res. Not. IMRN 2017, doi:10.1093/imrn/rnx116.Google Scholar
Deligne, P., Formes modulaires et représentations l-adiques. In Séminaire Bourbaki, Vol. 1968/69, Exposés 347–363 (Lecture Notes in Mathematics 179 ), Springer (Berlin, 1971), 139–172.Google Scholar
Deligne, P., La conjecture de Weil. I. Publ. Math. Inst. Hautes Études Sci. 43 1974, 273–307.Google Scholar
Deligne, P. and Serre, J.-P., Formes modulaires de poids 1. Ann. Sci. Éc. Norm. Supér. (4) 7 1974, 507–530.Google Scholar
Deshouillers, J.-M. and Iwaniec, H., Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70(2) 1982–1983, 219–288.Google Scholar
Dunster, T. M., Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21(4) 1990, 995–1018.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, Vol. II, Robert E. Krieger (Melbourne, FL, 1981); based on notes left by Harry Bateman, reprint of the 1953 original.Google Scholar
Ganguly, S. and Sengupta, J., Sums of Kloosterman sums over arithmetic progressions. Int. Math. Res. Not. IMRN 2012(1) 2012, 137–165.Google Scholar
Hoffstein, J. and Lockhart, P., Coefficients of Maass forms and the Siegel zero. Ann. of Math. (2) 140(1) 1994, 161–181; with an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman.Google Scholar
Humphries, P., Density theorems for exceptional eigenvalues for congruence subgroups. Algebra Number Theory 12(7) 2018, 1581–1610.Google Scholar
Iwaniec, H., Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics 53 ), 2nd edn., American Mathematical Society/Revista Matemática Iberoamericana, Madrid (Providence, RI, 2002).Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53 ), American Mathematical Society (Providence, RI, 2004).Google Scholar
Jutila, M., Convolutions of Fourier coefficients of cusp forms. Publ. Inst. Math. (Beograd) (N.S.) 65(79) 1999, 31–51.Google Scholar
Kim, H. H., Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 . J. Amer. Math. Soc. 16(1) 2003, 139–183; with Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Kim and Peter Sarnak.Google Scholar
Kiral, E. M. and Young, M. P., The fifth moment of modular $L$ -functions. Preprint, 2017, arXiv:1701.07507.Google Scholar
Kuznetsov, N. V., The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums. Mat. Sb. 111(153(3)) 1980, 334–383, 479.Google Scholar
Michel, P., Analytic number theory and families of automorphic L-functions. In Automorphic Forms and Applications (IAS/Park City Mathematics Series 12 ), American Mathematical Society (Providence, RI, 2007), 181–295.Google Scholar
Palm, M. R., Explicit $\text{GL}_{}(2)$ trace formulas and uniform, mixed Weyl laws. Preprint, 2012, arXiv:1212.4282.Google Scholar
Proskurin, N. V., On the general Kloosterman sums. In Analytical Theory of Numbers and Theory of Functions. Part 19 (Zap. Nauchn. Sem. POMI 302 ), POMI (St. Petersburg, 2003), 107–134.Google Scholar
Sardari, N. T., Optimal strong approximation for quadratic forms. Duke Math. J. (to appear). Preprint, 2015, arXiv:1510.00462.Google Scholar
Sarnak, P. and Tsimerman, J., On Linnik and Selberg’s conjecture about sums of Kloosterman sums. In Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin. Vol. II (Progress in Mathematics 270 ), Birkhäuser (Boston, MA, 2009), 619–635.Google Scholar
Topacogullari, B., The shifted convolution of divisor functions. Q. J. Math. 67(2) 2016, 331–363.Google Scholar
Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press/Macmillan (Cambridge, UK/New York, 1944).Google Scholar