×

Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms. (English) Zbl 1314.11038

Mem. Am. Math. Soc. 1055, v, 132 p. (2013).
The idea of a Fourier trace formula for \(\mathrm{GL}(2)\) comes from H. Petersson’s computation of the Fourier coefficients of Poincaré series in 1932 [Acta Math. 58, 169–215 (1932; Zbl 0003.35002; JFM 58.1110.01)] and his introduction of the inner product in 1939 [Jahresber. Dtsch. Math.-Ver. 49, 49–75 (1939; JFM 65.0355.01)]. This Fourier trace formula for \(\mathrm{GL}(2)\) is an identity between a product of two Fourier coefficients, averaged over a family of automorphic forms on \(\mathrm{GL}(2)\), and a series involving Kloosterman sums and the Bessel \(J\)-function. Because of the existence of the Weil bound on Kloosterman sums and its relation with the Bessel function, the Petersson formula is useful for approximating expressions involving Fourier coefficients of cusp forms. A. Selberg used the Petersson formula to obtain a nontrivial bound on Fourier coefficients [Proc. Sympos. Pure Math. 8, 1–15 (1965; Zbl 0142.33903)]. Selberg extended his method to the case of Maass forms. This idea was the beginning of an independent work by R. W. Bruggeman [Invent. Math. 45, 1–18 (1978; Zbl 0351.10019)] and N. V. Kuznetsov [Mat. Sb., Nov. Ser. 111(153), 334–383 (1980; Zbl 0427.10016)] in late 70’s. In this technical article, the authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on \(\mathrm{GL}(2)\) over \(\mathbb Q\). Their results contain a Hecke eigenvalue in addition to the two Fourier coefficients on the spectral side. A proof of a Weil bound for the generalized twisted Kloosterman sums is given as well. As an application, they show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed in the limit relative to the Sato-Tate measure.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F30 Fourier coefficients of automorphic forms
11L05 Gauss and Kloosterman sums; generalizations
11F25 Hecke-Petersson operators, differential operators (one variable)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Andersson, Summation formulae and zeta functions, Doctoral dissertation, Stockholm University, 2006.
[2] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001
[3] James Arthur, The Selberg trace formula for groups of \(F\)-rank one, Ann. of Math. (2) 100 (1974), 326-385. · Zbl 0257.20033
[4] James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in \(G({\mathbf Q})\), Duke Math. J. 45 (1978), no. 4, 911-952. · Zbl 0499.10032
[5] Roger C. Baker, Kloosterman sums and Maass forms. Vol. I, Kendrick Press, Heber City, UT, 2003. · Zbl 1028.11034
[6] Valentin Blomer and Gergely Harcos, Twisted \(L\)-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1-52. · Zbl 1221.11121 · doi:10.1007/s00039-010-0063-x
[7] V. Bykovsky, N. Kuznetsov, and A. Vinogradov, Generalized summation formula for inhomogeneous convolution, Automorphic functions and their applications (Khabarovsk, 1988) Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990, pp. 18-63. · Zbl 0758.11028
[8] Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29-98. · Zbl 1264.11044 · doi:10.2977/PRIMS/31
[9] Roelof W. Bruggeman and Roberto J. Miatello, Sum formula for \(\mathrm SL_2\) over a totally real number field, Mem. Amer. Math. Soc. 197 (2009), no. 919, vi+81. · Zbl 1230.11066 · doi:10.1090/memo/0919
[10] R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), no. 1, 1-18. · Zbl 0351.10019
[11] Farrell Brumley, Effective multiplicity one on \({\mathrm GL}_N\) and narrow zero-free regions for Rankin-Selberg \(L\)-functions, Amer. J. Math. 128 (2006), no. 6, 1455-1474. · Zbl 1137.11058
[12] Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. · Zbl 0868.11022
[13] P. Cartier, Some numerical computations relating to automorphic functions, in “Computers in Number Theory” , 37-48, Academic Press, London, 1971. · Zbl 0236.65069
[14] William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-314. · Zbl 0239.10015
[15] S. Chowla, On Kloosterman’s sum, Norske Vid. Selsk. Forh. (Trondheim) 40 (1967), 70-72. · Zbl 0157.09001
[16] Keith Conrad, On Weil’s proof of the bound for Kloosterman sums, J. Number Theory 97 (2002), no. 2, 439-446. · Zbl 1035.11063 · doi:10.1016/S0022-314X(02)00011-2
[17] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1-26. · Zbl 0668.10044 · doi:10.1515/crll.1989.399.1
[18] J. B. Conrey, W. Duke, and D. W. Farmer, The distribution of the eigenvalues of Hecke operators, Acta Arith. 78 (1997), no. 4, 405-409. · Zbl 0876.11020
[19] B. Conrey, H. Iwaniec, and K. Soundararajan, Critical zeros of Dirichlet \(L\)-functions, preprint, 2011. · Zbl 1357.11070
[20] James W. Cogdell and Ilya Piatetski-Shapiro, The arithmetic and spectral analysis of Poincaré series, Perspectives in Mathematics, vol. 13, Academic Press, Inc., Boston, MA, 1990. · Zbl 0714.11032
[21] W. Duke, J. B. Friedlander, and H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math. (2) 141 (1995), no. 2, 423-441. · Zbl 0840.11003 · doi:10.2307/2118527
[22] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219-288. · Zbl 0502.10021 · doi:10.1007/BF01390728
[23] Michel Duflo and Jean-Pierre Labesse, Sur la formule des traces de Selberg, Ann. Sci. École Norm. Sup. (4) 4 (1971), 193-284. · Zbl 0277.12011
[24] W. Duke, The critical order of vanishing of automorphic \(L\)-functions with large level, Invent. Math. 119 (1995), no. 1, 165-174. · Zbl 0838.11035 · doi:10.1007/BF01245178
[25] T. Estermann, On Kloosterman’s sum, Mathematika 8 (1961), 83-86. · Zbl 0114.26302
[26] Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. · Zbl 0924.28001
[27] Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. · Zbl 0841.35001
[28] Tobias Finis and Erez Lapid, On the Arthur-Selberg trace formula for \({\mathrm GL}(2)\), Groups Geom. Dyn. 5 (2011), no. 2, 367-391. · Zbl 1268.11069 · doi:10.4171/GGD/132
[29] Tobias Finis, Erez Lapid, and Werner Müller, On the spectral side of Arthur’s trace formula-absolute convergence, Ann. of Math. (2) 174 (2011), no. 1, 173-195. · Zbl 1242.11036 · doi:10.4007/annals.2011.174.1.5
[30] Stephen Gelbart and Hervé Jacquet, Forms of \({\mathrm GL}(2)\) from the analytic point of view, Automorphic forms, representations and \(L\)-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 213-251. · Zbl 0409.22013
[31] Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Basic classes of linear operators, Birkhäuser Verlag, Basel, 2003. · Zbl 1065.47001
[32] Dorian Goldfeld, Automorphic forms and \(L\)-functions for the group \({\mathrm GL}(n,\mathbf R)\), Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. · Zbl 1108.11039
[33] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). · Zbl 1208.65001
[34] Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. · Zbl 0483.10001
[35] A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. · Zbl 0715.11045
[36] Henryk Iwaniec, Promenade along modular forms and analytic number theory, Topics in analytic number theory (Austin, Tex., 1982) Univ. Texas Press, Austin, TX, 1985, pp. 221-303.
[37] Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. · Zbl 1006.11024
[38] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. · Zbl 1059.11001
[39] Henryk Iwaniec, Wenzhi Luo, and Peter Sarnak, Low lying zeros of families of \(L\)-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55-131 (2001). · Zbl 1012.11041
[40] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of \(L\)-functions, Geom. Funct. Anal. Special Volume (2000), 705-741. GAFA 2000 (Tel Aviv, 1999). · Zbl 0996.11036 · doi:10.1007/978-3-0346-0425-3_6
[41] Henryk Iwaniec and Peter Sarnak, The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros. part A, Israel J. Math. 120 (2000), no. part A, 155-177. · Zbl 0992.11037 · doi:10.1007/s11856-000-1275-9
[42] Hervé Jacquet, Automorphic spectrum of symmetric spaces, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 443-455. · Zbl 0888.11020 · doi:10.1090/pspum/061/1476509
[43] David Joyner, On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp, Math. Z. 203 (1990), no. 1, 59-104. · Zbl 0701.11020 · doi:10.1007/BF02570723
[44] A. W. Knapp, Theoretical aspects of the trace formula for \({\mathrm GL}(2)\), Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 355-405. · Zbl 0887.11023 · doi:10.1090/pspum/061/1476505
[45] Andrew Knightly and Charles Li, A relative trace formula proof of the Petersson trace formula, Acta Arith. 122 (2006), no. 3, 297-313. · Zbl 1095.11028 · doi:10.4064/aa122-3-5
[46] Andrew Knightly and Charles Li, Traces of Hecke operators, Mathematical Surveys and Monographs, vol. 133, American Mathematical Society, Providence, RI, 2006. · Zbl 1120.11024
[47] Andrew Knightly and Charles Li, Petersson’s trace formula and the Hecke eigenvalues of Hilbert modular forms, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 145-187. · Zbl 1228.11062 · doi:10.1017/CBO9780511543371.011
[48] E. Kowalski, P. Michel, and J. VanderKam, Non-vanishing of high derivatives of automorphic \(L\)-functions at the center of the critical strip, J. Reine Angew. Math. 526 (2000), 1-34. · Zbl 1020.11033 · doi:10.1515/crll.2000.074
[49] Henry H. Kim, Functoriality for the exterior square of \({\mathrm GL}_4\) and the symmetric fourth of \({\mathrm GL}_2\), J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1
[50] N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334-383, 479 (Russian).
[51] Robert P. Langlands, Beyond endoscopy, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 611-697. · Zbl 1078.11033
[52] Edmund Landau, Elementary number theory, Chelsea Publishing Co., New York, N.Y., 1958. Translated by J. E. Goodman. · Zbl 0079.06201
[53] Serge Lang, \({\mathrm SL}_2({\mathbf R})\), Graduate Texts in Mathematics, vol. 105, Springer-Verlag, New York, 1985. Reprint of the 1975 edition.
[54] Erez M. Lapid, On the fine spectral expansion of Jacquet’s relative trace formula, J. Inst. Math. Jussieu 5 (2006), no. 2, 263-308. · Zbl 1195.11071 · doi:10.1017/S1474748005000289
[55] Charles C. C. Li, Kuznietsov trace formula and weighted distribution of Hecke eigenvalues, J. Number Theory 104 (2004), no. 1, 177-192. · Zbl 1039.11025 · doi:10.1016/S0022-314X(03)00149-5
[56] Xiaoqing Li, Arithmetic trace formulas and Kloostermania, Automorphic forms and the Langlands program, Adv. Lect. Math. (ALM), vol. 9, Int. Press, Somerville, MA, 2010, pp. 199-235. · Zbl 1275.11089
[57] Yuk-Kam Lau and Yingnan Wang, Quantitative version of the joint distribution of eigenvalues of the Hecke operators, J. Number Theory 131 (2011), no. 12, 2262-2281. · Zbl 1268.11057 · doi:10.1016/j.jnt.2011.05.014
[58] Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141-183 (German). · Zbl 0033.11702
[59] Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. · Zbl 0701.11014
[60] Yoichi Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, vol. 127, Cambridge University Press, Cambridge, 1997. · Zbl 0878.11001
[61] Y. Motohashi, Sums of Kloosterman sums revisited, The Conference on \(L\)-Functions, World Sci. Publ., Hackensack, NJ, 2007, pp. 141-163. · Zbl 1183.11044
[62] —, Chapter 6: Appendix, post-publication appendix to [Mo1], arXiv:0810.2847, 2008.
[63] M. Ram Murty and Kaneenika Sinha, Effective equidistribution of eigenvalues of Hecke operators, J. Number Theory 129 (2009), no. 3, 681-714. · Zbl 1234.11055 · doi:10.1016/j.jnt.2008.10.010
[64] M. Ram Murty, On the estimation of eigenvalues of Hecke operators, Rocky Mountain J. Math. 15 (1985), no. 2, 521-533. Number theory (Winnipeg, Man., 1983). · Zbl 0588.10027 · doi:10.1216/RMJ-1985-15-2-521
[65] Philippe Michel and Akshay Venkatesh, The subconvexity problem for \({\mathrm GL}_2\), Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171-271. · Zbl 1376.11040 · doi:10.1007/s10240-010-0025-8
[66] Hans Petersson, Über die Entwicklungskoeffizienten der automorphen Formen, Acta Math. 58 (1932), no. 1, 169-215 (German). · Zbl 0003.35002 · doi:10.1007/BF02547776
[67] —, Uber eine Metrisierung der ganzen Modulformen, Jahresber. Dtsch. Math.-Ver. 49 (1939), 49-75. · Zbl 0021.02502
[68] K. Ramachandra, Theory of numbers: a textbook, Alpha Science, Oxford, 2007. · Zbl 1112.11001
[69] J. Rogawski, Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence, appendix in “Discrete Groups, Expanding Graphs and Invariant Measures,” by A. Lubotzky, Birkhäuser, Basel, 1994, pp. 135-176.
[70] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. · Zbl 0459.46001
[71] Hans Salié, Über die Kloostermanschen Summen \(S(u,v;q)\), Math. Z. 34 (1932), no. 1, 91-109 (German). · Zbl 0002.12801 · doi:10.1007/BF01180579
[72] Peter Sarnak, Selberg’s eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272-1277. · Zbl 1042.11517
[73] Peter Sarnak, Statistical properties of eigenvalues of the Hecke operators, Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 321-331.
[74] A. Selberg, Über die Fourierkoeffizienten elliptischer Modulformen negativer Dimension, C. R. Neuvième Congrès Math. Scandinaves, Helsingfors (1938), 320-322. · JFM 65.0352.02
[75] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87. · Zbl 0072.08201
[76] Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1-15. · Zbl 0142.33903
[77] —, Göttingen lecture notes, Collected Papers Vol. 1, Springer-Verlag, Berlin, (1989), 626-674.
[78] Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_p\), J. Amer. Math. Soc. 10 (1997), no. 1, 75-102 (French). · Zbl 0871.11032 · doi:10.1090/S0894-0347-97-00220-8
[79] F. Strömberg, Newforms and spectral multiplicites for \(\Gamma _0(9)\), Proc. Lond. Math. Soc., to appear.
[80] Akshay Venkatesh, Limiting forms of the trace formula, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)-Princeton University. · Zbl 1008.11019
[81] Akshay Venkatesh, “Beyond endoscopy” and special forms on GL(2), J. Reine Angew. Math. 577 (2004), 23-80. · Zbl 1061.22019 · doi:10.1515/crll.2004.2004.577.23
[82] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. · Zbl 0063.08184
[83] André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 204-207. · Zbl 0032.26102
[84] Eiji Yoshida, Remark on the Kuznetsov trace formula, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 377-382. · Zbl 0973.11058 · doi:10.1017/CBO9780511666179.026
[85] Don Zagier, Eisenstein series and the Selberg trace formula. I, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 303-355. · Zbl 0484.10020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.