Spectral square moments of a resonance sum for Maass forms

Abstract

Let fbe a Maass cusp form for {\Gamma }_{\mathrm{0}}(N) with Fourier coefficients {\lambda }_f (n)and Laplace eigenvalue {\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+k^{\mathrm{2}}.For real \alpha \ne \mathrm{0} and β>0,consider the sum S_X(f; \alpha , \beta ) ={\displaystyle {\sum }_n{\lambda }_f(n)e(\alpha n^{\beta })\varphi (n/X)},where φis a smooth function of compact support. We prove bounds for the second spectral moment of S_X(f; \alpha ,\hspace{0.17em}\beta ),with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X.This implies that if fhas its eigenvalue beyond X^{\frac{\mathrm{1}}{\mathrm{2}}+\epsilon },the standard resonance main term for S_X(f;\pm 2\sqrt{q},\mathrm{ 1}/2), q\in {\mathbb{Z}}_{+},cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2)×GL(2).It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K^{\epsilon }\le L\le K^{1-\epsilon }. The same bounds can be proved in a similar way for holomorphic cusp forms.

Keywords

Cusp form / Maass form / Fourier coefficient of cusp form / Kuznetsov trace formula / resonance sum / first derivative test / weighted stationary phase

References

[1] BatemanH. Tables of Integral Transforms.New York: McGraw-Hill, 1954

[2] BlomerV, KhanR, YoungM. Distribution of mass of holomorphic cusp forms.Duke Math J, 2013, 162(14): 2609–2644

[3] CzarneckiK. Resonance sums for Rankin–Selberg products of Maass cusp forms.J Number Theory, 2016, 163: 359–374

[4] Ernvall-HytönenA M. On certain exponential sums related to GL(3) cusp forms.C R Math Acad Sci Paris, 2010, 348: 5–8

[5] Ernvall-HytönenA M, JääsaariJ, VesalainenE V. Resonances and Ω-results for exponential sums related to Maass forms for SL(n, Z).J Number Theory, 2015, 153: 135–157

[6] HafnerJ L. Some remarks on odd Maass wave forms (and a correction to: Zeros of L-functions attached to Maass forms [Math Z, 1985, 190(1): 113–128] by Epstein, C., Hafner, J. L., Sarnak, P.). Math Z, 1987, 196(1): 129–132

[7] HuxleyM N. Area, Lattice Points, and Exponential Sums.London Math Soc Monogr New Ser, Vol 13. Oxford: Clarendon Press, 1996

[8] IwaniecH. Introduction to the Spectral Theory of Automorphic Forms.Bibl Rev Mat Iberoamericana. Madrid, 1995

[9] IwaniecH, KowalskiE. Analytic Number Theory.Amer Math Soc Colloq Publ, Vol 53. Providence: Amer Math Soc, 1997

[10] IwaniecH, LuoW Z, SarnakP. Low lying zeros of families of L-functions.Publ Math Inst Hautes Études Sci, 2000, 91: 55–131

[11] KaczorowskiJ, PerelliA. On the structure of the Selberg class VI: non-linear twists.Acta Arith, 2005, 116: 315–341

[12] KuznetsovN V. Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture.Sums of Kloosterman sums. Math USSR Sbornik, 1981, 29: 299–342

[13] LauY-K, LiuJ Y, YeY B.Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups.J Number Theory, 2006, 121: 204–223

[14] LauY-K, LiuJ Y, YeY B. A new bound k23+ε for Rankin-Selberg L-functions for Hecke congruence subgroups.Int Math Res Pap, 2006, Article ID 35090, 78pp

[15] LiX Q. Bounds for GL(2)×GL(3) L-functions and GL(3) L-functions.Ann of Math, 2011, 173(1): 301–336

[16] LiuJ Y, YeY B. Subconvexity for Rankin-Selberg L-functions of Maass forms.Geom Funct Anal, 2002, 12: 1296–1323

[17] LiuJ Y, YeY B. Petersson and Kuznetsov trace formulas. In: Lie Groups and Automorphic Forms. AMS/IP Stud Adv Math, Vol 37. Providence: Amer Math Soc, 2006, 147–168

[18] McKeeM, SunH W, YeY B. Weighted stationary phase of higher orders.Front Math China, 2017, 12(3): 675–702

[19] MichelP. Analytic number theory and families of automorphic L-functions. In: Sarnak P, Shahidi F, eds. Automorphic Forms and Applications. IAS/Park City Math Ser, Vol 12. Providence: Amer Math Soc and Inst Adv Study, 2007, 179–295

[20] MillerS D, SchmidW. The highly oscillatory behavior of automorphic distributions for SL(2).Lett Math Phys, 2004, 69: 265–286

[21] RenX M, YeY B. Resonance between automorphic forms and exponential functions.Sci China Math, 2010, 53(9): 2463–2472

[22] RenX M, YeY B. Asymptotic Voronoi’s summation formulas and their duality for SL₃(Z).In: Kanemitsu S, Li H Z, Liu J Y, eds. Number Theory: Arithmetic in Shangri-La—Proceedings of the 6th China-Japan Seminar on Number Theory held in Shanghai Jiaotong University, August 15–17, 2011. Ser on Number Theory and Its Appl, Vol 8. Singapore: World Scientific, 2012, 213–236

[23] RenX M, YeY B. Sums of Fourier coefficients of a Maass form for SL₃(X) twisted by exponential functions.Forum Math, 2014, 26(1): 221–238

[24] RenX M, YeY B. Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL_m(Z).Sci China Math, 2015, 58(10): 2105–2124

[25] RenX M, YeY B. Resonance of automorphic forms for GL(3).Trans Amer Math Soc, 2015, 367(3): 2137–2157

[26] RenX M, YeY B. Hyper-Kloosterman sums of different moduli and their applications to automorphic forms for SL_m(Z).Taiwanese J Math, 2016, 20(6): 1251–1274

[27] SarnakP. Estimates for Rankin-Selberg L-functions and quantum unique ergodicity.J Funct Anal, 2001, 184: 419–453

[28] SavalaP. Computing the Laplace eigenvalue and level for Maass cusp forms.J Number Theory, 2017, 173: 1–22

[29] SunQ F. On cusp form coefficients in nonlinear exponential sums.Quart J Math, 2010, 61(3): 363–372

[30] SunQ F, WuY Y. Exponential sums involving Maass forms.Front Math China, 2014, 9(6): 1349–1366

[31] WolffT H. Lectures on Harmonic Analysis.Edited by Laba I, Shubin C. Univ Lecture Ser, Vol 29. Providence: Amer Math Soc, 2003

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany