Resonance of automorphic forms for 𝐺𝐿(3)

Ren, Xiumin; Ye, Yangbo

doi:10.1090/S0002-9947-2014-06208-9

Resonance of automorphic forms for $GL(3)$
HTML articles powered by AMS MathViewer

by Xiumin Ren and Yangbo Ye PDF

Trans. Amer. Math. Soc. 367 (2015), 2137-2157

Abstract:

Let $f$ be a Maass form for $SL_3(\mathbb Z)$ with Fourier coefficients $A_f(m,n)$. A smoothly weighted sum of $A_f(m,n)$ against an exponential function $e(\alpha n^\beta )$ of fractional power $n^\beta$ for $X\leq n\leq 2X$ is proved to have a main term of size $X^{2/3}$ when $\beta =1/3$ and $\alpha$ is close to $3\ell ^{1/3}$ for some integer $\ell \neq 0$. The sum becomes rapidly decreasing if $\beta <1/3$. If such a sum is not smoothly weighted, the main term can only be detected under a conjectured bound toward the Ramanujan conjecture. The existence of such a main term manifests the vibration and resonance behavior of individual automorphic forms $f$ for $GL(3)$. Applications of these results include a new modularity test on whether a two dimensional array $a(m,n)$ comes from Fourier coefficients $A_f(m,n)$ of a Maass form $f$ for $SL_3(\mathbb Z)$. Techniques used in the proof include a Voronoi summation formula, its asymptotic expansion, and the weighted stationary phase.

References

Ce Bian, Computing $\textrm {GL}(3)$ automorphic forms, Bull. Lond. Math. Soc. 42 (2010), no. 5, 827–842. MR 2721743, DOI 10.1112/blms/bdq038
Andrew R. Booker, A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums, Experiment. Math. 9 (2000), no. 4, 571–581. MR 1806292
Andrew R. Booker, Numerical tests of modularity, J. Ramanujan Math. Soc. 20 (2005), no. 4, 283–339. MR 2193217
Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
Dorian Goldfeld and Xiaoqing Li, Voronoi formulas on $\textrm {GL}(n)$, Int. Math. Res. Not. , posted on (2006), Art. ID 86295, 25. MR 2233713, DOI 10.1155/IMRN/2006/86295
Dorian Goldfeld and Xiaoqing Li, The Voronoi formula for $\textrm {GL}(n,\Bbb R)$, Int. Math. Res. Not. IMRN 2 (2008), Art. ID rnm144, 39. MR 2418857, DOI 10.1093/imrn/rnm144
M. N. Huxley, Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1420620
Aleksandar Ivić, On the ternary additive divisor problem and the sixth moment of the zeta-function, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995) London Math. Soc. Lecture Note Ser., vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 205–243. MR 1635762, DOI 10.1017/CBO9780511526091.015
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). MR 1828743
J. Kaczorowski and A. Perelli, On the structure of the Selberg class. VI. Non-linear twists, Acta Arith. 116 (2005), no. 4, 315–341. MR 2110507, DOI 10.4064/aa116-4-2
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, Appendix to Kim \cite{Kim}.
Xiannan Li, Additive twists of Fourier coefficients of $GL(3)$ Maass forms, Proc. Amer. Math. Soc. 142 (2014), no. 6, 1825–1836. MR 3182004, DOI 10.1090/S0002-9939-2014-11909-5
Xiaoqing Li, The central value of the Rankin-Selberg $L$-functions, Geom. Funct. Anal. 18 (2009), no. 5, 1660–1695. MR 2481739, DOI 10.1007/s00039-008-0692-5
Xiaoqing Li, Bounds for $\textrm {GL}(3)\times \textrm {GL}(2)$ $L$-functions and $\textrm {GL}(3)$ $L$-functions, Ann. of Math. (2) 173 (2011), no. 1, 301–336. MR 2753605, DOI 10.4007/annals.2011.173.1.8
Xiaoqing Li and Matthew P. Young, Additive twists of Fourier coefficients of symmetric-square lifts, J. Number Theory 132 (2012), no. 7, 1626–1640. MR 2903173, DOI 10.1016/j.jnt.2011.12.017
Stephen D. Miller, Cancellation in additively twisted sums on $\textrm {GL}(n)$, Amer. J. Math. 128 (2006), no. 3, 699–729. MR 2230922
Stephen D. Miller and Wilfried Schmid, Automorphic distributions, $L$-functions, and Voronoi summation for $\textrm {GL}(3)$, Ann. of Math. (2) 164 (2006), no. 2, 423–488. MR 2247965, DOI 10.4007/annals.2006.164.423
A. Perelli, Non-linear twists of $L$-functions: a survey, Milan J. Math. 78 (2010), no. 1, 117–134. MR 2684775, DOI 10.1007/s00032-010-0119-2
XiuMin Ren and YangBo Ye, Resonance between automorphic forms and exponential functions, Sci. China Math. 53 (2010), no. 9, 2463–2472. MR 2718840, DOI 10.1007/s11425-010-3150-4
Xiumin Ren and Yangbo Ye, Sums of Fourier coefficients of a Maass form for $\textrm {SL}_3(\Bbb Z)$ twisted by exponential functions, Forum Math. 26 (2014), no. 1, 221–238. MR 3176629, DOI 10.1515/form.2011.157
Xiumin Ren and Yangbo Ye, Asymptotic Voronoi’s summation formulas and their duality for $SL_3(\mathbb Z)$, in Number Theory Arithmetic in Shangri-La Proceedings of the 6th China-Japan Seminar Shanghai, China 15-17 August 2011, edited by Shigeru Kanemitsu, Hongze Li, and Jianya Liu, Series on Number Theory and Its Applications, volume 8, 213–236, World Scientific, Singapore, 2013.
Peter Sarnak, Notes on the generalized Ramanujan conjectures, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 659–685. MR 2192019
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550