Resonances and \(\Omega\)-results for exponential sums related to Maass forms for \(\mathrm{SL}(n, \mathbb{Z})\). (English) Zbl 1357.11067
The \(\varphi\) be a \(\mathrm{GL}(n)\) automorphic form that is invariant under \(\mathrm{SL}(n,{\mathbb Z})\), and \(a(m_1,\cdots,m_{n-1})\) denote its Fourier coefficients. Then the standard \(L\)-function of \(\varphi\) is given for \(\mathrm{Re}(s)\gg0\) by \(L(s,\varphi)=\sum_{n\geq1} a(m,1,\dots,1) m^{-s}\). In this paper the authors obtain resonances for short exponential sums weighted by the Fourier coefficients \(a(m,1,\dots,1)\). Their main result is an estimate for
\[
\sum_{M\leq m\leq M+\Delta} a(m,1,\dots,1) w(m) \exp(d^{1/n} m /M^{1-1/n})
\]
for a suitable weight function \(w\) supported in \([M,M+\Delta]\). Here \(M^{1-1/n+\varepsilon}\ll \Delta\ll M\) and \(d\) is a fixed positive integer. To do so they derive asymptotics for integrals appearing in the (untwisted) \(\mathrm{GL}(n)\) Voronoi summation formula. As an application, they prove an \(\Omega\)-result for short unweighted sums of these Fourier coefficients, that is, for \(\sum_{M\leq m\leq M+\Delta} a(m,1,\dots,1)\). And as another consequence, they show that given any sequence of \(M_1,M_2,\dots\) of positive real numbers tending to infinity and any \(\varepsilon>0\), the coefficients \(a(m,1,\dots,1)\) for \(m\) in the union of intervals \([M_\ell,M_\ell+M_\ell^{1-1/n+\varepsilon}]\) uniquely determine the automorphic form \(\varphi\).
Reviewer: Solomon Friedberg (Chestnut Hill)
MSC:
| 11L03 | Trigonometric and exponential sums (general theory) |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
Keywords:
Fourier coefficients of autormorphic forms on \(\mathrm{GL}(n)\); resonance; short exponential sum; Voronoi summation formulaSoftware:
GL(n)packReferences:
| [1] | Brumley, F., Effective multiplicity one on \(GL_n\) and narrow zero-free regions for Rankin-Selberg \(L\)-functions, Amer. J. Math., 128, 1455-1474 (2006) · Zbl 1137.11058 |
| [2] | Deligne, P., La conjecture de Weil I, Inst. Hautes Études Sci. Publ. Math., 43, 273-307 (1974) · Zbl 0287.14001 |
| [3] | Epstein, C.; Hafner, J. L.; Sarnak, P., Zeros of \(L\)-functions attached to Maass forms, Math. Z., 190, 113-128 (1985) · Zbl 0565.10026 |
| [4] | Ernvall-Hytönen, A.-M., A relation between Fourier coefficients of holomorphic cusp forms and exponential sums, Publ. Inst. Math. (Beograd) (N.S.), 86, 97-105 (2009) · Zbl 1279.11083 |
| [5] | Ernvall-Hytönen, A.-M., A note concerning certain exponential sums related to cusp forms, Šiauliai Math. Semin., 4, 75-82 (2009) · Zbl 1211.11095 |
| [6] | Ernvall-Hytönen, A.-M., On certain exponential sums related to \(GL(3)\) cusp forms, C. R. Acad. Sci. Paris, Ser. I, 348, 5-8 (2010) · Zbl 1221.11173 |
| [7] | Ernvall-Hytönen, A.-M.; Karppinen, K., On short exponential sums involving Fourier coefficients of holomorphic cusp forms, Int. Math. Res. Not. IMRN, 1-44 (2008), article ID rnn022 · Zbl 1247.11106 |
| [8] | Ernvall-Hytönen, A.-M.; Lepistö, A., Bounds and computational results for exponential sums related to cusp forms, Acta Math. Univ. Ostrav., 17, 81-90 (2009) · Zbl 1246.11138 |
| [9] | Godber, D., Additive twists of Fourier coefficients of modular forms, J. Number Theory, 133, 83-104 (2013) · Zbl 1277.11040 |
| [10] | Goldfeld, D., Automorphic Forms and \(L\)-Functions for the Group \(GL(n, R)\), Cambridge Stud. Adv. Math., vol. 99 (2006), Cambridge University Press · Zbl 1108.11039 |
| [11] | Goldfeld, D.; Li, X., Voronoi formulas on \(GL(n)\), Int. Math. Res. Not. IMRN, 1-25 (2006), article ID 86295 · Zbl 1144.11039 |
| [12] | Goldfeld, D.; Li, X., The Voronoi formula for \(GL(n, R)\), Int. Math. Res. Not. IMRN, 1-39 (2008), article ID rnm144 · Zbl 1225.11058 |
| [13] | (Greaves, G. R.H.; Harman, G.; Huxley, M. N., Sieve Methods, Exponential Sums, and Their Applications in Number Theory. Proceedings of a Symposium Held in Cardiff. Sieve Methods, Exponential Sums, and Their Applications in Number Theory. Proceedings of a Symposium Held in Cardiff, July 1995. Sieve Methods, Exponential Sums, and Their Applications in Number Theory. Proceedings of a Symposium Held in Cardiff. Sieve Methods, Exponential Sums, and Their Applications in Number Theory. Proceedings of a Symposium Held in Cardiff, July 1995, London Math. Soc. Lecture Note Ser., vol. 237 (1997), Cambridge University Press) · Zbl 0910.00038 |
| [14] | Hafner, J. L., Some remarks on odd Maass wave forms, Math. Z., 196, 129-132 (1987), (and a correction to [3]) · Zbl 0628.10033 |
| [15] | Hafner, J. L.; Ivić, A., On sums of Fourier coefficients of cusp forms, Enseign. Math. (2), 35, 375-382 (1989) · Zbl 0696.10020 |
| [17] | Ivić, A., On the divisor function and the Riemann zeta-function in short intervals, Ramanujan J., 19, 207-224 (2009) · Zbl 1226.11086 |
| [18] | Iwaniec, H.; Luo, W.; Sarnak, P., Low-lying zeros of families of \(L\)-functions, Publ. Math. Inst. Hautes Études Sci., 91, 55-131 (2000) · Zbl 1012.11041 |
| [19] | Jacquet, H.; Shalika, J., On Euler products and the classification of automorphic representations I, Amer. J. Math., 103, 499-588 (1981) · Zbl 0473.12008 |
| [20] | (Ji, L., Geometry and Analysis, No. 2. Geometry and Analysis, No. 2, Adv. Lectures Math., vol. 18 (2011), International Press) |
| [21] | Jutila, M., On the divisor problem for short intervals, Ann. Univ. Turkuensis Ser. AI, 186, 23-30 (1984) · Zbl 0536.10032 |
| [22] | Jutila, M., On exponential sums involving the Ramanujan function, Proc. Indian Acad. Sci. Math. Sci., 97, 157-166 (1987) · Zbl 0658.10043 |
| [23] | Jutila, M.; Motohashi, Y., Uniform bound for Hecke \(L\)-functions, Acta Math., 195, 61-115 (2005) · Zbl 1098.11034 |
| [24] | Kim, H. H.; Sarnak, P., Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16, 1, 175-183 (2003) |
| [25] | Lebedev, N. N., Special Functions & Their Applications (1972), Dover Publications · Zbl 0271.33001 |
| [26] | Li, X., The central values of the Rankin-Selberg \(L\)-functions, Geom. Funct. Anal., 18, 1660-1695 (2009) · Zbl 1188.11021 |
| [27] | Li, X.; Young, M. P., Additive twists of Fourier coefficients of symmetric-square lifts, J. Number Theory, 132, 1626-1640 (2012) · Zbl 1319.11025 |
| [28] | Luo, W.; Rudnick, Z.; Sarnak, P., On the generalized Ramanujan conjecture for \(GL(n)\), Proc. Sympos. Pure Math., 66, 2, 301-310 (1999) · Zbl 0965.11023 |
| [29] | Miller, S. D., Cancellation in additively twisted sums on \(GL(n)\), Amer. J. Math., 128, 699-729 (2006) · Zbl 1142.11033 |
| [30] | Miller, S. D.; Schmid, W., Automorphic distributions, \(L\)-functions, and Voronoi summation for \(GL(3)\), Ann. of Math., 164, 423-488 (2006) · Zbl 1162.11341 |
| [32] | Olver, F. W.J., Asymptotics and Special Functions (1997), A K Peters · Zbl 0303.41035 |
| [33] | Ramakrishnan, D., Appendix: A refinement of the strong multiplicity one theorem for \(GL(2)\), Invent. Math., 116, 645-649 (1994) · Zbl 0823.11021 |
| [34] | Ren, X. M.; Ye, Y. B., Resonance between automorphic forms and exponential functions, Sci. China Math., 53, 9, 2463-2472 (2010) · Zbl 1250.11078 |
| [35] | Ren, X. M.; Ye, Y. B., Sums of Fourier coefficients of a Maass form for \(SL_3(Z)\) twisted by exponential functions, Forum Math., 26, 1, 221-238 (2014) · Zbl 1360.11070 |
| [36] | Ren, X. M.; Ye, Y. B., Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for \(GL_m(Z)\), Sci. China Math., 58 (2015) |
| [37] | Wilton, J. R., A note on Ramanujan’s arithmetical function \(\tau(n)\), Proc. Cambridge Philos. Soc., 25, 121-129 (1929) · JFM 55.0709.02 |
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.
