On signs of Fourier coefficients of Hecke-Maass cusp forms on 𝐺𝐿₃

Jääsaari, Jesse

doi:10.1090/tran/9012

On signs of Fourier coefficients of Hecke-Maass cusp forms on $\mathrm {GL}_3$
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by Jesse Jääsaari

Trans. Amer. Math. Soc. 376 (2023), 8193-8223

DOI: https://doi.org/10.1090/tran/9012

Published electronically: August 14, 2023

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Abstract:

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm {SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg _\varepsilon X^{5/6-\varepsilon }$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm {GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\mathrm {GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.

References

S. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkov 13 (1912), 1–2.
Valentin Blomer, Applications of the Kuznetsov formula on $GL(3)$, Invent. Math. 194 (2013), no. 3, 673–729. MR 3127065, DOI 10.1007/s00222-013-0454-3
Valentin Blomer, Jack Buttcane, and Nicole Raulf, A Sato-Tate law for $\rm GL (3)$, Comment. Math. Helv. 89 (2014), no. 4, 895–919. MR 3284298, DOI 10.4171/CMH/337
Jack Buttcane and Fan Zhou, Plancherel distribution of Satake parameters of Maass cusp forms on $\textrm {GL}_3$, Int. Math. Res. Not. IMRN 5 (2020), 1417–1444. MR 4073945, DOI 10.1093/imrn/rny061
W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
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, Eric Stade, and Michael Woodbury, An orthogonality relation for $\rm GL(4, \Bbb R)$ (with an appendix by Bingrong Huang), Forum Math. Sigma 9 (2021), Paper No. e47, 83. MR 4271357, DOI 10.1017/fms.2021.39
Andrew Granville, Dimitris Koukoulopoulos, and Kaisa Matomäki, When the sieve works, Duke Math. J. 164 (2015), no. 10, 1935–1969. MR 3369306, DOI 10.1215/00127094-3120891
João Guerreiro, An orthogonality relation for a thin family of $\textrm {GL}(3)$ Maass forms, Int. J. Number Theory 11 (2015), no. 8, 2277–2294. MR 3420744, DOI 10.1142/S1793042115501031
D. R. Heath-Brown and K. Tsang, Sign changes of $E(T)$, $\Delta (x)$, and $P(x)$, J. Number Theory 49 (1994), no. 1, 73–83. MR 1295953, DOI 10.1006/jnth.1994.1081
Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker, Sign changes of coefficients and sums of coefficients of $L$-functions, J. Number Theory 177 (2017), 112–135. MR 3629237, DOI 10.1016/j.jnt.2017.01.007
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Yujiao Jiang and Guangshi Lü, Fourth power moment of coefficients of automorphic $L$-functions for $\textrm {GL}(m)$, Forum Math. 29 (2017), no. 5, 1199–1212. MR 3692033, DOI 10.1515/forum-2016-0097
Yujiao Jiang, Guangshi Lü, and Zhiwei Wang, Exponential sums with multiplicative coefficients without the Ramanujan conjecture, Math. Ann. 379 (2021), no. 1-2, 589–632. MR 4211098, DOI 10.1007/s00208-020-02108-z
Shin-ichi Kato, Spherical functions and a $q$-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), no. 3, 461–468. MR 662602, DOI 10.1007/BF01389223
H. H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc. 16 (2003), 175–183.
Winfried Kohnen, Yuk-Kam Lau, and Igor E. Shparlinski, On the number of sign changes of Hecke eigenvalues of newforms, J. Aust. Math. Soc. 85 (2008), no. 1, 87–94. MR 2460867, DOI 10.1017/S1446788708000323
E. Kowalski, Y.-K. Lau, K. Soundararajan, and J. Wu, On modular signs, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 3, 389–411. MR 2726725, DOI 10.1017/S030500411000040X
Yuk-Kam Lau, Jianya Liu, and Jie Wu, Local behavior of arithmetical functions with applications to automorphic $L$-functions, Int. Math. Res. Not. IMRN 16 (2017), 4815–4839. MR 3687117, DOI 10.1093/imrn/rnw148
Yuk-Kam Lau, Jianya Liu, and Jie Wu, Sign changes of the coefficients of automorphic $L$-functions, Number theory—arithmetic in Shangri-La, Ser. Number Theory Appl., vol. 8, World Sci. Publ., Hackensack, NJ, 2013, pp. 141–181. MR 3089015, DOI 10.1142/9789814452458_{0}009
Yuk-Kam Lau, Ming Ho Ng, Emmanuel Royer, and Yingnan Wang, A large sieve inequality of Elliott-Montgomery-Vaughan type for Maass forms on $\textrm {GL}(n,\Bbb R)$ with applications, Rev. Mat. Iberoam. 37 (2021), no. 4, 1539–1552. MR 4269408, DOI 10.4171/rmi/1238
Y.-K. Lau, M. Ho Ng, and Y. Wang, Average bound toward the generalized Ramanujan conjecture and its applications on Sato-Tate laws for $\mathrm {GL}(n)$, IMRN (2020).
Yuk-kam Lau, Emmanuel Royer, and Jie Wu, Sign of Fourier coefficients of modular forms of half-integral weight, Mathematika 62 (2016), no. 3, 866–883. MR 3521358, DOI 10.1112/S0025579316000103
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. MR 2832823, DOI 10.1016/j.jnt.2011.05.014
Y.-K. Lau and J. Wu, The number of Hecke eigenvalues of same signs, Math. Z. 263 (2009), no. 4, 959–970. MR 2551607, DOI 10.1007/s00209-008-0448-9
Stephen Lester, Kaisa Matomäki, and Maksym Radziwiłł, Small scale distribution of zeros and mass of modular forms, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 7, 1595–1627. MR 3807308, DOI 10.4171/JEMS/794
Stephen Lester and Maksym Radziwiłł, Signs of Fourier coefficients of half-integral weight modular forms, Math. Ann. 379 (2021), no. 3-4, 1553–1604. MR 4238273, DOI 10.1007/s00208-020-02123-0
Yongxiao Lin, Ramon Nunes, and Zhi Qi, Strong subconvexity for self-dual $\rm GL(3)$ $L$-functions, Int. Math. Res. Not. IMRN 13 (2023), 11453–11470. MR 4609788, DOI 10.1093/imrn/rnac153
Jianya Liu, Yan Qu, and Jie Wu, Two Linnik-type problems for automorphic $L$-functions, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 2, 219–227. MR 2823131, DOI 10.1017/S0305004111000314
Jianya Liu, Yonghui Wang, and Yangbo Ye, A proof of Selberg’s orthogonality for automorphic $L$-functions, Manuscripta Math. 118 (2005), no. 2, 135–149. MR 2177681, DOI 10.1007/s00229-005-0563-4
Jianya Liu and Jie Wu, The number of coefficients of automorphic $L$-functions for $GL_m$ of same signs, J. Number Theory 148 (2015), 429–450. MR 3283188, DOI 10.1016/j.jnt.2014.09.020
Kaisa Matomäki, On signs of Fourier coefficients of cusp forms, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 2, 207–222. MR 2887873, DOI 10.1017/S030500411100034X
Kaisa Matomäki and Maksym Radziwiłł, Sign changes of Hecke eigenvalues, Geom. Funct. Anal. 25 (2015), no. 6, 1937–1955. MR 3432161, DOI 10.1007/s00039-015-0350-7
Kaisa Matomäki and Maksym Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056. MR 3488742, DOI 10.4007/annals.2016.183.3.6
Jasmin Matz and Nicolas Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on $\textrm {SL}(n, \Bbb R )\slash \textrm {SO}(n)$, Algebra Number Theory 15 (2021), no. 6, 1343–1428. MR 4324829, DOI 10.2140/ant.2021.15.1343
Jaban Meher and M. Ram Murty, Oscillations of coefficients of Dirichlet series attached to automorphic forms, Proc. Amer. Math. Soc. 145 (2017), no. 2, 563–575. MR 3577861, DOI 10.1090/proc/13264
M. Ram Murty and Kaneenika Sinha, Effective equidistribution of eigenvalues of Hecke operators, J. Number Theory 129 (2009), no. 3, 681–714. MR 2488597, DOI 10.1016/j.jnt.2008.10.010
Yan Qu, Linnik-type problems for automorphic $L$-functions, J. Number Theory 130 (2010), no. 3, 786–802. MR 2584855, DOI 10.1016/j.jnt.2009.08.015
Dinakar Ramakrishnan, An exercise concerning the selfdual cusp forms on $\textrm {GL}(3)$, Indian J. Pure Appl. Math. 45 (2014), no. 5, 777–785. MR 3286086, DOI 10.1007/s13226-014-0088-1
B. Saffari and R. C. Vaughan, On the fractional parts of $x/n$ and related sequences. III, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 2, v, 31–36 (English, with French summary). MR 480390
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. MR 1018385
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Takuro Shintani, On an explicit formula for class-$1$ “Whittaker functions” on $GL_{n}$ over $P$-adic fields, Proc. Japan Acad. 52 (1976), no. 4, 180–182. MR 407208
A. Steiger, Some aspects of families of cusp forms, PhD. thesis, ETH Zürich, 2014.
Xuanxuan Xiao and Zhao Xu, A large sieve inequality of Elliott-Montgomery-Vaughan type for automorphic forms on $\textrm {GL}_3$, Rev. Mat. Iberoam. 35 (2019), no. 6, 1693–1714. MR 4029780, DOI 10.4171/rmi/1098
Fan Zhou, Weighted Sato-Tate vertical distribution of the Satake parameter of Maass forms on $\textrm {PGL}(N)$, Ramanujan J. 35 (2014), no. 3, 405–425. MR 3274875, DOI 10.1007/s11139-013-9535-6