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On signs of Fourier coefficients of cusp forms

Published online by Cambridge University Press:  05 December 2011

KAISA MATOMÄKI
KAISA MATOMÄKI*
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland. e-mail: [email protected]
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Abstract

We consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 2 , March 2012 , pp. 207 - 222
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Barnet–Lamb, T., Geraghty, D., Harris, M. and Taylor, R.A family of Calabi–Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.CrossRefGoogle Scholar
[2]Chong, K. M. and Rice, N. M.Equimeasurable rearrangements of functions. Queen's papers in pure and applied mathematics (Queen's University, Kingston, 1971).Google Scholar
[3]Cogdell, J. and Michel, P.On the complex moments of symmetric power L-functions at s = 1. Int. Math. Res. Not. 31 (2004), 1561–1617.CrossRefGoogle Scholar
[4]Duke, W. and Kowalski, E.A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. Invent. Math. 139 (1) (2000), 1–39. With an appendix by Dinakar Ramakrishnan.CrossRefGoogle Scholar
[5]Gelbart, S. and Jacquet, H.A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. École Norm. Sup. (4) 11 (4) (1978), 471–542.CrossRefGoogle Scholar
[6]Granville, A. and Soundararajan, K.The spectrum of multiplicative functions. Ann. Math. 153 (2001), 407–470.CrossRefGoogle Scholar
[7]Iwaniec, H., Kohnen, W. and Sengupta, J.The first negative Hecke eigenvalue. Int. J. Number Theory 3 (2007), 355–363.CrossRefGoogle Scholar
[8]Iwaniec, H. and Kowalski, E.Analytic number theory. American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, Rhode Island, 2004).Google Scholar
[9]Kim, H. H.Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2. J. Amer. Math. Soc. 16 (2003), 139–183. With appendices by Ramakrishnan, D. and by Kim, H. H. and Sarnak, P..Google Scholar
[10]Kim, H. H. and Shahidi, F.Cuspidality of symmetric powers with applications. Duke Math. J. 112 (1) (2002), 177–197.CrossRefGoogle Scholar
[11]Kim, H. H. and Shahidi, F.Functorial products for GL2 × GL3 and the symmetric cube for GL2. Ann. Math. (2) 155 (3) (2002), 837–893. With an appendix by Bushnell, Colin J. and Henniart, Guy.CrossRefGoogle Scholar
[12]Kohnen, W. and Sengupta, J.On the first sign change of Hecke eigenvalues of newforms. Math. Z. 254 (1) (2006), 173–184.CrossRefGoogle Scholar
[13]Kohnen, W. and Sengupta, J.Signs of Fourier coefficients of two cusp forms of different weights. Proc. Amer. Math. Soc. 137 (11) (2009), 3563–3567.CrossRefGoogle Scholar
[14]Kowalski, E., Lau, Y.-K., Soundararajan, K. and Wu, J.On modular signs. Math. Proc. Camb. Phil. Soc. 149 (3) (2010), 389–411.CrossRefGoogle Scholar
[15]Lau, Y.-K., Liu, J.-Y. and Wu, J. The first negative coefficient of symmetric square L-functions. Ramanujan J., to appear.Google Scholar
[16]Matomäki, K. A note on signs of Kloosterman sums. Bull. Soc. Math. France, to appear.Google Scholar
[17]Michel, P. and Venkatesh, A.The subconvexity problem for GL2. Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271.CrossRefGoogle Scholar
[18]Mœglin, C. and Waldspurger, J.-L.Le spectre résiduel de GL(n). Ann. Sci. École Norm. Sup. (4) 22 (4) (1989), 605–674.CrossRefGoogle Scholar
[19]Ramakrishnan, D.Modularity of the Rankin–Selberg L-series, and multiplicity one for SL(2). Ann. Math (2) 152 (2000), 45–111.CrossRefGoogle Scholar