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On modular signs

Published online by Cambridge University Press:  19 July 2010

E. KOWALSKI ,
Y.-K. LAU ,
K. SOUNDARARAJAN  and
J. WU
E. KOWALSKI
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland. e-mail: [email protected]
Y.-K. LAU
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. e-mail: [email protected]
K. SOUNDARARAJAN
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail: [email protected]
J. WU
Affiliation:
Institut Elie Cartan, Nancy-Université, INRIA, Boulevard des Aiguillettes, B.P. 239, 54506 Vandœuvre-lès-Nancy, France. e-mail: [email protected]
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Abstract

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 3 , November 2010 , pp. 389 - 411
Copyright
Copyright © Cambridge Philosophical Society 2010

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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, preprint (2009), available at http://www.math.harvard.edu/~rtaylor/Google Scholar
[2]Barton, J. T., Montgomery, H. L. and Vaaler, J. D.Note on a diophantine inequality in several variables, Proc. Amer. Math. Soc. 129 (2001), 337–345.CrossRefGoogle Scholar
[3]Duke, W. and Kowalski, E.A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, With an appendix by Dinakar Ramakrishnan. Invent. Math. 139 (2000), no. 1, 1–39.CrossRefGoogle Scholar
[4]Goldfeld, D. and Hoffstein, J. On the number of Fourier coefficients that determine a modular form, in: A tribute to Emil Grosswald: number theory and related analysis. Contemp. Math. 143 (Amer. Math. Soc., 1993), 385–393.Google Scholar
[5]Granville, A. and Soundararajan, K.The spectrum of multiplicative functions. Ann. of Math. 153 (2001), no. 2, 407–470.CrossRefGoogle Scholar
[6]Iwaniec, H., Kohnen, W. and Sengupta, J.The first negative Hecke eigenvalue. Internat. J. Number Theory 3 (2007), No. 3, 355–363.CrossRefGoogle Scholar
[7]Iwaniec, H. and Kowalski, E.Analytic Number Theory. American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, 2004), xii+615.Google Scholar
[8]Iwaniec, H., Luo, W. and Sarnak, P.Low-lying zeros of families of L-functions. Publ. Math. Inst. Hautes. Études. Sci. 91 (2000), 55–131.CrossRefGoogle Scholar
[9]Kowalski, E.Variants of recognition problems for modular forms. Arch. Math. (Basel) 84 (2005), no. 5, 57–70.CrossRefGoogle Scholar
[10]Kim, H. and Shahidi, F.Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), 177–197.CrossRefGoogle Scholar
[11]Lamzouri, Y. The two-dimensional distribution of values of ζ(1 + it). Internat. Math. Res. Notices, Vol. 2008, Article ID rnn106, 48 pages.CrossRefGoogle Scholar
[12]Lau, Y.–K. and Wu, J.On the least quadratic non-residue. Internat. J. Number Theory 4 (2008), No 3, 423–435.CrossRefGoogle Scholar
[13]Lau, Y.–K. and Wu, J.A large sieve inequality of Elliott-Montgomery-Vaughan type and two applications, IMRN, Vol. 2008, Number 5, Article ID rnm 162, 35 pages.Google Scholar
[14]Matomäki, K.On signs of Fourier coefficients of cusp forms, preprint (2010).Google Scholar
[15]Mazur, B.Finding meaning in error terms. Bull. Amer. Math. Soc. 45 (2008), 185–228.CrossRefGoogle Scholar
[16]Michel, P. and Venkatesh, A. The subconvexity problem for GL 2, arXiv:0903.3591v1.Google Scholar
[17]Murty, M. R.Congruences between modular forms, in: “Analytic number theory (Kyoto, 1996)”, London Math. Soc. Lecture Note Ser. 247 (Cambridge University Press, 1997), 309–320.Google Scholar
[18]Murty, V. K.On the Sato-Tate conjecture, in Progr. Math. 26 (1982), p. 195–205.CrossRefGoogle Scholar
[19]Ramakrishnan, D.Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000), no. 1, 45–111.CrossRefGoogle Scholar
[20]Royer, E.Facteurs ℚ-simples de J 0(N) de grande dimension et de grand rang. Bull. Soc. Math. France 128 (2000), 219–248.CrossRefGoogle Scholar
[21]Sarnak, P.Statistical properties of eigenvalues of the Hecke operators, in “Analytic Number Theory and Diophantine Problems” (Stillwater, OK, 1984), Progr. Math. 70Birkhäuser (1987), 321–331.Google Scholar
[22]Serre, J.-P.Répartition asymptotique des valeurs propres de l'opérateur de Hecke Tp J. Amer. Math. Soc. 10 (1997), 75–102.CrossRefGoogle Scholar
[23]Shahidi, F.Symmetric power L-functions for GL(2), in: “Elliptic curves and related topics”, edited by Kishilevsky, E. and Ram Murty, M., CRM Proc. and Lecture Notes 4, 1994, 159–182.Google Scholar
[24]Tenenbaum, G.Cribler les entiers sans grand facteur premier. Philos. Trans. Roy. Soc. London Ser. A 345 (1993), no. 1676, 377–384.Google Scholar
[25]Tenenbaum, G.Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics 46, (Cambridge University Press, 1995).Google Scholar
[26]Tenenbaum, G., in collaboration with J. Wu Exercices corrigés de théorie analytique et probabiliste des nombres, Cours spécialisés, n°2 (Société Mathématique de France 1996), xiv + 251 pp.Google Scholar
[27]Tenenbaum, G. and Wu, J.Moyennes de certaines fonctions multiplicatives sur les entiers friables. J. Reine Angew. Math. 564 (2003), 119–166.Google Scholar