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SIGN OF FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT

Part of: Discontinuous groups and automorphic forms Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  17 May 2016

Yuk-kam Lau ,
Emmanuel Royer  and
Jie Wu
Yuk-kam Lau
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong email yklau@maths.hku.hk
Emmanuel Royer
Affiliation:
Université Clermont Auvergne, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand, France CNRS, UMR 6620, LM, F-63178 Aubière, France email emmanuel.royer@math.univ-bpclermont.fr
Jie Wu
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, F-54506 Vandœuvre-lès-Nancy, France Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, F-54506 Vandœuvre-lès-Nancy, France email jie.wu@univ-lorraine.fr
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Abstract

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11M41: Other Dirichlet series and zeta functions 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 866 - 883
Copyright
Copyright © University College London 2016 

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References

Blomer, V., Harcos, G. and Michel, P., A Burgess-like subconvex bound for twisted L-functions. Forum Math. 19(1) 2007, 61–105, Appendix 2 by Z. Mao; MR 2296066 (2008i:11067).Google Scholar
Bruinier, J. H. and Kohnen, W., Sign changes of coefficients of half integral weight modular forms. In Modular Forms on Schiermonnikoog, Cambridge University Press (Cambridge, 2008), 57–65; MR 2512356 (2010k:11072).CrossRefGoogle Scholar
Conrey, J. B. and Iwaniec, H., The cubic moment of central values of automorphic L-functions. Ann. of Math. (2) 151(3) 2000, 1175–1216; MR 1779567 (2001g:11070).Google Scholar
Hulse, T. A., Kiral, E. M., Kuan, C. I. and Lim, L.-M., The sign of Fourier coefficients of half-integral weight cusp forms. Int. J. Number Theory 8(3) 2012, 749–762; MR 2904928.Google Scholar
Inam, I. and Wiese, G., Equidistribution of signs for modular eigenforms of half integral weight. Arch. Math. (Basel) 101(4) 2013, 331–339; MR 3116654.CrossRefGoogle Scholar
Inam, I. and Wiese, G., A short note on the Bruinier–Kohnen sign equidistribution conjecture and Halász’ theorem. Int. J. Number Theory 12 2016, 357–360, doi:10.1142/S1793042116500214; MR 3461436.Google Scholar
Iwaniec, H., Topics in Classical Automorphic Forms (Graduate Studies in Mathematics 17 ), American Mathematical Society (Providence, RI, 1997); MR 1474964 (98e:11051).Google Scholar
Iwaniec, H., Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics 53 ), 2nd edn., American Mathematical Society and Revista Matemática Iberoamericana (Providence, RI and Madrid, 2002); MR 1942691 (2003k:11085).Google Scholar
Kohnen, W., A short note on Fourier coefficients of half-integral weight modular forms. Int. J. Number Theory 6(6) 2010, 1255–1259; MR 2726580 (2011i:11070).Google Scholar
Kohnen, W., Lau, Y.-K. and Wu, J., Fourier coefficients of cusp forms of half-integral weight. Math. Z. 273(1–2) 2013, 29–41; MR 3010150.Google Scholar
Kohnen, W. and Zagier, D., Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64 1981, 175–198, doi:10.1007/BF01389166; MR 629468.CrossRefGoogle Scholar
Kubota, T., Elementary Theory of Eisenstein Series (Kodansha Scientific Books), John Wiley & Sons (1973); MR 0429749 (55 #2759).Google Scholar
Niwa, S., Modular forms of half integral weight and the integral of certain theta-functions. Nagoya Math. J. 56 1975, 147–161; MR 0364106 (51 #361).Google Scholar
Ono, K., The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series (CBMS Regional Conference Series in Mathematics 102 ), American Mathematical Society (Providence, RI, 2004); MR 2020489 (2005c:11053).Google Scholar
Shimura, G., On modular forms of half integral weight. Ann. of Math. (2) 97 1973, 440–481; MR 0332663 (48 #10989).Google Scholar
Soundararajan, K., Smooth numbers in short intervals, Preprint, 2010, arXiv:1009.1591 [math.NT].Google Scholar
Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory (Cambridge Studies in Advanced Mathematics 46 ), Cambridge University Press (Cambridge, 1995); translated from the second French edition 1995 by C. B. Thomas; MR 1342300 (97e:11005b).Google Scholar
Waldspurger, J.-L., Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9) 60(4) 1981, 375–484; MR 646366 (83h:10061).Google Scholar