Abstract
Let \({{\mathfrak f}}\) be a cusp form of half integral weight whose Fourier coefficients \({{\mathfrak a}_{\mathfrak f}(n)}\) are all real. We study the sign change problem of \({{\mathfrak a}_{\mathfrak f}(n)}\) , when n runs over some specific sets of integers. Lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs. These give an improvement on some recent results of Bruinier and Kohnen (Modular forms on Schiermonnikoong. Cambridge University Press, Cambridge 57–66, 2008) and Kohnen (Int. J. Number. Theory 6:1255–1259, 2010).
Similar content being viewed by others
References
Atkin A., Lehner J.: Hecke operators on Γ0(m). Math. Ann. 185, 134–160 (1970)
Bruinier J.-H., Kohnen W. et al.: Sign changes of coefficients of half integral weight modular forms. In: Edixhoven, B. (eds) Modular forms on Schiermonnikoong., pp. 57–66. Cambridge University Press, Cambridge (2008)
Cipra B.A.: On the Niwa–Shintani theta-kernel lifting of modular forms. Nagoya Math. J. 91, 49–117 (1983)
Duke W., Iwaniec H.: Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes. Math. Ann. 286, 783–802 (1990)
Duke, W., 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, no. 1, 1–39 (2000)
Erdős P.: On the difference of consecutive terms of sequences, defined by divisibility properties. Acta Arith. 12, 175–182 (1966)
Iwaniec H., Kohnen W., Sengupta J.: The first sign change of Hecke eigenvalue. Int. J. Number. Theory 3(3), 355–363 (2007)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence. xii+615 (2004)
Kim H.H., Shahidi F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112(1), 177–197 (2002)
Knopp M., Kohnen W., Pribitkin W.: On the signs of Fourier coefficients of cusp forms. Rankin memorial issues. Ramanujan J. 7(1-3), 269–277 (2003)
Kohnen W.: Newforms of half-integral weight. J. Reine Angew. Math. 333, 32–72 (1982)
Kohnen W.: Fourier coefficients of modular forms of half integral weight. Math. Ann. 271, 237–268 (1985)
Kohnen W.: A short note on Fourier coefficients of half-integral weight modular forms. Int. J. Number. Theory 6, 1255–1259 (2010)
Kohnen W., Lau Y.-K., Shparlinski I.E.: On the number of sign changes of Hecke eigenvalues of newforms. J. Aust. Math. Soc. 85(1), 87–94 (2008)
Kohnen W., Sengupta J.: On the first sign change of Hecke eigenvalues of newforms. Math. Z. 254, 173–184 (2006)
Kohnen W., Zagier D.: Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64, 175–198 (1981)
Kowalski E., Lau Y.-K., Soundararajan K., Wu J.: On modular signs. Math. Proc. Camb. Phil. Soc 149, 389–411 (2010)
Lau Y.-K., Wu J.: The number of Hecke eigenvalues of same signs. Math. Z. 263, 957–970 (2009)
Lehmer D.H.: The vanishing of Ramanujan’s function τ(n). Duke Math. J. 14, 429–433 (1974)
Ramakrishnan, D.: Recovering modular forms from squares, Invent. Math. 139 (2000), no. 1, 29–39, Appendix of [5]
Serre J.-P.: Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math. 54, 323–401 (1981)
Shimura G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973)
Waldspurger J.-L.: Sur les coefficients de Fourier des forms modulaires de poids demi-entier. J. Math. Pures et Appl. 60, 375–484 (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kohnen, W., Lau, YK. & Wu, J. Fourier coefficients of cusp forms of half-integral weight. Math. Z. 273, 29–41 (2013). https://doi.org/10.1007/s00209-012-0994-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-0994-z
Keywords
- Forms of half-integer weight
- Fourier coefficients of automorphic forms
- \({\fancyscript{B}}\) -free numbers