A note on small gaps between nonzero Fourier coefficients of cusp forms
HTML articles powered by AMS MathViewer
- by Soumya Das and Satadal Ganguly
- Proc. Amer. Math. Soc. 144 (2016), 2301-2305
- DOI: https://doi.org/10.1090/proc/12887
- Published electronically: October 1, 2015
- PDF | Request permission
Abstract:
It is shown that there are infinitely many primitive cusp forms $f$ of weight $2$ with the property that for all $X$ large enough, every interval $(X, X+cX^{1/4})$, where $c>0$ depends only on the form, contains an integer $n$ such that the $n$-th Fourier coefficient of $f$ is nonzero.References
- Emre Alkan, Nonvanishing of Fourier coefficients of modular forms, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1673–1680. MR 1953571, DOI 10.1090/S0002-9939-02-06758-8
- Emre Alkan, On the sizes of gaps in the Fourier expansion of modular forms, Canad. J. Math. 57 (2005), no. 3, 449–470. MR 2134398, DOI 10.4153/CJM-2005-019-7
- Emre Alkan and Alexandru Zaharescu, On the gaps in the Fourier expansion of cusp forms, Ramanujan J. 16 (2008), no. 1, 41–52. MR 2407238, DOI 10.1007/s11139-007-9091-z
- Antal Balog and Ken Ono, The Chebotarev density theorem in short intervals and some questions of Serre, J. Number Theory 91 (2001), no. 2, 356–371. MR 1876282, DOI 10.1006/jnth.2001.2694
- R. P. Bambah and S. Chowla, On numbers which can be expressed as a sum of two squares, Proc. Nat. Inst. Sci. India 13 (1947), 101–103. MR 22879
- Soumya Das and Satadal Ganguly, Gaps between nonzero Fourier coefficients of cusp forms, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3747–3755. MR 3251716, DOI 10.1090/S0002-9939-2014-12164-2
- Fred Diamond and John Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR 1357209
- Noam D. Elkies, Distribution of supersingular primes, Astérisque 198-200 (1991), 127–132 (1992). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144318
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- Kazuyuki Hatada, Eigenvalues of Hecke operators on $\textrm {SL}(2,\,\textbf {Z})$, Math. Ann. 239 (1979), no. 1, 75–96. MR 516060, DOI 10.1007/BF01420494
- Emmanuel Kowalski, Olivier Robert, and Jie Wu, Small gaps in coefficients of $L$-functions and $\mathfrak {B}$-free numbers in short intervals, Rev. Mat. Iberoam. 23 (2007), no. 1, 281–326. MR 2351136, DOI 10.4171/RMI/496
- Penny C. Ridgdill, On the frequency of finitely anomalous elliptic curves, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of Massachusetts Amherst. MR 2941460
- Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Bibliographic Information
- Soumya Das
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
- Email: soumya.u2k@gmail.com
- Satadal Ganguly
- Affiliation: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India
- Email: sgisical@gmail.com
- Received by editor(s): January 25, 2015
- Received by editor(s) in revised form: June 29, 2015
- Published electronically: October 1, 2015
- Communicated by: Kathrin Bringmann
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2301-2305
- MSC (2010): Primary 11F30; Secondary 11F11, 11G05
- DOI: https://doi.org/10.1090/proc/12887
- MathSciNet review: 3477047