A note on small gaps between nonzero Fourier coefficients of cusp forms

Das, Soumya; Ganguly, Satadal

doi:10.1090/proc/12887

A note on small gaps between nonzero Fourier coefficients of cusp forms
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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

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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