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On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

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Abstract

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).

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Acknowledgments

The author would like to thank the referee for pointing out the references which are relevant to this problem and also for his/her suggestions which improved the presentation of the article. The author would also like to thank Dr. Satadal Ganguly for his suggestions during the preparation of this article. The basic idea of this article stems from the work of Das and Ganguly [12] and Rasmussen’s thesis [19] which provided us a way to look for the examples that we need.

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  1. Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, India

    Narasimha Kumar

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  1. Narasimha Kumar
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Correspondence to Narasimha Kumar.

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This work was supported by IIT Hyderabad through Institute’s startup research grant.

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Kumar, N. On the gaps between non-zero Fourier coefficients of cusp forms of higher weight. Ramanujan J 45, 95–109 (2018). https://doi.org/10.1007/s11139-016-9837-6

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  • DOI: https://doi.org/10.1007/s11139-016-9837-6

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