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Ramanujan-style congruences for prime level

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Abstract

We establish Ramanujan-style congruences modulo certain primes \(\ell \) between an Eisenstein series of weight k, prime level p and a cuspidal newform in the \(\varepsilon \)-eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for \(\Gamma _0(p)\). Under a mild assumption, this refines a result of Gaba–Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler–Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by \(\ell .\) The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

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Notes

  1. Some authors use y-smooth. Friable is an adjective meaning easily crumbled or broken.

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Acknowledgements

The authors would like to thank Shaunak Deo for several useful discussions, going through the paper carefully and giving useful suggestions. We also thank Jaban Meher for sharing his ideas in the proof of Theorem 1.4 and Dan Fretwell for his useful comments. Matteo Bordignon explained the authors how to compute \(\rho (u)\) using SAGE, Zhiwei Wang updated them on the literature on friable shifted primes and Nicolas Billerey and Ricardo Menares pointed out the existence of [3]. The third author thanks Younes Nikdelan for his meticulous proofreading of earlier versions and frequent discussions on his related preprint [29]. The authors thank the anonymous referees for a careful reading of the manuscript and for giving valuable suggestions. The research of the first author was supported by the grant no. 692854 provided by the European Research Council (ERC) while the second author was supported by Israeli Science Foundation grant 1400/19. This work was carried out when these two authors were postdoctoral fellows at Hebrew University of Jerusalem and Bar–Ilan University respectively. The open-source mathematics software SAGE (www.sagemath.org) has been used for the numerical computations in this work.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Jammu, Jagti NH-44, PO Nagrota, Jammu, 181221, India

    Arvind Kumar

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Moni Kumari & Pieter Moree

  3. School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Sujeet Kumar Singh

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  1. Arvind Kumar
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Kumar, A., Kumari, M., Moree, P. et al. Ramanujan-style congruences for prime level. Math. Z. 303, 19 (2023). https://doi.org/10.1007/s00209-022-03159-5

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