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Upper bounds for constant slope p-adic families of modular forms

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Abstract

We study p-adic families of eigenforms for which the p-th Hecke eigenvalue \(a_p\) has constant p-adic valuation (“constant slope families”). We prove two separate upper bounds for the size of such families. The first is in terms of the logarithmic derivative of \(a_p\) while the second depends only on the slope of the family. We also investigate the numerical relationship between our results and the former Gouvêa–Mazur conjecture.

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Notes

  1. We do not explain Pollack’s method here. It is generally based on combining the Mazur–Tate–Teitelbaum conjecture with calculations of p-adic L-functions. The reader may also be interested in the recent works [1, 29].

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Acknowledgements

The research reported on here was partially supported by NSF award DMS-1402005. The author thanks James Newton, Robert Pollack, Sandra Rozensztajn, and an anonymous referee for helpful discussions and comments. There are some computer calculations given below that were made by Pollack; we also thank him for access to this data. In addition, during the elaboration of this work we also benefited from short visits to, and the hospitality at, Imperial College (London), the Institut des Hautes Études Scientifiques (Bures-sur-Yvette), and the Max-Planck-Institut für Mathematik (Bonn). The staff and members of these institutions are duly thanked.

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  1. Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA, 19010, USA

    John Bergdall

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Bergdall, J. Upper bounds for constant slope p-adic families of modular forms. Sel. Math. New Ser. 25, 59 (2019). https://doi.org/10.1007/s00029-019-0505-8

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