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Journal Article

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

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Bergdall,  John
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1708.03663.pdf
(Preprint), 348KB

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Citation

Bergdall, J. (2019). Upper bounds for constant slope p-adic families of modular forms. Selecta Mathematica, 25(4): 59. doi:10.1007/s00029-019-0505-8.


Cite as: http://hdl.handle.net/21.11116/0000-0005-4334-5
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 Gouv\^ea--Mazur conjecture.