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arXiv · 1708.03663

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

John Bergdall
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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 Gouv\^ea--Mazur conjecture.

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John Bergdall. 2017-08-11. Upper bounds for constant slope $p$-adic families of modular forms. https://doi.org/10.1007/s00029-019-0505-8

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