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

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.

22 pages. 4 tables. Minor revision. Final version. To appear in Selecta Mathematica

References in corpus (3)

• The eigencurve over the boundary of weight space
• Arithmetic properties of Fredholm series for p-adic modular forms
• Slopes of modular forms and the ghost conjecture