Author's Final Manuscript
We study p-adic families of eigenforms for which the p-th Hecke eigenvalue 𝑎𝑝 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 𝑎𝑝 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.
Bergdall, John. 2019. "Upper bounds for constant slope p-adic families of modular forms." Selecta Mathematica 25.4.