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

TL;DR

This paper establishes upper bounds on constant slope $p$-adic families of modular forms, linking their size to properties of the $p$-th Hecke eigenvalue and slope, and explores their relation to the Gouv extsuperscript{e}a--Mazur conjecture.

Contribution

It provides two novel upper bounds for constant slope $p$-adic families, one based on the logarithmic derivative of $a_p$ and another on the slope, advancing understanding of their structure.

Findings

Derived upper bounds in terms of logarithmic derivatives and slopes.

Analyzed the numerical relationship with the Gouv extsuperscript{e}a--Mazur conjecture.

Enhanced bounds contribute to the theory of $p$-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 Gouv\^ea--Mazur conjecture.