Eisenstein series, p-adic modular functions, and overconvergence

TL;DR

This paper establishes explicit overconvergence rates for p-adic Eisenstein families, enhancing understanding of p-adic modular functions and extending key results in Coleman's theory of p-adic modular forms.

Contribution

It provides explicit overconvergence rates for Eisenstein family members, generalizes Coleman–Wan's theorem, and applies findings to primes 5 and 7.

Findings

Explicit overconvergence rates for Eisenstein family members

Extension of Coleman–Wan's theorem on overconvergence

Applications to primes 5 and 7

Abstract

Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast generalization of a theorem of Coleman--Wan regarding the rate of overconvergence of $V(E_{p-1})/E_{p-1}$. We also comment on previous results in the literature. These include applications of our results for the primes $5$ and $7$.