Eisenstein series, p-adic modular functions, and overconvergence

Abstract

Let p be a prime $\ge 5$. We establish explicit rates of overconvergence for some members of the “Eisenstein family”, notably for the p-adic modular function $V(E_{(1,0)}^{*})/E_{(1,0)}^{*}$ (V the p-adic Frobenius operator) that plays a pivotal 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.