Eisenstein series, p-adic modular functions, and overconvergence

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