Eisenstein series, p-adic modular functions, and overconvergence

2020/10/03 by Kiming, Ian, Rustom, Nadim
#11F33 #11F85 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.01325

Abstract

Let p be a prime ≥ 5. We establish explicit rates of overconvergence for 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 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.

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