Eisenstein series, $p$-adic modular functions, and overconvergence, II

TL;DR

This paper extends previous work on overconvergence rates of p-adic modular functions involving Eisenstein series, providing explicit bounds for primes 2 and 3, and unifying the method across all primes.

Contribution

It offers explicit overconvergence bounds for Eisenstein-based modular functions at small primes and introduces a uniform approach applicable to all primes.

Findings

Improved overconvergence bounds for primes 2 and 3.

Unified method for all primes p.

Numerical examples illustrating theoretical results.

Abstract

Let $p$ be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form $\frac{E_k^{\ast}}{V(E_k^{\ast})}$ where $E_k^{\ast}$ is a classical, normalized Eisenstein series on $\Gamma_0(p)$ and $V$ the $p$-adic Frobenius operator. In particular, we extend our previous paper to the primes $2$ and $3$. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes $p\ge 5$. Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples.