I will talk about a series of works from recent years on the rate of overconvergence of certain modular functions that arise from classical Eisenstein series. The question arises because these rates have implications for the finer structure of the Coleman-Mazur eigencurve. Specifically, proving explicit bounds for the rates played a pivotal role in the proofs of "halo" type conjectures for p=2 and p=3 by Buzzard-Kilford and Roe, respectively. Those theorems lead Coleman to formulate a general conjecture for all primes p, and a recent (2025) theorem by myself together with B. Advocaat and G. Wiese proved a version of this conjecture. This new work builds on and uses previous work by myself together with N. Rustom.

My aim in this talk will primarily be to explain the content of these results. I will start the talk at a very basic level, talking about overconvergent modular forms and "Katz expansions" in very concrete terms that are accessible to anyone who knows about modular forms. At the end I might be able to sketch some of the ideas that go into the proofs.