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

Abstract

AbstractLet 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 $$\frac{E^*_k}{V(E^*_k)}$$ E k ∗ V ( E k ∗ ) where $$E^*_k$$ E k ∗ is a classical, normalized Eisenstein series on $$\Gamma _0(p)$$ Γ 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 $$p\ge 5$$ p ≥ 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.