P-adic Banach spaces and families of modular forms

Spectral Expansions of Overconvergent Modular Functions

“…It has long been known that these objects satisfy many interesting congruence relations. One very powerful method for studying the congruences obeyed by modular forms modulo powers of a fixed prime p is to embed this space into the p-adic Banach space S k (Γ 1 (N ), r) of r-overconvergent p-adic cusp forms, defined as in [Kat73] using sections of ω ⊗k on certain affinoid subdomains of X 1 (N ) obtained by removing discs of radius p −r around the supersingular points; this space has been used to great effect by Coleman and others ([Col96,Col97]). …”

On the coefficients of the characteristic series of the U -operator

“…In other words, I give a conceptual proof of the part of theorem B6.1, when p is odd, which is evident from the explicit formulas (see appendix I of ref. 1) and which asserts that the coefficients of this series lie in the Iwasawa algebra ⌳ ϭ Z p ͓͓Z* p ͔͔. I also prove that this series analytically continues to a larger space.…”

Overconvergent modular forms

“…Dans [3], Buzzard a axiomatisé la construction de [5] et [6]. Sa "machine" permet d'associer une variété de Hecke à la donnée (A, M, H, U p ) à condition de vérifier que M satisfait une hypothèse (Pr) que nous allons rappeler.…”