Overconvergent Modular Forms
The overconvergent modular forms are a large subset of the p-adic modular forms on which the U_p operator acts compactly.
Overconvergent modular forms of weight 0 are defined to be sections of the structure sheaf of (small) affinoid neighbourhoods of the connected component of the ordinary locus of X_0(p^m) containing the cusp infinity. They form a Banach space.
We can define forms of higher weight and nontrivial character by multiplication by a suitable Eisenstein series.
There are Hecke operators~$T_l$ and~$U_p$ which act on these forms in the normal way.