Spaces of p-adic modular forms enlarge and interpolate spaces of classical modular forms, and have been a useful tool since their introduction by Serre in the early 1970’s. An important question in this theory is how to recognize classical modular forms in the vast sea of p-adic modular forms. For example, for GL(2), a p-adic modular eigenform whose Hecke eigenvalues agrees with those of a classical eigenform is in fact a classical eigenform. For SL(2), Judith Ludwig discovered, by an existence proof, that this need not be the case. The goal of my talk will be to explain how to understand and quantify this phenomenon using the geometry of moduli spaces of Galois representation. This is joint work in progress with Judith Ludwig.