I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k\_1 \equiv k\_2 \pmod {p^{n}(p-1)}$ for some integer $n\geq \alpha$. then $d(k\_1,\alpha)= d(k\_2,\alpha)$ where $d(k,\alpha)$ is the dimension of slope $\alpha$ subspace of $U\_p$ acting on cusp forms on weight k. One possible approach in proving this : one wants to embed the classical modular forms in p-adic modular forms. In a latter paper by Gouvea and Mazur "On the characteristic power series of U operator" they prove a certain continuity property of the U operator. In fact they show under the above hypothesis the coefficients of the power series of U acting on the spaces of overconvergent modular forms of weight $k\_1 ,k\_2$ are p-adically close. But how can one hope to prove the above conjecture from such a p-adic statement even if one knows that any overconvergent modular form of small slope is classical. Since the conjecture relates to cusp forms not the whole space of modular forms. My source of confusion is at the end of the paper "On the characteristic power series of U operator" they talk about the same conjecture but now they consider the space of all modular forms. Are these conjectures equivalent?



Gouvea-Mazur conjecture
The distinction between the spaces of cusp forms and of all modular forms is not important for the Gouvea-Mazur conjecture, since it's very easy to show that the Eisenstein series vary in p-adic families (and hence the dimension of the slope $\alpha$ subspace of the space of Eisenstein forms is trivially locally constant).


