I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k_1 \equiv k_2 \pmod {p^{n}(p1)}$ 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 padic 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 padically close. But how can one hope to prove the above conjecture from such a padic 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?
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4$\begingroup$ Not an answer to your question, but you might be interested in the paper "A counterexample to the GouvêaMazur conjecture", by Buzzard and Calegari. $\endgroup$ – Laurent Berger Jul 25 '11 at 16:17

1$\begingroup$ By "the space of all modular forms" you probably mean the space of classical modular forms (which is actually much smaller than the space of overconvergent padic forms). The relation between the two is provided by Coleman's "low slope implies classical" theorems. See his papers "Classical and overconvergent modular forms" and "classical and overconvergent modular forms of higher level." $\endgroup$ – Ramsey Jul 25 '11 at 16:27

$\begingroup$ @ prof. Berger and Prof. Ramsey Thanks a lot for the comments. In fact I had looked at that paper by Buzzard and Calegari. I remember a result by Coleman where he shows that the GoveaMazur conjecture is true if $k_i = O(\sqrt{m})$. Is this the best possible bound known? $\endgroup$ – Arijit Jul 26 '11 at 1:45

$\begingroup$ I don't think Coleman's work gives any explicit bounds at all. The quadratic bound you mention comes from work of Daqing Wan, "Dimension variation of classical and padic modular forms". I don't know of any subsequent improvements on this. $\endgroup$ – David Loeffler Jul 26 '11 at 7:37

$\begingroup$ @Arijit: I think Wan's theorem is something like: the size of the slope $\alpha$ spaces will be the same if $k_1$ and $k_2$ are congruent mod $(p1)p^M$ with $M=O(\alpha^2)$. It's perhaps also worth remarking that the implied constant in the $O$ depends on the level that you're working at. The classical and overconvergent conjectures are basically the same now, after Coleman's classicality result. $\endgroup$ – Kevin Buzzard Jul 26 '11 at 17:54
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The distinction between the spaces of cusp forms and of all modular forms is not important for the GouveaMazur conjecture, since it's very easy to show that the Eisenstein series vary in padic families (and hence the dimension of the slope $\alpha$ subspace of the space of Eisenstein forms is trivially locally constant).

$\begingroup$ Thanks I completely forgot about Serre's result. That clears a lot of things. $\endgroup$ – Arijit Jul 26 '11 at 1:42