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?
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4$\begingroup$ Not an answer to your question, but you might be interested in the paper "A counterexample to the Gouvêa-Mazur conjecture", by Buzzard and Calegari. $\endgroup$– Laurent BergerCommented Jul 25, 2011 at 16:17
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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 p-adic 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." $\endgroup$– RamseyCommented Jul 25, 2011 at 16:27
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$\begingroup$ @ 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 Govea-Mazur conjecture is true if $k_i = O(\sqrt{m})$. Is this the best possible bound known? $\endgroup$– ArijitCommented Jul 26, 2011 at 1:45
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$\begingroup$ 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 p-adic modular forms". I don't know of any subsequent improvements on this. $\endgroup$– David LoefflerCommented Jul 26, 2011 at 7:37
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$\begingroup$ @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 $(p-1)p^M$ 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 BuzzardCommented Jul 26, 2011 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 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).
answered Jul 25, 2011 at 17:47
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$\begingroup$ Thanks I completely forgot about Serre's result. That clears a lot of things. $\endgroup$– ArijitCommented Jul 26, 2011 at 1:42