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The eigencurve $\mathcal{E}$ is a rigid-analytic space parametrizing certain $p$-adic families of modular forms and associated Galois representations. By constructing an auxiliary reduced rigid curve locally isomorphic to $\mathcal{E}$ it was shown that $\mathcal{E}$ is a curve. There is a morphism of rigid spaces $f: \mathcal{E} \to \mathcal{W}$ assigning to a modular form its weight-character. In general this is not a proper morphism of rigid spaces, but a "valuative criterion" was proven in some cases.

As it is a "moduli space," one should be able to deduce certain properties about $p$-adic modular forms from the geometric structure of $\mathcal{E}$. What are some potential number-theoretic consequences of the "properness" of $f: \mathcal{E} \to \mathcal{W}$? How about other unknown geometric properties?

I hope you can explain what this object is about for non-experts like me. Thank you!

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The eigencurve is an honest moduli space---it parametrises families of finite slope overconvergent modular eigenforms (or more precisely, of systems of overconvergent finite slope Hecke eigenvalues)---but I know of no "natural" properties of p-adic modular forms that one can deduce from any geometric structure, other than e.g. the consequences of the fact that the eigencurve is 1-dimensional.

Properness just means that you can't have a family of overconvergent modular forms of finite slope that degenerates to an infinite slope form, which somehow isn't too surprising if you think about what the associated $(\phi,\Gamma)$-modules might look like. I think Alex Paulin has been thinking about making this train of thought rigorous. The fact that the eigencurve is 1-dimensional can be regarded as a statement about deformations of Galois representations---an overconvergent finite slope eigenform gives rise to a Galois representation with a crystalline period and one can now consider deforming such things and I guess one will end up proving that a certain $H^1$ is 1-dimensional.

As for "unknown geometric properties", I don't know of any reasonable conjectural statements about the eigencurve that have not already been conjectured. I made the observation that near the boundary of weight space the eigencurve seemed to be a disjoint union of annuli; Kilford and I proved a special case of this and Roe proved another one---and this has classical consequences for slopes of the $U_p$ operator. For example the eigenvalues of $U_2$ on weight $k$ level $\Gamma_1(4)$ modular forms, when $k$ is odd, have $2$-adic valuations which form an arithmetic progression (whatever the value of $k$). But I think this is nothing more than a curiosity at the moment.

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