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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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We do not know so much about the geometry of eigenvarieties. We do not even know if the eigenvarieties has finitely many irreducible components or whether the eigenvarieties are proper over the weight space, in the sense of non existence of hole, but there is an interesting preprint of S.Hattori '' On a properness of the Hilbert eigenvariety at integral weights: the case of quadratic residue fields''. – Adel Lille Jan 28 at 19:08
up vote 11 down vote accepted

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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