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!