A Sunday question for specialists of eigenverieties:
In their important paper "the eigencurve", Coleman and Mazur globalized the earlier construction of Coleman's families, constructing a beautiful eponymous rigid analytic space that parametrizes all systems of Hecke eigenvalues of finite-slope overconvergent modular forms. Since them many generalizations of this construction have been performed (the "eigenvarieties") which each time appear as globalizations of local constructions generalizing Coleman's families. This process of globalization has even been axiomatized by Buzzard ("the eigenvariety machine").
Yet, I wonder:
What are the benefits of working with a global object (which is considerably more difficult to construct and to deal with) rather than just the local objects with which it is constructed (the families of Coleman and their generalizations) ?
Of course, having a global, canonical, object is much more satisfying on esthetic grounds. As a mathematician formed after Grothendieck's revolution, this reason alone would be for me a sufficient one to consent the effort of constructing global eigenvarieties. But my question is meant to be understood a little bit more specifically:
What are the applications, or expected applications (to our knowledge of the arithmetic of automorphic forms, Galois representations, L-functions, etc.) of the global existence and geometry of eigenverities that are not already consequences of the existence and geometry of their local pieces?
Of course, there are already an enormous amount, still growing fast, of arithmetic informations obtained from the local pieces of eigenvarieties. But what for the global structure? Let me mention the only one I know: the global existence of the eigencurve (say) is necessary to be able to attached to any overconvergent finite slope modular form a Galois representation. With Coleman's families alone, we could construct those representations only for these forms having a weight sufficiently close p-adically to a non-negative integer (for example the one with negative weights). Yet I find this application not very convincing, as why do we care about overconvergent form with weight far away from integers except for their being the "flesh" of the eigencurve?
So what other applications do you have in mind?
(edited for one typo)