I recently realized some kind of analogy when considering modularity results (such as the modularity of elliptic curves over Q). The analogy comes from algebraic groups. Take one point (say, the origin) of an algebraic group, then something over the one point (for example, a tangent vector) can be extended to the whole algebraic group by group translations (so we get a translation invariant vector field). Now, modular curves are moduli spaces for elliptic curves. One elliptic curve is like one point on the moduli space, so probably things over that one point (for example, the first etale cohomology, which is the same as the Tate module Galois representation) could be extended to the whole modular curve, and indeed, that is the modular form, which also lives in some cohomology of the modular curve.
The modularity results have always been very mysterious and surprising for me, since it links two very "natural" and "intuitive" but "seems-far-away" objects together (the Tate module, and a modular form). But the above point of view might be a good reason for such things to be true.
This leads me to think more about the general theory of moduli spaces, which I don't know very much. It seems that there is a quite well developped theory of moduli spaces of curves with fixed genre, and also there is some special kind of moduli spaces such as Hilbert moduli space. There is also a very important moduli space in number theory,namely, moduli space of p-divisible groups, which recently has an important work by Mark Kisin. But it seems to be that the techniques and ideas used in Kisin's work (of course as well as in Breuil and other people's work) look quite different from that of traditional geometrical moduli spaces.
So my question is, can someone give some motivation about studying p-divisible groups and their moduli spaces? Also, is the study of such objects analogus to that of the traditional moduli spaces?
Breuil ''Groupes p-divisibles, groupes finis et modules filtrés'', Annals of Math. 151, 2000, 489-549. http://www.ihes.fr/~breuil/PUBLICATIONS/p-divisibles.pdf
Kisin, Moduli of finite flat group schemes and modularity -- Annals of Math. 170(3) (2009), 1085-1180. http://www.math.harvard.edu/~kisin/dvifiles/bt.dvi