# moduli space and modularity

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?

references:

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

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I like the questions at the end (and my ability to answer is limited to "Serre-Tate theorem" and "yes"), but I'm having trouble understanding the mathematics in the first paragraph. Aren't there essential things like level structures, Jacobians, and Hecke operators? –  S. Carnahan Jan 20 '10 at 6:06
yes, I was only being (too) brief. –  natura Jan 20 '10 at 7:25

Kisin's work is fairly technical, and is devoted to studying deformations of Galois representations which arise by taking $\overline{K}$-valued points of a finite flat group over $\mathcal O_K$ (where $K$ is a finite extension of $\mathbb Q_p$).

The subtlety of this concept is that when $K$ is ramified over $\mathbb Q_p$ (more precisely, when $e \geq p-1$, where $e$ is the ramification degree of $K$ over $\mathbb Q_p$), there can be more than one finite flat group scheme modelling a given Galois represenation. E.g. if $p = 2$ and $K = {\mathbb Q}\_2$ (so that $e = 1 = 2 - 1$), the trivial character with values in the finite field $\mathbb F_2$ has two finite flat models over $\mathbb Z_2$; the constant etale group scheme $\mathbb Z/2 \mathbb Z$, and the group scheme $\mu_2$ of 2nd roots of unity.

In general, as $e$ increases, there are more and more possible models. Kisin's work shows that they are in fact classified by a certain moduli space (the "moduli of finite flat group schemes" of the title). He is able to get some control over these moduli spaces, and hence prove new modularity lifting theorems; in particular, with this (and several other fantastic ideas) he is able to extend the Taylor--Wiles modularity lifting theorem to the context of arbitrary ramification at $p$, provided one restricts to a finite flat deformation problem. This result plays a key role in the proof of Serre's conjecture by Khare, Wintenberger, and Kisin.

The detailed geometry of the moduli spaces is controlled by some Grassmanian--type structures that are very similar to ones arising in the study of local models of Shimura varieties. However, there is not an immediately direct connection between the two situations.

EDIT: It might be worth remarking that, in the study of modularity of elliptic curves, the fact that the modular forms classifying elliptic curves over $\mathbb Q$ are themselves functions on the moduli space of elliptic curves is something of a coincidence.

One can already see this from the fact that lots of the other objects over $\mathbb Q$ that are not elliptic curves are also classified by modular forms, e.g. any abelian variety of $GL_2$-type.

When one studies more general instances of the Langlands correspondence, it becomes increasingly clear that these two roles of elliptic curves (providing the moduli space, and then being classified by modular forms which are functions on the moduli space) are independent of one another.

Of course, historically, it helped a lot that the same theory that was developed to study the Diophantine properties of elliptic curves was also available to study the Diophantine properties of the moduli spaces (which again turn out to be curves, though typically not elliptic curves) and their Jacobians (which are abelian varieties, and so can be studied by suitable generalizations of many of the tools developed in the study of elliptic curves). But this is a historical relationship between the two roles that elliptic curves play, not a mathematical one.

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Thank you! but what do you mean by the two roles that elliptic curves play? As for higher dimensional abelian varieties, the corresponding automorphic representation living in the cohomology of the Shimura variety (the moduli space of abelian varieties) is not a function on the moduli. Is that what you mean by "coincidence" (since it does not hold for higher-dimension case)? –  natura Jan 20 '10 at 7:34
I am pretty sure he means the following: consider the modular curve X_0(N). This plays not one but two roles in the theory of elliptic curves. One is that its non-cuspidal points parametrise (in some sense that can be made rigorous) elliptic curves with a cyclic subgroup of order N. That statement is geometric---for example it works over the complexes, over finite fields and so on. The other role X_0(N) plays is that if E is an elliptic curve over the rational numbers then there is a non-constant map X_0(N)-->E for some N (the conductor of the curve). That is an arithmetic statement. –  Kevin Buzzard Jan 20 '10 at 11:04
...and it's just a coincidence that both statements about elliptic curves. They have a very different flavour though: for example if E is an elliptic curve over the complexes, then it still gives a point on some X_0(N) if you choose a cyclic subgroup of order N, but you don't expect it to be the image of some X_0(N) under some holomorphic map (for example if the j-invariant is transcendental). –  Kevin Buzzard Jan 20 '10 at 11:05
Thanks Kevin; that's what I meant. –  Emerton Jan 20 '10 at 13:58