Let’s start today by recalling that every elliptic curve over is modular and every rigid Calabi-Yau threefold over is modular as well. This just means that the Galois representation that arises from the etale cohomology is the same as the Galois representation of a modular form. It has also been found that many, many other types of varieties are modular. This leads us to a major theme of number theory which is to make precise and prove some variant on the question: Is every representation that comes from geometry modular?
Serre’s conjecture (now proven) tells us that if we start with an absolutely irreducible odd two-dimensional residual representation then there is a modular form such that . So one approach to studying the above question would be to figure out which deformations of residual representations are modular.
Let’s put ourselves in this situation. We could get massively bogged down in the details at this point, so we’ll just set things up by analogy to the earlier posts on modular forms. Fix a DVR that is a finite extension of . Now we want to consider . We haven’t discussed , but it is similar to in that it is just matrices that are upper-triangular mod with the added property that the diagonal entries must be . Roughly, you can think of as modular forms where the coefficients of the -expansion are allowed to be in . Our other way works as well where we consider a moduli space of elliptic curves with level -structure and take the -valued points. As before, we actually only want to consider certain eigenforms so that the theory works out.
To any such we can attach (very non-obvious step here!!) a Galois representation where . Suppose is the maximal ideal of . To get to our residual representation we can just take the reduction mod followed by the semisimplification . Now we can finally say something interesting. Suppose and live in completely different places, for example by altering and . As long as the coefficients of the -expansion are the same mod it is easy to see that . Thus we can produce lots of different deformations over of a given residual representation.
Let’s just reiterate a few things from last time. Take our as above and suppose that the attached residual representation is absolutely irreducible. We know in this situation that the deformation functor is representable and by a Galois cohomology calculation (supposing unobstructedness) there is a universal deformation ring . In order for the theory to be flexible enough for such amazing conjectures as given in the first paragraph we need to expand the class of modular forms we consider. We’ll just say that we want to be able to define modular forms over -adically complete separated rings. It turns out that the classical modular forms are dense in this enlarged space and so bootstrapping off that we can make a universal modular deformation even when our original was not a classical modular form.
We won’t worry what this ring is. The take away from this is that even though we allowed ourselves to move away from classical modular forms, the universal modular form cuts out a subspace of the deformation space consisting of the -adic modular forms. Or in other words it is the Zariski closure of the set of classical modular forms. Since in many cases we can explicitly determine what this subspace is, we haven’t lost much by allowing ourselves this larger class of modular forms. In fact, by imposing extra deformation conditions it can sometimes restrict us to exactly classical modular forms, and this is exactly a key technique used to proved the Taniyama-Shimura conjecture.
Although all this is really interesting and fun, I’ve already skipped so many technicalities and complication that it would be pointless to drag this on much further. I think I’ll probably do one more post on this stuff since the main idea of what gets used in Taniyama-Shimura is in this post.