6. Stating the modularity lifting theorems
I think that a nice and accessible goal (which will maybe take a month or two) would be to state the modularity lifting theorems which we'll be formalising. There are in fact two; one (the "minimal case") is proved using an extension of the original Taylor--Wiles techniques, and the other is deduced from it using various more modern tricks which were developed later. This chapter (currently work in progress) will contain a detailed discussion of all the things involved in the statement of the theorem.
6.1. Automorphic forms and analysis
Modular forms were historically the first nontrivial examples of automorphic
forms, but by the 1950s or so it was realised that they were special cases of
a very general notion of an automorphic form, as were Dirichlet characters!
Modular forms are holomorphic automorphic forms for the group \GL_2/\Q, and
Dirichlet characters are automorphic forms for the group \GL_1/\Q. It's
possible to make sense of the notion of an automorphic form for the group
G/k. Here k is a "global field" -- that is, a field which is either a
finite extension of \Q (a number field) or a finite extension of
(\Z/p\Z)(T) (a function field), and G is a connected reductive group
variety over k.
The reason that the definition of a modular form involves some analysis (they
are holomorphic functions) is that if you quotient out the group \GL_2(\R)
by its centre and the maximal compact subgroup O_2(\R), you get something
which can be naturally identified with the upper half plane, a symmetric space
with lots of interesting differential operators associated to it (for example a
Casimir operator). However if you do the same thing with \GL_1(\R) then you
get a one point set, which is why a Dirichlet character is just a combinatorial
object; it's a group homomorphism (\Z/N\Z)^\times\to\bbC^\times where N
is some positive integer. It turns out that there are many other connected
reductive groups where the associated symmetric space is 0-dimensional, and
in these cases the definition of an automorphic form is again combinatorial. An
example would be the group variety associated to the units of a totally
definite quaternion algebra over a totally real field. In this case, the
analogue of \GL_2(\R) would be the units \bbH^\times in the Hamilton
quaternions, a maximal compact subgroup would be the quaternions of norm 1
(homeomorphic to the 3-sphere S^3) and quotienting out \bbH^\times by
its centre \R^\times and S^3 again just gives you 1 point.
Before we talk about quaternion algebras, let's talk about central simple algebras.
6.2. Central simple algebras
Convention: in this section, fields are commutative, but algebras over a field may not be.
Recall that a central simple algebra over a field K is a nonzero
K-algebra D such that K is the centre of D and that D has no
nontrivial two-sided ideals.
Another way of saying that D has no nontrivial two-sided ideals: every
surjective ring homomorphism D\twoheadrightarrow A to any ring A is
either an isomorphism, or the zero map to the zero ring. Note that this latter
condition has nothing to do with K.
If n\geq1 then the n\times n matrices M_n(K) are a central simple
algebra over K.
We prove more generally that matrices with coefficients in K and indexed by
an arbitrary nonempty finite type are a central simple algebra over K.
They are clearly an algebra over K, with K embedded via scalar matrices as
usual (the injectivity of the map from K comes from nonemptiness of the finite
index type). The centre clearly contains K; to show that it equals K, we
argue as follows. Let e(i,j) be the matrix with a 1 in the ith row and
jth column, and zeros everywhere else. An element
Z=(Z_{s,t})_{s,t} of the centre commutes with all matrices e(i,j) for
i\not=j and these equations immediately imply that Z_{i,j}=0 if
i\not=j and that Z_{i,i}=Z_{j,j}.
It suffices to prove that any nonzero two-sided ideal I is all of
M_n(K). So say 0\not=M\in I and let us fix (i,j) such that
M_{i,j}\not=0. One easily checks that
M_{i,j} \mathrm{id} = \sum_k e(k,i)\times M\times e(j,k)\in I, where
\mathrm{id}\in M_n(K) is the identity matrix. Therefore,
\mathrm{id}\in I, so I=M_n(K).
The definition also requires that the ring be nonzero, but this follows from the index type being nonempty.
If D is a central simple algebra over K and L/K is a field extension,
then L\otimes_KD is a central simple algebra over L.
This is not too hard: it's lemma b of section 12.4 in Peirce's
"Associative algebras".
Next: define trace and norm.