Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) that would help the novice in better
understanding what's going on?
Here's what I have in mind. Let
$\rho: G_{\mathbb Q} \longrightarrow {\mathbb C}^\times$
be a 1-dimensional representation of the absolute Galois group
of the rationals factoring over some finite extension. Given a
Dirichlet character
$\chi: GL_1({\mathbb Z}/N{\mathbb Z}) \longrightarrow {\mathbb C}^\times$,
we can find representations
$\rho_\chi: Gal(K/{\mathbb Q}) \longrightarrow {\mathbb C}^\times$
for any cyclotomic extension $K = {\mathbb Q}(\zeta_N)$.
Call $\rho$ modular if there is a $\chi$ such that $\rho = \rho_\chi$.
The statement that every $\rho$ coming from an abelian extension is
modular is the theorem of Kronecker-Weber, and in this form it can be
proved using Galois deformations along the lines of Wiles' proof
(see Tunnell's proof in Kowalski's notes).
BTW if anyone knows a source for this result that is more readable
than Kowalski's notes (which I discovered just a couple of days
ago and haven't studied in detail yet) I'm all ears.
Question: Are there other similarly "elementary" questions, for
example in embedding problems or inverse Galois theory, that can
be described in terms of Galois deformations?
Best Answer
Late answer in continuation to #4: if I remember right, at first Tunnell thought he would need Poitou-Tate to complete the Kronecker-Weber proof in this manner (which would make things rather unsatisfactory, as people have said), but in the end -- for Kronecker-Weber only -- he saw that he only needed the local Kronecker-Weber theorem to complete the argument, which he then proved in the class (or maybe he just sketched the proof?).