MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

12 TOBY GEE
want to prove that ρ is modular. For this to be true, we will have to assume
some conditions. Firstly:
◦ ρ is unramified at almost all primes, and is de Rham at places
dividing p, with distinct Hodge-Tate weights.
◦ ρ is odd, i.e. det ρ(cv) = −1 for any complex conjugation cv.
These conditions are conjecturally enough. However, we can’t prove a theorem
anything like as strong as this. There are a number of assumptions
that we need to make that appear to be intrinsic to the method.
◦ ρ is modular, say ρ ∼ = ¯rp(π).
◦ ρ|G F (ζp) is irreducible.
◦ π has the same weight as ρ, in a sense that we will make precise
below.
There is another assumption that we will make that can be considerably
relaxed, but not as yet removed.
◦ If v|p is a place of F , then Fv/Qp is unramified, and the Hodge-Tate
weights of ρ are all {0, k − 1} with 0 ≤ k ≤ p − 1. Furthermore πv
is unramified for all v|p, and π has weight (k, 0).
We begin by making a number of simplifications. Firstly, we use Grothendieck’s
monodromy theorem, local-global compatibility, and solvable base change,
which together show that we may assume that:
◦ At every finite place v ∤ p of F , ρ|GFv is either unramified, or has
unipotent ramification.
◦ Each πv is either unramified or is Sp2(χ) with χ unramified.
Less obviously, but explained in Ken Ribet’s talk, we can assume that
◦ Each πv is unramified.
[Sketch: base change so that if v is a place with πv Steinberg, than Nv ≡ 1
(mod p). Then produce a congruence between the Steinberg and a ramified
principal series, and then make another base change to replace this with an
unramified principal series.] For the method we’re going to use, we will also
make a further base change and assume that:
◦ If v ∤ p is a place at which ρ|GFv is ramified, then Nv ≡ 1 (mod p),
and ρ(GFv) = 1.
We can finally also assume that det ρ = det rp(π); the ratio of these characters
is everywhere unramified and residually trivial, thus has finite p-power
order, and in particular is necessarily trivial on each complex conjugation,
so can be trivialised by base change.
By a standard argument using the Baire category theorem and the fact
that GF is compact, we may assume that ρ : GF → GL2(K), K/Qp finite.
Again using the compactness of GF , after conjugation we may assume that
ρ : GF → GL2(O), where O is the ring of integers of K, with maximal ideal λ
and residue field O/λ = k. We will sometimes assume that K is sufficiently