MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

16 TOBY GEE
Define R� as above, except that we no longer require the deformations
?,QN
to be unramified at q ∈ QN. R� is still generated by u elements over
?,QN
Rloc ? , by another duality calculation.
�
Write ∆QN = q∈QN Sylp((Z/qZ) × ). For simplicity of notation, let’s suppose
that this is isomorphic to (Z/pNZ) u (again, the general case works in
the same way, but the notation is slightly more complicated). By the above,
∼= χα ⊕ χβ, and we have a homomor-
runiv |GQq is reducible, say runiv
?,QN ?,QN |GQq
phism Sylp((Z/qZ) × ) → (Runiv ?,QN )× induced by local class field theory and
χα|IQq . Taking the product over q ∈ QN, we have a map ∆QN → (Runiv
?,QN )× .
So R� ?,QN is a module over O[∆QN ], and if we let aN denote the augmentation
ideal of O[∆QN ], then R� ?,QN /aN = Runiv ? .
We now have to introduce modular forms into the picture. At this point
we will be rather brief, as to give the full details would take a prohibitive
amount of space. This is also the one place where the theory is better
developed over a totally real field other than Q; indeed, the optimal choice
of field is a field F with [F : Q] even. We then choose a division algebra
D with S(D) = ∞. We now define spaces of “algebraic modular forms” for
GD(A∞ ) ∼ = GL2(A∞ F ). These are an integral p-adic avatar of the spaces of
modular forms considered above, and we can use them to study congruences
between modular forms, and in particular to relate the global deformation
rings we are considering to Hecke algebras.
Let U = �
v Uv ⊂ GL2(A∞ F ) be a compact open subgroup, with Uv =
GL2(OFv) for all v|p. Fix χ0 : (A∞ F )× /F × → O × a continuous character.
Let Λk denote the representation of GL2(OF,l) given by
⊗τ:F ↩→K Sym k−2 O 2 .
If A is a finite O-module, we let Sk,χ0 (U, A) denote the space of functions
satisfying
φ : D × \ GL2(A ∞ F ) → Λ ⊗O A
φ(guz) = χ0(z)u −1
p φ(g)
for all g ∈ GL2(A∞ F ), u ∈ U, z ∈ (A∞ F )× . We will sometimes write S(U, A)
for Sk,χ0 (U, A). Note that S(U, O) is a finite free O-module.
We have an action of Hecke operators on S(U, A); if we let S be the set
of places v of F which either divide p or for which Uv �= GL2(OFv), then
we have an action of the Hecke operators Tv, Sv at all places v /∈ S, via
the usual formulas we found above. Let TU denote the O-subalgebra of
EndO(Sk,χ0 (U, O)) generated by these operators; this is a commutative Oalgebra,
and is finite and free as an O-module. It is not so hard to relate
these spaces to the spaces of modular forms we considered previously.
∼
Fact 3.8. Fix an isomorphism ı : Qp −→ C. Then the ı-linear ring homomorphisms
θ : TU → C are in natural bijection with the cuspidal automorphic
representations of GD(A∞ F ) of weight (k, 0) such that πv is unramified