MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

10 TOBY GEE
If π is a cuspidal automorphic representation of GD(A ∞ ) then there is a
unique ideal n such that π U1(n) is one-dimensional, and π U1(m) �= 0 if and
only if n|m. We call n the conductor of π.
Analogous to the theory of admissible representations of GL2(K), K/Qp
finite that we sketched above, there is a theory of admissible representations
of D × , D a nonsplit quaternion algebra over K. Since D × /K × is
compact, any irreducible smooth representation of D × is finite-dimensional.
There is a bijection JL from the irreducible smooth representations of D ×
to the discrete series representations of GL2(K), determined by a character
identity.
Fact 2.8. (The global Jacquet-Langlands correspondence)
(i) The only finite-dimensional cuspidal automorphic representations of
GD(A∞ ) occur if S(D) ⊃ S∞ and kv = 2 for all v ∈ S∞, in which
case there are 1-dimensional representations, which factor through
the reduced determinant.
(ii) There is a bijection JL from the infinite-dimensional cuspidal automorphic
representations of GD(A∞ ) of weight (k, η) to the cuspidal
automorphic representations π of GL2(A∞ F ) of weight (k, η)
such that πv is discrete series for all finite places v ∈ S(D). Furthermore
if v /∈ S(D) then JL(π)v = πv, and if v ∈ S(D) then
JL(π)v = JL(πv).
Fact 2.9. (The existence of Galois representations associated to cuspidal
modular forms) Let π be an irreducible admissible representation of GL2(AF ).
Then there is a CM field Lπ and for each finite place λ of Lπ a continuous
irreducible Galois representation
such that
rλ(π) : GF → GL2(Lπ,λ)
(i) if πv is unramified and v does not divide the residue characteristic of
λ, then rλ(π)|GFv is unramified, and the characteristic polynomial
of Frobv is X2 − tvX + (#kv)sv, where tv and sv are the eigen-
GL2(OFv )
values of Tv and Sv respectively on πv . [Note that by the
Chebotarev density theorem, this already characterises rλ(π) up to
isomorphism.]
(ii) More generally, WD(rλ(π)|GFv )F −ss ∼ = recFv(πv ⊗ | det | −1/2 ).
(iii) If v divides the residue characteristic of λ then rλ(π)|GFv is de Rham
with τ-Hodge-Tate weights ητ , ητ + kτ − 1, where τ : F ↩→ Lπ ⊂ C
is an embedding lying over v. If πv is unramified then rλ(π)|GFv is
crystalline.
(iv) If cv is a complex conjugation, then det rλ(π)(cv) = −1.
Fact 2.10. (Base change) Let E/F be a cyclic extension of totally real fields
of prime degree. Let Gal(E/F ) = 〈σ〉 and let Gal(E/F ) ∨ = 〈δ E/F 〉. Let π be