MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

8 TOBY GEE
If in addition we have F = Q, then we furthermore demand that for
all g ∈ GD(A ∞ ), h∞ ∈ GL2(R) + the function φ(gh∞)|ℑ(h∞i)| k/2
is bounded on C − R.
There is a natural action of GD(A ∞ ) on SD,k,η by right-translation, i.e.
(gφ)(x) := φ(xg).
Exercise 2.1. While this definition may at first sight appear rather mysterious,
it is just a generalisation of the familiar spaces of cuspidal modular
forms. For example, take F = Q, S(D) = ∅, k∞ = k, and η∞ = 0. Define
U1(N) = {g ∈ GL2( ˆ � �
∗ ∗
Z)|g ≡ (mod N)}.
0 1
(i) Let GL2(Q) + be the subgroup of GL2(Q) consisting of matrices with
positive determinant. Show that the intersection of GL2(Q) + and
U1(N) inside GL2(A∞ � �
) is Γ1(N), the matrices in SL2(Z) congruent
1 ∗
to (mod N). [Hint: what is
0 1
ˆ Z × ∩ Q × ?]
(ii) Use the facts that GL2(A) = GL2(Q)U1(N) GL2(R) + [which follows
from strong approximation for SL2 and the fact that det U1(N) =
ˆZ × ] and that A × = Q × Z ˆ× ×
R >0 to show that S U1(N)
D,k,0 can naturally be
identified with a space of functions
satisfying
φ : Γ1(N)\ GL2(R) + → C
φ(gu∞) = j(u∞, i) −k φ(g)
for all g ∈ GL2(R) + , u∞ ∈ R × >0 SO(2).
(iii) Show that the stabiliser of i ∈ in GL2(R) + is R × >0 SO(2). Hence deduce
a natural isomorphism between S U1(N)
D,k,0 and Sk(Γ1(N)), which
takes a function φ as above to the function (gi ↦→ j(g, i) kφ(g), g ∈ GL2(R) + .
The case that S∞ ⊂ S(D) is particularly simple; then if U ⊂ GD(A∞ ) is
open, SU D,2,0 is just the set of C-valued functions on
GD(Q)/GD(A)/GD(R)U,
which is a finite set.
We will now examine the action of Hecke operators on these spaces.
Choose an order OD ⊂ D (that is, a Z-subalgebra of D which is finitely
generated as a Z-module and for which OD ⊗Z Q ∼ −→ D). For example, if
D = M2(F ), one may take OD = M2(OF ).
For all but finite many finite places v of F we can choose an isomorphism
Dv ∼ = M2(Fv) such that this isomorphism induces an isomorphism OD ⊗OF
∼
OFv −→ M2(OFv). Then GD(A∞ �
) is the subset of elements g = (gv) ∈
v∤∞ GD(Fv) such that gv ∈ GL2(OFv) for almost all v.