5Modular forms and representation theory
IV Topics in Number Theory
5 Modular forms and representation theory
Recall that a modular form is a holomorphic function f : H → C such that
f(z) = j(γ, z)
−k
f(γ(z))
for all γ in some congruence subgroup of SL
2
(Z), where
γ =
a b
c d
, γ(z) =
az + b
cz + d
, j(γ, z) = cz + d.
Let M
k
be the set of all such f.
Consider the group
GL
2
(Q)
+
= {g ∈ GL
2
(Q) : det g > 0}.
This acts on M
k
on the left by
g : f 7→ j
1
(g
−1
, z)
−k
f(g
−1
(z)), j
1
(g, z) = |det g|
−1/2
j(g, z).
The factor of
|det g|
−1/2
makes the diagonal
diag
(
Q
×
>0
) act trivially. Note that
later we will consider some
g
with negative determinant, so we put the absolute
value sign in.
For any
f ∈ M
k
, the stabilizer contains some Γ(
N
), which we can think of
as some continuity condition. To make this precise, we can form the completion
of
GL
2
(
Q
)
+
with respect to
{
Γ(
N
) :
N ≥
1
}
, and the action extends to a
representation π
0
of this completion. In fact, this completion is
G
0
= {g ∈ GL
2
(A
∞
Q
) : det(g) ∈ Q
×
>0
}.
This is a closed subgroup of G = GL
2
(A
∞
Q
), and in fact
G = G
0
·
ˆ
Z 0
0 1
.
In fact G is a semidirect product of the groups.
The group
G
0
seems quite nasty, since the determinant condition is rather
unnatural. It would be nice to get a representation of
G
itself, and the easy way
to do so is by induction. What is this? By definition, it is
Ind
G
G
0
(M
k
) = {ϕ : G → M
k
: ∀h ∈ G
0
, ϕ(hg) = π
0
(h)ϕ(g)}.
Equivalently, this consists of functions
F
:
H × G → C
such that for all
γ ∈
GL
2
(Q)
+
, we have
j
1
(γ, z)
−k
F (γ(z), γg) = F (z, g),
and for every g ∈ G, there is some open compact K ⊆ G such that
F (z, g) = F(z, gh) for all h ∈ K,
and that F is holomorphic in z (and at the cusps).
To get rid of the plus, we can just replace
GL
2
(
Q
)
+
, H
with
GL
2
(
Q
)
, C \R
=
H
±
. These objects are called adelic modular forms.
If
F
is an adelic modular form, and we fix
g
, then the function
f
(
z
) =
F
(
z, g
) is a usual modular form. Conversely, if
F
invariant under
ker
(
G
2
(
ˆ
Z
)
→
GL
2
(
Z/N Z
)), then
F
corresponds to a tuple of Γ(
N
)-modular forms indexed
by (
Z/N Z
)
×
. This has an action of
G
=
Q
0
p
GL
2
(
Q
p
) (which is the restricted
product with respect to GL
2
(Z
p
)).
The adelic modular forms contain the cusp forms, and any
f ∈ M
k
generates
a subrepresentation of the space of adelic forms.
Theorem.
(i)
The space
V
f
of adelic cusp forms generated by
f ∈ S
k
(Γ
1
(
N
)) is irreducible
iff f is a T
p
eigenvector for all p - n.
(ii)
This gives a bijection between irreducible
G
-invariant spaces of adelic cusp
forms and Atkin–Lehner newforms.
Note that it is important to stick to cusp forms, where there is an inner
product, because if we look at the space of adelic modular forms, it is not
completely decomposable.
Now suppose (
π, V
) is an irreducible admissible representation of
GL
2
(
A
∞
Q
) =
Q
0
G
p
=
GL
2
(
Q
p
), and take a maximal compact subgroups
K
0
p
=
GL
2
(
Z
p
)
⊆
GL
2
(
Q
p
). Then general facts about irreducible representations of products imply
irreducibility (and admissibility) is equivalent to the existence of irreducible
admissible representations (π
p
, V
p
) of G
p
for all p such that
(i)
For almost all
p
,
dim V
K
0
p
p
≥
1 (for
G
p
=
GL
n
(
Q
p
), this implies the
dimension is 1). Fix some non-zero f
0
p
∈ V
K
0
p
p
.
(ii) We have
π = ⊗
0
p
π
p
,
the restricted tensor product, which is generated by
N
p
v
p
with
v
p
=
f
0
p
for almost all p. To be precise,
⊗
0
p
π
p
= lim
−→
finite S
O
p∈S
π
p
.
The use of
v
p
is to identify smaller tensor products with larger tensor
products.
Note that (i) is equivalent to the assertion that (
π
p
, V
p
) is an irreducible
principal series representation
Ind
G
p
B
p
(
χ
1
, χ
2
) where
χ
i
are unramified characters.
These unramified characters are determined by χ
p
(p) = α
p,i
.
If
f
=
P
a
n
q
n
∈ S
k
(Γ
1
(
N
)) is a normalized eigenform with character
ω
:
(
Z/N Z
)
×
→ C
×
, and if
f
corresponds to
π
=
N
0
π
p
, then for every
p - N
, we
have π
p
= Ind
G
p
B
p
(χ
1
, χ
2
) is an unramified principal series, and
a
p
= p
(k−1)/2
(α
p,1
+ α
p,2
)
ω(p)
−1
= α
p,1
α
p,2
.
We can now translate difficult theorems about modular forms into representation
theory.
Example. The Ramanujan conjecture (proved by Deligne) says
|a
p
| ≤ 2p
(k−1)/2
.
If we look at the formulae above, since
ω
(
p
)
−1
is a root of unity, this is equivalent
to the statement that |α
p,i
| = 1. This is true iff π
p
is tempered.
We said above that if
dim V
K
0
p
p
≥
1, then in fact it is equal to 1. There is a
generalization of that.
Theorem
(Local Atkin–Lehner theorem)
.
If (
π, V
) is an irreducible representa-
tion of
GL
2
(
F
), where
F/Q
p
and
dim V
=
∞
, then there exists a unique
n
π
>
0
such that
V
K
n
=
(
0 n < n
π
one-dimensional n = n
π
, K
n
=
g ≡
∗ ∗
0 1
mod $
n
.
Taking the product of these invariant vectors for
n
=
n
π
over all
p
gives Atkin–
Lehner newform.
What about the other primes? i.e. the primes at infinity?
We have a map
f
:
H
±
×GL
2
(
A
∞
Q
)
→ C
. Writing
H
=
SL
2
(
R
)
/SO
(2), which
has an action of Γ, we can convert this to an action of
SO
(2) on Γ
\SL
2
(
R
).
Consider the function
Φ
f
: GL
2
(R) ×GL
2
(A
∞
Q
) = GL
2
(A
Q
) → C
given by
Φ
f
(h
∞
, h
∞
) = j
1
(h
∞
, i)
−k
f(h
∞
(i), h
∞
).
Then this is invariant under GL
2
(Q)
+
, i.e.
Φ
f
(γh
∞
, γh
∞
) = Φ(h
∞
, h
∞
).
Now if we take
k
θ
=
cos θ sin θ
−sin θ cos θ
∈ SO(2),
then
Φ
f
(h
∞
k
θ
, h
∞
) = e
ikθ
Φ
f
(h
∞
, h
∞
).
So we get invariance under
γ
, but we need to change what
SO
(2) does. In other
words, Φ
f
is now a function
Φ
f
: GL
2
(Q)\G
2
(A
Q
) → C
satisfying various properties:
– It generates a finite-dimensional representation of SO(2)
– It is invariant under an open subset of GL
2
(A
∞
Q
)
– It satisfies growth condition and cuspidality, etc.
By the Cauchy–Riemann equations, the holomorphicity condition of
f
says Φ
f
satisfies some differential equation. In particular, that says Φ
f
is an eigenfunction
for the Casimir in the universal enveloping algebra of
sl
2
. These conditions
together define the space of automorphic forms.
Example. Take the non-holomorphic Eisenstein series
E(z, s) =
X
(c,d)6=(0,0)
1
|cz + d|
2s
.
This is a real analytic function on Γ
\H → C
. Using the above process, we get
an automorphic form on
GL
2
(
Q
)
\GL
2
(
A
Q
) with
k
= 0. So it actually invariant
under SO(2). It satisfies
∆E = s(1 −s)E.
There exist automorphic cusp forms which are invariant under
SO
(2), known
as Maass forms. They also satisfy a differential equation with a Laplacian
eigenvalue
λ
. A famous conjecture of Selberg says
λ ≥
1
4
. What this is equivalent
to is that the representation of GL
2
(R) they generate is tempered.
In terms of representation theory, this looks quite similar to Ramanujan’s
conjecture!