2L-functions
IV Topics in Number Theory
2.3 Non-abelian L-functions
Let K be a number field. Then we have a reciprocity isomorphism
Art
K
: C
K
/C
0
K
∼
→ Γ
ab
K
.
If
χ
:
C
K
→ C
0
K
→ C
×
is a Hecke character of finite order, then we can view it
as a map ψ = χ ◦ Art
−1
K
: Γ
K
→ C
×
. Then
L(χ, s) =
Y
finite v unramified
1
1 − χ
v
(π
v
)q
−s
v
−1
=
Y
1
1 − ψ(Frob
v
)q
−s
v
,
where
Frob
v
∈
Γ
K
v
/I
K
v
is the geometric Frobenius, using that
ψ
(
I
K
v
) = 1.
Artin generalized this to arbitrary complex representations of Γ
K
.
Let ρ : Γ
K
→ GL
n
(C) be a representation. Define
L(ρ, s) =
Y
finite v
L(ρ
v
, s),
where
ρ
v
is the restriction to the decomposition group at
v
, and depends only
on the isomorphism class of
ρ
. We first define these local factors for non-
Archimedean fields:
Definition.
Let
F
be local and non-Archimedean. Let
ρ
:
W
F
→ GL
C
(
V
) be a
representation. Then we define
L(ρ, s) = det(1 − q
−s
ρ(Frob
F
)|
V
I
F
)
−1
,
where V
I
F
is the invariants under I
F
.
Note that in this section, all representations will be finite-dimensional and
continuous for the complex topology (so in the case of
W
F
, we require
ker σ
to
be open).
Proposition.
(i) If
0 → (ρ
0
, V
0
) → (ρ, V ) → (ρ
00
, V
0
) → 0
is exact, then
L(ρ, s) = L(ρ
0
, s) · L(ρ
00
, s).
(ii)
If
E/F
is finite separable,
ρ
:
W
E
→ GL
C
(
V
) and
σ
=
Ind
W
F
W
E
ρ
:
W
F
→
GL
C
(U), then
L(ρ, s) = L(σ, s).
Proof.
(i) Since ρ has open kernel, we know ρ(I
F
) is finite. So
0 → (V
0
)
I
F
→ V
I
F
→ (V
0
)
I
F
→ 0
is exact. Then the result follows from the multiplicativity of det.
(ii) We can write
U = {ϕ : W
F
→ V : ϕ(gx) = ρ(g)ϕ(x) for all g ∈ W
E
, x ∈ W
F
}.
where W
F
acts by
σ(g)ϕ(x) = ϕ(xg).
Then we have
U
I
F
= {ϕ : W
F
/I
F
→ V : ···}.
Then whenever ϕ ∈ U
I
F
and g ∈ I
E
, then
σ(g)ϕ(x) = ϕ(xg) = ϕ((xgx
−1
)x) = ϕ(x).
So in fact ϕ takes values in V
I
E
. Therefore
U
I
F
= Ind
W
F
/I
F
W
E
/I
E
V
I
E
.
Of course, W
F
/I
F
∼
=
Z, which contains W
E
/I
E
as a subgroup. Moreover,
Frob
d
F
= Frob
E
,
where d = [k
E
: k
F
]. We note the following lemma:
Lemma.
Let
G
=
hgi ⊇ H
=
hh
=
g
d
i
,
ρ
:
H → GL
C
(
V
) and
σ
=
Ind
G
H
ρ
.
Then
det(1 − t
d
ρ(h)) = det(1 − tσ(g)).
Proof.
Both sides are multiplicative for exact sequences of representations
of
H
. So we can reduce to the case of
dim V
= 1, where
ρ
(
h
) =
λ ∈ C
×
.
We then check it explicitly.
To complete the proof of (ii), take
g
=
Frob
F
and
t
=
q
−s
F
so that
t
d
= q
−s
E
.
For Archimedean
F
, we define
L
(
ρ, s
) in such a way to ensure that (i) and
(ii) hold, and if dim V = 1, then
L(ρ, s) = L(χ, s),
where if
ρ
:
W
ab
F
→ C
×
, then
χ
Is the corresponding character of
F
×
under the
Artin map.
If
F ' C
, then this is rather easy, since every irreducible representation of
W
F
∼
=
C
×
is one-dimensional. We then just define for
ρ
1-dimensional using
W
ab
F
∼
=
F
×
and extend to all
ρ
by (i). The Jordan–H¨older theorem tells us this
is well-defined.
If F ' R, then recall that
W
R
= hC
×
, s : s
2
= −1 ∈ C
×
, szs
−1
= ¯zi.
Contained in here is W
(1)
R
= hU(1), si. Then
W
R
= W
(1)
R
× R
×
>0
.
It is then easy to see that the irreducible representations of W
R
are
(i) 1-dimensional ρ
W
R
; or
(ii) 2-dimensional, σ = Ind
W
R
C
ρ, where ρ 6= ρ
s
: C
×
→ C
×
.
In the first case, we define
L(ρ, s) = L(χ, s)
using the Artin map, and in the second case, we define
L(σ, s) = L(ρ, s)
using (ii).
To see that the properties are satisfied, note that (i) is true by construction,
and there is only one case to check for (ii), which is if ρ = ρ
s
, i.e.
ρ(z) = (z¯z)
t
.
Then
Ind
W
R
C
×
ρ
is reducible, and is a sum of characters of
W
ab
R
∼
=
R
×
, namely
x 7→ |x|
t
and x 7→ sgn(x)|x|
t
= x
−1
|x|
t+1
. Then (ii) follows from the identity
Γ
R
(s)Γ
R
(s + 1) = Γ
C
(s) = 2(2π)
−s
Γ(s).
Now let
K
be global, and let
ρ
: Γ
K
→ GL
C
(
V
). For each
v ∈
Σ
K
, choose
¯
k
of
¯
K
over
v
. Let Γ
v
∼
=
Γ
K
v
be the decomposition group at
¯v
. These contain
I
V
,
and we have the geometric Frobenius
Frob
v
∈
Γ
v
/I
V
. We define
ρ
v
=
ρ|
Γ
v
, and
then set
L(ρ, s) =
Y
v-∞
L(ρ
v
, s) =
Y
v-∞
det(1 − q
−s
v
Frob
v
|
V
I
v
)
−1
Λ(ρ, s) = LL
∞
L
∞
=
Y
v|∞
L(ρ
v
, s).
This is well-defined as the decomposition groups
¯v | v
are conjugate. If
dim V
= 1,
then ρ = χ ◦ Art
−1
K
for a finite-order Hecke character χ, and then
L(ρ, s) = L(χ, s).
The facts we had for local factors extend to global statements
Proposition.
(i) L(ρ ⊕ ρ
0
, s) = L(ρ, s)L(ρ
0
, s).
(ii) If L/K is finite separable and ρ : Γ
L
→ GL
C
(V ) and σ = Ind
Γ
K
Γ
L
(ρ), then
L(ρ, s) = L(σ, s).
The same are true for Λ(ρ, s).
Proof.
(i) is clear. For (ii), we saw that if
w ∈
Σ
L
over
v ∈
Σ
K
and consider the
local extension L
w
/K
v
, then
L(ρ
w
, s) = L(Ind
Γ
K
v
Γ
L
w
ρ
w
).
In the global world, we have to take care of the splitting of primes. This boils
down to the fact that
Ind
Γ
K
Γ
L
ρ
Γ
K
v
=
M
w|v
Ind
Γ
K
v
Γ
L
w
(ρ|
Γ
L
w
). (∗)
We fix a valuation
¯v
of
¯
K
over
v
. Write Γ
¯v/v
for the decomposition group in Γ
K
.
Write
¯
S for the places of
¯
K over v, and S the places of L over v.
The Galois group acts transitively on
¯
S, and we have
¯
S
∼
=
Γ
K
/Γ
¯v/v
.
We then have
S
∼
=
Γ
L
\Γ
K
/Γ
¯v/v
,
which is compatible with the obvious map
¯
S → S.
For ¯w = g¯v, we have
Γ
¯w/v
= gΓ
¯v/v
g
−1
.
Conjugating by
g
−1
, we can identify this with Γ
¯v/v
. Similarly, if
w
=
¯w|
L
, then
this contains
Γ
¯w/w
= gΓ
¯v/v
g
−1
∩ Γ
L
,
and we can identify this with Γ
¯v/v
∩ g
−1
Γ
L
g.
There is a theorem, usually called Mackey’s formula, which says if
H, K ⊆ G
are two subgroups of finite index, and
ρ
:
H → GL
C
(
V
) is a representation of
H
.
Then
(Ind
G
H
V )|
K
∼
=
M
g∈H\G/K
Ind
K
K∩g
−1
Hg
(
g
−1
V ),
where
g
−1
V
is the
K ∩ g
−1
Hg
-representation where
g
−1
xg
acts by
ρ
(
x
). We
then apply this to G = Γ
K
, H = Γ
L
, K = Γ
¯v/v
.
Example. If ρ is trivial, then
L(ρ, s) =
Y
v
(1 − q
−s
v
)
−1
=
X
aCO
K
1
Na
s
= ζ
K
(s).
This is just the Dedekind ζ-function of K.
Example.
Let
L/K
be a finite Galois extension with Galois group
G
. Consider
the regular representation
r
L/K
on
C
[
G
]. This decomposes as
L
ρ
d
i
i
, where
{ρ
i
}
run over the irreducible representations of G of dimension d
i
. We also have
r
L/K
= Ind
Γ
K
Γ
L
(1).
So by the induction formula, we have
ζ
L
(s) = L(r
L/K
, s) =
Y
i
L(ρ
i
, s)
d
i
.
Example. For example, if L/K = Q(ζ
N
)/Q, then
ζ
Q(ζ
N
)
(s) =
Y
χ
L(χ, s),
where the product runs over all primitive Dirichlet characters mod
M | N
. Since
ζ
Q(ζ
N
)
, ζ
Q
have simple poles at s = 1, we know that L(χ, 1) 6= 0 if χ 6= χ
0
.
Theorem
(Brauer induction theorem)
.
Suppose
ρ
:
G → GL
N
(
C
) is a represen-
tation of a finite group. Then there exists subgroups
H
j
⊆ G
and homomorphisms
χ
j
: H
j
→ C
×
and integers m
i
∈ Z such that
tr ρ =
X
j
m
j
tr Ind
G
H
j
χ
j
.
Note that the
m
j
need not be non-negative. So we cannot quite state this as a
statement about representations.
Corollary.
Let
ρ
: Γ
K
→ GL
N
(
C
). Then there exists finite separable
L
j
/K
and χ
j
: Γ
L
j
→ C
×
of finite order and m
j
∈ Z such that
L(ρ, s) =
Y
j
L(χ
j
, s)
m
j
.
In particular,
L
(
ρ, s
) has meromorphic continuation to
C
and has a functional
equation
Λ(ρ, s) = L · L
∞
= ε(ρ, s)L(˜ρ, 1 − s)
where
ε(ρ, s) = AB
s
=
Y
ε(χ
j
, s)
m
j
,
and ˜ρ(g) =
t
ρ(g
−1
).
Conjecture
(Artin conjecture)
.
If
ρ
does not contain the trivial representation,
then Λ(ρ, s) is entire.
This is closely related to the global Langlands conjecture.
In general, there is more than one way to write
ρ
as an sum of virtual
induced characters. But when we take the product of the
ε
factors, it is always
well-defined. We also know that
ε(χ
j
, s) =
Y
ε
v
(χ
j,v
, s)
is a product of local factors. It turns out the local decomposition is not indepen-
dent of the decomposition, so if we want to write
ε(ρ, s) =
Y
v
ε
v
(ρ
v
, s),
we cannot just take
ε
v
(
ρ
v
, s
) =
Q
ε
v
(
χ
j,v
, s
), as this is not well-defined. However,
Langlands proved that there exists a unique factorization of
ε
(
ρ, s
) satisfying
certain conditions.
We fix F a non-Archimedean local field, χ : F
×
→ C
×
and local ε factors
ε(χ, ψ, µ),
where
µ
is a Haar measure on
F
and
ψ
:
F →
U(1) is a non-trivial character.
Let n(ψ) be the least integer such that ψ(π
n
F
O
F
) = 1. Then
ε(χ, ψ, µ) =
(
µ(O
F
) χ unramified, n(ψ) = 0
R
F
×
χ
−1
· ψ dµ χ ramified
Since
χ
and
ψ
are locally constant, the integral is actually sum, which turns out
to be finite (this uses the fact that χ is ramified).
For a ∈ F
×
and b > 0, we have
ε(χ, ψ(ax), bµ) = χ(a)|a|
−1
bε(χ, ψ, µ).
Theorem
(Langlands–Deligne)
.
There exists a unique system of local constants
ε(ρ, ψ, µ) for ρ : W
F
→ GL
C
(V ) such that
(i) ε
is multiplicative in exact sequences, so it is well-defined for virtual
representations.
(ii) ε(ρ, ψ, bµ) = b
dim V
ε(ρ, ψ, µ).
(iii)
If
E/F
is finite separable, and
ρ
is a virtual representation of
W
F
of degree
0 and σ = Ind
W
F
W
E
ρ, then
ε(σ, ψ, µ) = ε(ρ, ψ ◦ tr
E/F
, µ
0
).
Note that this is independent of the choice of
µ
and
µ
0
, since “
dim V
= 0”.
(iv) If dim ρ = 1, then ε(ρ) is the usual abelian ε(χ).