2L-functions
IV Topics in Number Theory
2.2 Abelian L-functions
We are now going to define
L
-functions for Hecke characters. Recall that amongst
all other things, an
L
-function is a function in a complex variable. Here we are
going to do things slightly differently. For any Hecke character
χ
, we will define
L(χ), which will be a number. We then define
L(χ, s) = L(| · |
s
A
χ).
We shall define
L
(
χ
) as an Euler product, and then later show it can be written
as a sum.
Definition
(Hecke
L
-function)
.
Let
χ
:
C
K
→ C
×
be a Hecke character. For
v ∈ Σ
K
, we define local L-factors L(χ
v
) as follows:
– If v is non-Archimedean and χ
v
unramified, i.e. χ
v
|
O
×
K
v
= 1, we set
L(χ
v
) =
1
1 −χ
v
(π
v
)
.
– If v is non-Archimedean and χ
v
is ramified, then we set
L(χ
v
) = 1.
– If v is a real place, then χ
v
is of the form
χ
v
(x) = x
−N
|x|
s
v
,
where N = 0, 1. We write
L(χ
v
) = Γ
R
(s) = π
−s/2
Γ(s/2).
– If v is a complex place, then χ
v
is of the form
χ
v
(x) = σ(x)
−N
|x|
s
v
,
where σ is an embedding of K
v
into C and N ≥ 0. Then
L(χ
v
) = Γ
C
(s) = 2(2π)
−s
Γ(s)
We then define
L(χ
v
, s) = L(χ
v
· | · |
s
v
).
So for finite unramified v, we have
L(χ
v
, s) =
1
1 − χ
v
(π
v
)q
−s
v
,
where q
v
= |O
K
v
/(π
v
)|.
Finally, we define
L(χ, s) =
Y
v-∞
L(χ
v
, s)
Λ(χ, s) =
Y
v
L(χ
v
, s).
Recall that the kernel of the idelic norm
| · |
A
:
C
K
→ R
×
>0
is compact. It is
then not hard to see that for every
χ
, there is some
t ∈ R
such that
χ · | · |
t
A
is
unitary. Thus,
L
(
χ, s
) converges absolutely on some right half-plane. Observe
that
Λ(χ| · |
t
A
, s) = Λ(χ, t + s).
Theorem (Hecke–Tate).
(i)
Λ(
χ, s
) has a meromorphic continuation to
C
, entire unless
χ
=
| · |
t
A
for
some t ∈ C, in which case there are simple poles at s = 1 − t, −t.
(ii) There is some function, the global ε-factor,
ε(χ, s) = AB
s
for some A ∈ C
×
and B ∈ R
>0
such that
Λ(χ, s) = ε(χ, s)Λ(χ
−1
, 1 − s).
(iii) There is a factorization
ε(χ, s) =
Y
v
ε
v
(χ
v
, µ
v
, ψ
v
, s),
where
ε
v
= 1 for almost all
v
, and
ε
v
depends only on
χ
v
and certain
auxiliary data ψ
v
, µ
v
. These are the local ε-factors.
Traditionally, we write
L(χ, s) =
Y
finite v
L(χ
v
, s),
and then
Λ(χ, s) = L(χ, s)L
∞
(χ, s).
However, Tate (and others, especially the automorphic people) use
L
(
χ, s
) for
the product over all v.
At first, Hecke proved (i) and (ii) using global methods, using certain Θ
functions. Later, Tate proved (i) to (iii) using local-global methods and espe-
cially Fourier analysis on
K
v
and
A
K
. This generalizes considerably, e.g. to
automorphic representations.
We can explain some ideas of Hecke’s method. We have a decomposition
K
∞
= K ⊗
Q
R
∼
=
R
r
1
× C
r
2
∼
=
C
n
,
and this has a norm
k · k
induces by the Euclidean metric on
R
n
. Let ∆
⊆ O
×
K,+
be a subgroup of totally positive units of finite index, which is
∼
=
Z
r
1
+r
2
−1
. This
has an embedding ∆ =
K
×,1
∞
, which extends to a continuous homomorphism
∆ ⊗ R → K
×,1
∞
. The key fact is
Proposition. Let x ∈ K
×
. Pick some invariant measure du on ∆ ⊗ R. Then
Z
∆⊗R
1
kuxk
2s
du =
stuff
|N
K/Q
(x)|
2s/n
,
where the stuff is some ratio of Γ factors and powers of π (and depends on s).
Exercise. Prove this when K = Q[
√
d] for d > 0. Then ∆ = hεi, and then
LHS =
Z
∞
−∞
1
|ε
t
x + ε
−t
x
0
|
2s
dt
RHS =
stuff
|xx
0
|
s
.
The consequence of this is that if a ⊆ K is a fractional ideal, then
X
06=x∈a mod ∆
1
|N
K/Q
(x)|
s
= stuff ·
X
06=x∈a mod ∆
Z
∆⊗R
1
kuxk
ns
du
= stuff ·
Z
∆⊗R/∆
X
06=x∈a
1
kuxk
ns
du
The integrand has a name, and is called the Epstein
ζ
-function of the lattice
(
a, ku · k
2
). By the Poisson summation formula, we get an analytic continuity
and functional equation for the epsilon
ζ
function. On the other hand, taking
linear combinations of the left gives
L
(
χ, s
) for
χ
:
Cl
(
K
)
→ C
×
. For more
general
χ
, we modify this with some extra factors. When the infinity type is
non-trivial, this is actually quite subtle.
Note that if
χ
is unramified outside
S
and ramified at
S
, recall we had a
homomorphism Θ : I
S
→ C
×
sending p
v
7→ χ
v
(π
v
)
−1
. So
L(χ, s) =
Y
finite v6∈S
1
1 − Θ(p
v
)
−1
(Np
v
)
−s
=
X
a∈O
K
prime to S
Θ(a)
−1
(Na)
s
.
This was Hecke’s original definition of the Hecke character.
If
K
=
Q
and
χ
:
C
Q
→ C
×
is of finite order, then it factors through
C
Q
→ C
Q
/C
0
Q
∼
=
ˆ
Z
×
→
(
Z/NZ
)
×
, and so
χ
is just some Dirichlet character
ϕ
: (
Z/nZ
)
×
→ C
×
. The associated
L
-functions are just Dirichlet
L
-functions.
Indeed, if p - N, then
χ
p
(p) = χ(1, . . . , 1, p, 1, . . .) = χ(p
−1
, . . . , p
−1
, 1, p
−1
, . . .) = ϕ(p mod N)
−1
.
In other words,
L
(
χ, s
) is the Dirichlet
L
-series of
ϕ
−1
(assuming
N
is chosen so
that χ ramifies exactly at v | N).
Tate’s method uses local
ε
-factors
ε
(
χ
v
, µ
v
, ψ
v
, s
), where
ψ
v
:
K
v
→
U(1) is
a non-trivial additive character, e.g. for v finite,
K
V
Q
p
Q
p
/Z
p
∼
=
Z[v
p
]/Z C
×
,
tr e
2πix
which we needed because Fourier transforms take in additive measures, and
µ
v
is a Haar measure on K
v
. The condition for (iii) to hold is
Y
ψ
v
: A
K
→ U(1)
is well-defined and trivial on
K ⊆ A
K
, and
µ
A
=
Q
µ
v
is a well-defined measure
on A
K
, i.e. µ
v
(O
v
) = 1 for all v and
Z
A
K
/K
µ
A
= 1.
There exists explicit formulae for these
ε
v
’s. If
χ
v
is unramified, then it is just
A
v
B
s
v
, and is usually 1; for ramified finite v, they are given by Gauss sums.