2L-functions

IV Topics in Number Theory



2.1 Hecke characters
Definition
(Hecke character)
.
A Hecke character is a continuous (not necessarily
unitary) homomorphism
χ : J
K
/K
×
C
×
.
These are also known as quasi-characters in some places, where character
means unitary. However, we shall adopt the convention that characters need not
be unitary. The German term Gr¨oßencharakter (or suitable variations) are also
used.
In this section, we will seek to understand Hecke characters better, and see
how Dirichlet characters arise as a special case where
K
=
Q
. Doing so is useful
if we want to write down actual Hecke characters. The theory of
L
-functions
will be deferred to the next chapter.
We begin with the following result:
Proposition.
Let
G
be a profinite group, and
ρ
:
G GL
n
(
C
) continuous.
Then ker ρ is open.
Of course, the kernel is always closed.
Proof.
It suffices to show that
ker ρ
contains an open subgroup. We use the fact
that
GL
n
(
C
) has “no small subgroups”, i.e. there is an open neighbourhood
U
of
1
GL
n
(
C
) such that
U
contains no non-trivial subgroup of
GL
n
(
C
) (exercise!).
For example, if n = 1, then we can take U to be the right half plane.
Then for such
U
, we know
ρ
1
(
U
) is open. So it contains an open subgroup
V
. Then
ρ
(
V
) is a subgroup of
GL
n
(
C
) contained in
U
, hence is trivial. So
V ker(ρ).
While the multiplicative group of a local field is not profinite, it is close
enough, and we similarly have
Exercise.
Let
F
be a local field. Then any continuous homomorphism
F
×
C
×
has an open kernel, i.e. χ(1 + p
N
F
) = 1 for some N 0.
Definition
(Unramified character)
.
If
F
is a local field, a character
χ
:
F C
×
is unramified if
χ|
O
×
F
= 1.
If F
=
R, we say χ : F
×
C
×
is unramified if χ(1) = 1.
Using the decomposition F
=
O
F
× hπ
F
i (for the local case), we see that
Proposition. χ is unramified iff χ(x) = |x|
s
F
for some s C.
We now return to global fields. We will think of Hecke characters as continuous
maps
J
K
C
×
that factor through
J
K
/K
×
, since it is easier to reason about
J
K
than the quotient. We can begin by discussing arbitrary characters
χ
:
J
K
C
×
.
Proposition.
The set of continuous homomorphisms
χ
:
J
K
=
Q
0
v
K
×
v
C
×
bijects with the set of all families (
χ
v
)
vΣ
k
,
χ
v
:
K
×
v
C
×
such that
χ
v
is
unramified for almost all (i.e. all but finitely many)
v
, with the bijection given
by χ 7→ (χ
v
), χ
v
= χ|
K
×
v
.
Proof.
Let
χ
:
J
K
C
×
be a character. Since
ˆ
O
K
J
K
is profinite, we know
ker χ|
ˆ
O
×
K
is an open subgroup. Thus, it contains
O
×
v
for all but finitely many
v
.
So we have a map from the LHS to the RHS.
In the other direction, suppose we are given a family (
χ
v
)
v
. We attempt to
define a character χ : J
K
C
×
by
χ(x
v
) =
Y
χ
v
(x
v
).
By assumption,
χ
v
(
x
v
) = 1 for all but finitely many
v
. So this is well-defined.
These two operations are clearly inverses to each other.
In general, we can write χ as
χ = χ
χ
, χ
=
Y
v-
χ
v
: K
×,
C
×
, χ
=
Y
v|∞
χ
v
: K
×
C
×
.
Lemma. Let χ be a Hecke character. Then the following are equivalent:
(i) χ has finite image.
(ii) χ
(K
×,0
) = 1.
(iii) χ
2
= 1.
(iv) χ(C
0
K
) = 1.
(v) χ factors through Cl
m
(K) for some modulus m.
In this case, we say χ is a ray class character.
Proof.
Since
χ
(
K
×,0
) is either 1 or infinite, we know (i)
(ii). It is clear that
(ii) (iii), and these easily imply (iv). Since C
K
/C
0
K
is profinite, if (iii) holds,
then
χ
factors through
C
K
/C
0
K
and has open kernel, hence the kernel contains
U
m
for some modulus
m
. So
χ
factors through
Cl
m
(
K
). Finally, since
Cl
m
(
K
) is
finite, (v) (i) is clear.
Using this, we are able to classify all Hecke characters when K = Q.
Example.
The idele norm
| · |
A
:
C
K
R
×
>0
is a character not of finite order.
In the case
K
=
Q
, we have
C
Q
=
R
×
>0
×
ˆ
Z
×
. The idele norm is then the
projection onto R
×
>0
.
Thus, if
χ
;
C
Q
C
×
is a Hecke character, then the restriction to
R
×
>0
is of
the form x 7→ x
s
for some s. If we write
χ(x) = |x|
s
A
· χ
0
(x)
for some
χ
0
, then
χ
0
vanishes on
R
×
>0
, hence factors through
ˆ
Z
×
and has finite
order. Thus, it factors as
χ
0
: C
Q
ˆ
Z
×
(Z/NZ)
×
C
×
for some N. In other words, it is a Dirichlet character.
For other fields, there can be more interesting Hecke characters.
For a general field
K
, we have finite order characters as we just saw. They
correspond to characters on
I
S
which are trivial on
P
m
. In fact, we can describe
all Hecke characters in terms of ideals.
There is an alternative way to think about Hecke characters. We can
think of Dirichlet characters as (partial) functions
Z C
×
that satisfy certain
multiplicativity properties. In general, a Hecke character can be thought of as a
function on a set of ideals of O
K
.
Pick a modulus
m
such that
χ
is trivial on
ˆ
K
×
U
m
. Let
S
be the set of
finite
v
such that
m
v
is positive, and let
I
S
be the set of fractional ideals prime
to S. We then define a homomorphism
Θ : I
S
C
×
p
v
7→ χ
v
(π
v
)
1
One would not expect Θ to remember much information about the infinite
part
χ
. However, once we know Θ and
χ
(and
m
), it is not difficult to see
that we can recover all of χ.
On the other hand, an arbitrary pair
, χ
) doesn’t necessarily come from
a Hecke character. Indeed, the fact that
χ
vanishes on
K
×
implies there is some
compatibility condition between Θ and χ
.
Suppose x K
×
is such that x 1 mod m. Then (x) I
S
, and we have
1 = χ(x) = χ
(x)
Y
v6∈S finite
χ
v
(x) = χ
(x)
Y
finite v6∈S
χ
v
(π
v
)
v(x)
.
Writing
P
m
for the set of principal ideals generated by these
x
, as we previously
did, we see that for all x P
m
,
χ
(x) = Θ(x).
One can check that given (Θ
, χ
) (and
m
) satisfying this compatibility condition,
there is a unique Hecke character that gives rise to this pair. This was Hecke’s
original definition of a Hecke character, which is more similar to the definition
of a Dirichlet character.
Example.
Take
K
=
Q
(
i
), and fix an embedding into
C
. Since
Cl
(
K
) = 1, we
have
C
K
=
C
×
×
ˆ
O
×
K
µ
4
= 1, ±i}
.
Let
v
2
be the place over 2, corresponding to the prime (1 +
i
)
O
K
. Then
K
v
2
is
a ramified extension over Q
2
of degree 2. Moreover,
O
K
(1 + i)
3
O
K
×
= µ
4
= 1, ±i}.
So we have a decomposition
O
×
v
2
= (1 + (1 + i)
3
O
v
2
) × µ
4
.
Thus, there is a natural projection
C
K
C
×
× O
×
v
2
µ
4
=
C
×
× (1 + (1 + i)
3
O
v
2
) C
×
.
This gives a Hecke character with χ
(z) = z, and is trivial on
Y
v6∈{v
2
,∞}
O
×
v
× (1 + (1 + i)
3
O
v
2
),
This has modulus
m = 3(v
2
).
In ideal-theoretic terms, if
p 6
= (1 +
i
) is a prime ideal of
K
, then
p
= (
π
p
)
for a unique π
p
O
K
with π
p
1 mod (1 + i)
3
. Then Θ sends p to π
p
.
This is an example of an algebraic Hecke character.
Definition
(Algebraic homomorphism)
.
A homomorphism
K
×
C
×
is alge-
braic if there exists integers n(σ) (for all σ : K C) such that
ϕ(x) =
Y
σ(x)
n(σ)
.
The first thing to note is that if
ϕ
is algebraic, then
ϕ
(
K
×
) is contained
in the Galois closure of
K
in
C
. In particular, it takes values in the number
field. Another equivalent definition is that it is algebraic in the sense of algebraic
geometry, i.e. if
K
=
L
Qe
i
for
i
= 1
, . . . , n
as a vector space, then we can view
K
as the
Q
-points of an
n
-dimensional affine group scheme. We can then define
R
K/Q
G
m
A
to be the set on which
X
is invertible, and then an algebraic
Hecke character is a homomorphism of algebraic groups (
T
K
)
/C G
m
/C
, where
T
K
= Res
K/Q
(G
m
).
If we have a real place
v
of
K
, then this corresponds to a real embedding
σ
v
:
K K
v
=
R
, and if
v
is a complex place, we have a pair of embedding
σ
v
, ¯σ
v
:
K K
v
' C
, picking one of the pair to be
σ
v
. So
ϕ
extends to a
homomorphism
ϕ : K
×
C
×
given by
ϕ(x
v
) =
Y
v real
x
n(σ
v
)
v
Y
v complex
x
n(σ
v
)
v
¯x
n(¯σ
v
)
v
Definition
(Algebraic Hecke character)
.
A Hecke character
χ
=
χ
χ
:
J
K
/K
×
C
×
is algebraic if there exists an algebraic homomorphism
ϕ
:
K
×
C
×
such that
ϕ
(
x
) =
χ
(
x
) for all
x K
×,0
, i.e.
χ
=
ϕ
Q
v real
sgn
e
v
v
for e
v
{0, 1}.
We say ϕ (or the tuple (n(σ))
σ
) is the infinite type of χ.
Example. The adelic norm | · |
A
: J
K
C
×
has
χ
=
Y
| · |
v
,
and so
χ
is algebraic, and the associated
ϕ
is just
N
K/Q
:
K
×
Q
×
C
×
, with
(n
σ
) = (1, . . . , 1).
Exercise.
Let
K
=
Q
(
i
), and
χ
from the previous example, whose associated
character of ideals was Θ :
p 7→ π
p
, where
π
p
1
mod
(2 + 2
i
). The infinity type
is the inclusion K
×
C
×
, i.e. it has type (1, 0).
Observe that the image of an algebraic homomorphism
ϕ
:
K
×
C
×
lies in
the normal closure of K. More generally,
Proposition.
If
χ
is an algebraic Hecke character, then
χ
takes values in
some number field. We write E = E(χ) for the smallest such field.
Of course, we cannot expect
χ
to take algebraic values, since
J
K
contains
copies of R and C.
Proof.
Observe that
χ
(
ˆ
O
×
K
) is finite subgroup, so is
µ
n
for some
n
. Let
x K
×
,
totally positive. Then
χ
(x) = χ
(x)
1
= ϕ(x)
1
K
cl
,
where
K
cl
is the Galois closure. Then since
K
×
>0
×
ˆ
O
×
K
ˆ
K
×
has finite cokernel
(by the finiteness of the class group), so
χ
(
ˆ
K
×
) =
d
a
i=1
z
i
χ
(K
×
>0
ˆ
O
×
K
),
where
z
d
i
χ
(
K
×
>0
ˆ
O
×
K
), and is therefore contained inside a finite extension of
the image of K
×
>0
×
ˆ
O
×
K
.
Hecke characters of finite order (i.e. algebraic Hecke characters with infinity
type (0
, . . . ,
0)) are in bijection with continuous homomorphisms Γ
K
C
×
,
necessarily of finite order. What we show now is how to associate to a general
algebraic Hecke character
χ
a continuous homomorphism
ψ
: Γ
K
E
(
χ
)
×
λ
Q
×
, where
λ
is a place of
E
(
χ
) over
. This is continuous for the
-adic topology
on
E
λ
. In general, this will not be of finite order. Thus, algebraic Hecke
characters correspond to -adic Galois representations.
The construction works as follows: since
χ
(
x
) =
ϕ
(
x
)
1
, we can restrict
the infinity type
ϕ
to a homomorphism
ϕ
:
K
×
E
×
. We define
˜χ
:
J
K
E
×
as follows: if x = x
x
K
×
K
,×
J
K
, then we set
˜χ(x) = χ(x)ϕ(x
)
1
.
Notice that this is not trivial on
K
×
in general. Then
˜χ
takes values in
1
}
.
Thus,
˜χ
takes values in
E
×
. Thus, we know that
˜χ
has open kernel, i.e. it is
continuous for the discrete topology on E
×
, and ˜χ|
K
×
= ϕ
1
.
Conversely, if
˜χ
:
K
×
E
×
is a continuous homomorphism for the discrete
topology on
E
×
, and
˜χ|
K
×
is an algebraic homomorphism, then it comes from
an algebraic Hecke character in this way.
Let
λ
be a finite place of
E
over
, a rational prime. Recall that
ϕ
:
K
×
E
×
is an algebraic homomorphism, i.e.
ϕ
X
x
i
e
i
= f(x), f E(X
1
, . . . , X
n
).
We can extend this to K
×
= (K
Q
Q
)
×
=
Q
v|
K
×
v
to get a homomorphism
ϕ
λ
: K
×
E
×
λ
This is still algebraic, so it is certainly continuous for the -adic topology.
Now consider the character ψ
λ
: J
K
E
×
λ
, where now
ψ
λ
((x
v
)) = ˜χ(x)ϕ
λ
((x
v
)
v|
).
This is then continuous for the
-adic topology on
E
×
λ
, and moreover, we see
that
ψ
λ
(
K
×
) =
{
1
}
as
˜χ|
K
×
=
ϕ
1
while
ϕ
λ
|
K
×
=
ϕ
. Since
˜χ
(
K
×,0
) =
{
1
}
, we
know that ψ
λ
it is in fact defined on C
K
/C
0
K
=
Γ
ab
K
.
Obviously, ψ
λ
determines ˜χ and hence χ.
Fact.
An
-adic character
ψ
:
C
K
/C
0
K
E
×
λ
comes from an algebraic Hecke
character in this way if and only if the associated Galois representation is
Hodge–Tate, which is a condition on the restriction to the decomposition groups
Gal(
¯
K
v
/K
v
) for the primes v | .
Example. Let K = Q and χ = | · |
A
, then
˜χ = sgn(x
)
Y
p
|x
p
|
p
.
So
ψ
((x
v
)) = sgn(x
)
Y
p6=
|x
p
|
p
· |x
|
· x
.
Note that |x
|
x
Z
×
. We have
C
Q
/C
0
Q
=
ˆ
Z
×
.
Under this isomorphism, the map
ˆ
Z
×
Q
×
is just the projection onto
Z
×
followed
by the inclusion, and by class field theory,
ψ
:
Gal
(
¯
Q/Q
)
Z
×
is just the
cyclotomic character of the field Q({ζ
n
}),
σ(ζ
n
) = ζ
ψ
`
(σ) mod
n
n
.
Example.
Consider the elliptic curve
y
2
=
x
3
x
with complex multiplication
over Q(i). In other words, End(E/Q(i)) = Z[i], where we let i act by
i · (x, y) 7→ (x, iy).
Its Tate module
T
E = lim E[
n
]
is a Z
[i]-module. If λ | , then we define
V
λ
E = T
E
Z
`
[i]
K
λ
.
Then Γ
K
act by Γ
K
: Aut
K
λ
V
λ
E = K
×
λ
.
We now want to study the infinity types of an algebraic Hecke character.
Lemma.
Let
K
be a number field,
ϕ
:
K
×
E
×
C
×
be an algebraic
homomorphism, and suppose E/Q is Galois. Then ϕ factors as
K
×
norm
(K E)
×
φ
0
E
×
.
Note that since E is Galois, the intersection K E makes perfect sense.
Proof. By definition, we can write
ϕ(x) =
Y
σ:KC
σ(x)
n(σ)
.
Then since ϕ(x) E, for all x K
×
and τ Γ
E
, we have
Y
τσ(x)
n(σ)
=
Y
σ(x)
n(σ)
.
In other words, we have
Y
σ
σ(x)
n(τ
1
σ)
=
Y
σ
σ(x)
n(σ)
.
Since the homomorphisms
σ
are independent, we must have
n
(
τσ
) =
n
(
σ
) for
all embeddings σ : K
¯
Q and τ Γ
E
. This implies the theorem.
Recall that if
m
is a modulus, then we defined open subgroups
U
m
J
K
,
consisting of the elements (
x
v
) such that if a real
v | m
, then
x
v
>
0, and if
v | m
for a finite v, then v(x
v
1) m
v
. We can write this as
U
m
= U
m,
× U
m
.
Proposition.
Let
ϕ
:
K
×
C
×
be an algebraic homomorphism. Then
ϕ
is
the infinity type of an algebraic Hecke character χ iff ϕ(O
×
K
) is finite.
Proof.
To prove the (
) direction, suppose
χ
=
χ
χ
is an algebraic Hecke
character with infinity type
ϕ
. Then
χ
(
U
m
) = 1 for some
m
. Let
E
m
=
K
×
U
m
O
×
K
, a subgroup of finite index. As
χ
(
E
m
) = 1 =
χ
(
E
m
), we know
χ
(E
m
) = 1. So ϕ(O
×
K
) is finite.
To prove (
), given
ϕ
with
ϕ
(
O
×
K
) finite, we can find some
m
such that
ϕ
(
E
m
) = 1. Then (
ϕ,
1) :
K
×
× U
m
C
×
is trivial on
E
m
. So we can extend
this to a homomorphism
K
×
U
m
K
×
K
×
=
K
×
U
m
E
m
C
×
,
since
E
m
=
K
×
U
m
. But the LHS is a finite index subgroup of
C
K
. So the
map extends to some χ.
Here are some non-standard terminology:
Definition
(Serre type)
.
A homomorphism
ϕ
:
K
×
C
×
is of Serre type if it
is algebraic and ϕ(O
×
K
) is finite.
These are precisely homomorphisms that occur as infinity types of algebraic
Hecke characters.
Note that the unit theorem implies that
O
×
K
K
×,1
= {x K
×
: |x|
A
= 1}
has compact cokernel. If
ϕ
(
O
×
K
) is finite, then
ϕ
(
K
×,1
) is compact. So it maps
into U(1).
Example. Suppose K is totally real. Then
K
×
= (R
×
)
{σ:KR}
.
Then we have
K
×,1
= {(x
σ
) :
Y
x
σ
= ±1}.
Then
ϕ
((
x
σ
)) =
Q
x
n(σ)
σ
, so
|ϕ
(
K
×,1
)
|
= 1. In other words, all the
n
σ
are equal.
Thus, ϕ is just a power of the norm map.
Thus, algebraic Hecke characters are all of the form
| · |
m
A
· (finite order character).
Another class of examples comes from CM fields.
Definition
(CM field)
. K
is a CM field if
K
is a totally complex quadratic
extension of a totally real number field K
+
.
This CM refers to complex multiplication.
This is a rather restrictive condition, since this implies
Gal
(
K/K
+
) =
{
1
, c}
=
Gal
(
K
w
/K
+
v
) for every
w | v |
. So
c
is equal to complex conjugation for every
embedding K C.
From this, it is easy to see that CM fields are all contained in
Q
CM
¯
Q C
,
given by the fixed field of the subgroup
h
1
: σ Γ
Q
i Γ
Q
.
For example, we see that the compositum of two CM fields is another CM field.
Exercise.
Let
K
be a totally complex
S
3
-extension over
Q
. Then
K
is not CM,
but the quadratic subfields is complex and is equal to K Q
CM
.
Example.
Let
K
be a CM field of degree 2
r
. Then Dirichlet’s unit theorem
tells us
rk O
×
K
= r 1 = rk O
×
K
+
.
So
O
×
K
is a finite index subgroup of
O
×
K
+
. So
ϕ
:
K
×
C
×
is of Serre type iff
it is algebraic and its restriction to
K
+,×
is of Serre type. In other words, we
need n(σ) + n(¯σ) to be independent of σ.
Theorem.
Suppose
K
is arbitrary, and
ϕ
:
K
×
E
×
C
×
is algebraic, and
we assume
E/Q
is Galois, containing the normal closure of
K
. Thus, we can
write
ϕ(x) =
Y
σ:KE
σ(x)
n(σ)
.
Then the following are equivalent:
(i) ϕ is of Serre type.
(ii) ϕ
=
ψ N
K/F
, where
F
is the maximal CM subfield and
ψ
is of Serre type.
(iii)
For all
c
0
Gal
(
E/Q
) conjugate to complex conjugation
c
, the map
σ 7→ n(σ) + n(c
0
σ) is constant.
(iv)
(in the case
K C
and
K/Q
is Galois with Galois group
G
) Let
λ
=
P
n(σ)σ Z[G]. Then for all τ G, we have
(τ 1)(c + 1)λ = 0 = (c + 1)(τ 1)λ.
Note that in (iii), the constant is necessarily
2
[K : Q]
X
σ
n(σ).
So in particular, it is independent of c
0
.
Proof.
(iii) (iv): This is just some formal symbol manipulation.
(ii) (i): The norm takes units to units.
(i)
(iii): By the previous lecture, we know that if
ϕ
is of Serre type,
then
|ϕ(K
×,1
)| = 1.
Now if (x
v
) K
×
, we have
|ϕ((x
v
))| =
Y
real v
|x
v
|
n(σ
v
)
Y
complex v
|x
v
|
n(σ
v
)+n(¯σ
v
)
=
Y
v
|x
v
|
1
2
(n(σ
v
)+n(¯σ
v
))
v
.
Here the modulus without the subscript is the usual modulus. Then
|ϕ
(
K
×,1
)
|
= 1 implies
n
(
σ
v
) +
n
(
¯σ
v
) is constant. In other words,
n
(
σ
) +
n() = m is constant.
But if
τ Gal
(
E/Q
), and
ϕ
0
=
τ ϕ
,
n
0
(
σ
) =
n
(
τ
1
σ
), then this is also of
Serre type. So
m = n
0
(σ) + n
0
() = n(τ
1
σ) + n(τ
1
) = n(τ
1
σ) + n((τ
1
)τ
1
σ).
(iii)
(ii): Suppose
n
(
σ
) +
n
(
c
0
σ
) =
m
for all
σ
and all
c
0
=
τ
1
. Then
we must have
n(c
0
σ) = n()
for all σ. So
n(σ) = n(
1
σ)
So
n
is invariant under
H
= [
c, Gal
(
E/Q
)]
Gal
(
E/Q
), noting that
c
has order 2. So
ϕ
takes values in the fixed field
E
H
=
E Q
CM
. By
the proposition last time, this implies
ϕ
factors through
N
K/F
, where
F = E
H
K = K Q
CM
.
Recall that a homomorphism
ϕ
:
K
×
C
×
is algebraic iff it is a character
of the commutative algebraic group
T
K
=
R
K/Q
G
m
, so that
T
K
(
Q
) =
K
×
, i.e.
there is an algebraic character
ϕ
0
:
T
K
/C G
m
/C
such that
ϕ
0
restricted to
T
K
(Q) is ϕ.
Then
ϕ
is of Serre type iff
ϕ
is a character of
K
S
0
=
T
K
/E
0
K
, where
E
K
is the
Zariski closure of
O
×
K
in
T
K
and
E
0
K
is the identity component, which is the same
as the Zariski closure of
O
×
K
, where is a sufficiently small finite-index
subgroup.
The group
K
S
0
is called the connected Serre group. We have a commutative
diagram (with exact rows)
1 K
×
J
K
J
K
/K
×
1
1
K
S
0 K
S Γ
ab
K
1
π
0
This
K
S is a projective limit of algebraic groups over Q. We have
Hom(
K
S, C
×
) = Hom(
K
S, G
m
/C) = {algebraic Hecke characters of K}
The infinity type is just the restriction to
K
S
0
.
Langlands created a larger group, the Tamiyama group, an extension of
Gal
(
¯
Q/Q
) by
K
S
0
, which is useful for abelian varieties with CM and conjugations
and Shimura varieties.