IV Topics in Number Theory

2L-functions

2.1 Hecke characters

Definition

(Hecke character)

A Hecke character is a continuous (not necessarily

unitary) homomorphism

χ : J

→ 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

. Doing so is useful

if we want to write down actual Hecke characters. The theory of

-functions

will be deferred to the next chapter.

We begin with the following result:

Proposition.

Let

be a profinite group, and

G → GL

(

) 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

(

) has “no small subgroups”, i.e. there is an open neighbourhood

∈ GL

(

) such that

contains no non-trivial subgroup of

(

) (exercise!).

For example, if n = 1, then we can take U to be the right half plane.

Then for such

, we know

−1

(

) is open. So it contains an open subgroup

. Then

(

) is a subgroup of

(

) contained in

, 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

be a local field. Then any continuous homomorphism

→ C

has an open kernel, i.e. χ(1 + p

) = 1 for some N  0.

Definition

(Unramified character)

is a local field, a character

F → C

is unramified if

χ|

= 1.

If F

∼

R, we say χ : F

→ C

is unramified if χ(−1) = 1.

Using the decomposition F

∼

× hπ

i (for the local case), we see that

Proposition. χ is unramified iff χ(x) = |x|

for some s ∈ C.

We now return to global fields. We will think of Hecke characters as continuous

maps

→ C

that factor through

, since it is easier to reason about

than the quotient. We can begin by discussing arbitrary characters

→ C

Proposition.

The set of continuous homomorphisms

→ C

bijects with the set of all families (

)

v∈Σ

→ C

such that

unramified for almost all (i.e. all but finitely many)

, with the bijection given

by χ 7→ (χ

), χ

= χ|

Proof.

Let

→ C

be a character. Since

⊆ J

is profinite, we know

ker χ|

is an open subgroup. Thus, it contains

for all but finitely many

So we have a map from the LHS to the RHS.

In the other direction, suppose we are given a family (

)

. We attempt to

define a character χ : J

→ C

χ(x

) =

By assumption,

(

) = 1 for all but finitely many

. So this is well-defined.

These two operations are clearly inverses to each other.

In general, we can write χ as

χ = χ

∞

, χ

∞

v-∞

: K

×,∞

→ C

, χ

∞

v|∞

: K

∞

→ C

Lemma. Let χ be a Hecke character. Then the following are equivalent:

(i) χ has finite image.

(ii) χ

∞

×,0

∞

) = 1.

(iii) χ

∞

= 1.

(iv) χ(C

) = 1.

(v) χ factors through Cl

(K) for some modulus m.

In this case, we say χ is a ray class character.

Proof.

Since

∞

(

×,0

∞

) is either 1 or infinite, we know (i)

⇒

(ii). It is clear that

(ii) ⇔ (iii), and these easily imply (iv). Since C

is profinite, if (iii) holds,

then

factors through

and has open kernel, hence the kernel contains

for some modulus

. So

factors through

(

). Finally, since

(

) is

finite, (v) ⇒ (i) is clear.

Using this, we are able to classify all Hecke characters when K = Q.

Example.

The idele norm

| · |

→ R

is a character not of finite order.

In the case

, we have

. The idele norm is then the

projection onto R

Thus, if

;

→ C

is a Hecke character, then the restriction to

is of

the form x 7→ x

for some s. If we write

χ(x) = |x|

· χ

(x)

for some

, then

vanishes on

, hence factors through

and has finite

order. Thus, it factors as

: C

→

→ (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

, we have finite order characters as we just saw. They

correspond to characters on

which are trivial on

. 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

Pick a modulus

such that

∞

is trivial on

∩ U

. Let

be the set of

finite

such that

is positive, and let

be the set of fractional ideals prime

to S. We then define a homomorphism

Θ : I

→ C

7→ χ

(π

)

−1

One would not expect Θ to remember much information about the infinite

part

∞

. However, once we know Θ and

∞

(and

), it is not difficult to see

that we can recover all of χ.

On the other hand, an arbitrary pair (Θ

, χ

∞

) doesn’t necessarily come from

a Hecke character. Indeed, the fact that

vanishes on

implies there is some

compatibility condition between Θ and χ

∞

Suppose x ∈ K

is such that x ≡ 1 mod m. Then (x) ∈ I

, and we have

1 = χ(x) = χ

∞

(x)

v6∈S finite

(x) = χ

∞

(x)

finite v6∈S

(π

)

v(x)

Writing

for the set of principal ideals generated by these

, as we previously

did, we see that for all x ∈ P

∞

(x) = Θ(x).

One can check that given (Θ

, χ

∞

) (and

) 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

(

), and fix an embedding into

. Since

(

) = 1, we

have

= {±1, ±i}

Let

be the place over 2, corresponding to the prime (1 +

)

. Then

a ramified extension over Q

of degree 2. Moreover,



(1 + i)



= µ

= {±1, ±i}.

So we have a decomposition

= (1 + (1 + i)

) × µ

Thus, there is a natural projection



× O

∼

× (1 + (1 + i)

)  C

This gives a Hecke character with χ

∞

(z) = z, and is trivial on

v6∈{v

,∞}

× (1 + (1 + i)

This has modulus

m = 3(v

In ideal-theoretic terms, if

p 6

= (1 +

) is a prime ideal of

, then

= (

)

for a unique π

∈ O

with π

≡ 1 mod (1 + i)

. Then Θ sends p to π

This is an example of an algebraic Hecke character.

Definition

(Algebraic homomorphism)

A homomorphism

→ C

is alge-

braic if there exists integers n(σ) (for all σ : K → C) such that

ϕ(x) =

σ(x)

n(σ)

The first thing to note is that if

is algebraic, then

(

) is contained

in the Galois closure of

. 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

for

= 1

, . . . , n

as a vector space, then we can view

as the

-points of an

-dimensional affine group scheme. We can then define

K/Q

⊆ A

to be the set on which

is invertible, and then an algebraic

Hecke character is a homomorphism of algebraic groups (

)

/C → G

, where

= Res

K/Q

If we have a real place

, then this corresponds to a real embedding

K → K

∼

, and if

is a complex place, we have a pair of embedding

, ¯σ

K → K

' C

, picking one of the pair to be

. So

extends to a

homomorphism

ϕ : K

∞

→ C

given by

ϕ(x

) =

v real

n(σ

)

v complex

n(σ

)

¯x

n(¯σ

)

Definition

(Algebraic Hecke character)

A Hecke character

∞

→ C

is algebraic if there exists an algebraic homomorphism

→ C

such that

(

) =

∞

(

) for all

x ∈ K

×,0

∞

, i.e.

∞

v real

sgn

for e

∈ {0, 1}.

We say ϕ (or the tuple (n(σ))

) is the infinite type of χ.

Example. The adelic norm | · |

: J

→ C

has

∞

| · |

and so

is algebraic, and the associated

is just

K/Q

→ Q

⊆ C

, with

) = (1, . . . , 1).

Exercise.

Let

(

), and

from the previous example, whose associated

character of ideals was Θ :

p 7→ π

, where

≡

mod

(2 + 2

). The infinity type

is the inclusion K

→ C

, i.e. it has type (1, 0).

Observe that the image of an algebraic homomorphism

→ C

lies in

the normal closure of K. More generally,

Proposition.

is an algebraic Hecke character, then

∞

takes values in

some number field. We write E = E(χ) for the smallest such field.

Of course, we cannot expect

to take algebraic values, since

contains

copies of R and C.

Proof.

Observe that

∞

(

) is finite subgroup, so is

for some

. Let

x ∈ K

totally positive. Then

∞

(x) = χ

∞

(x)

−1

= ϕ(x)

−1

∈ K

where

is the Galois closure. Then since

→

has finite cokernel

(by the finiteness of the class group), so

∞

(

) =

i=1

∞

where

∈ χ

∞

(

), and is therefore contained inside a finite extension of

the image of K

Hecke characters of finite order (i.e. algebraic Hecke characters with infinity

type (0

, . . . ,

0)) are in bijection with continuous homomorphisms Γ

→ C

necessarily of finite order. What we show now is how to associate to a general

algebraic Hecke character

a continuous homomorphism



: Γ

→ E

(

)

⊇



, where

is a place of

(

) over



. This is continuous for the



-adic topology

. In general, this will not be of finite order. Thus, algebraic Hecke

characters correspond to -adic Galois representations.

The construction works as follows: since

∞

(

) =

(

)

−1

, we can restrict

the infinity type

to a homomorphism

→ E

. We define

˜χ

→ E

as follows: if x = x

∞

∈ K

∞

∞,×

∈ J

, then we set

˜χ(x) = χ(x)ϕ(x

∞

)

−1

Notice that this is not trivial on

in general. Then

˜χ

∞

takes values in

{±

}

Thus,

˜χ

takes values in

. Thus, we know that

˜χ

has open kernel, i.e. it is

continuous for the discrete topology on E

, and ˜χ|

= ϕ

−1

Conversely, if

˜χ

→ E

is a continuous homomorphism for the discrete

topology on

, and

˜χ|

is an algebraic homomorphism, then it comes from

an algebraic Hecke character in this way.

Let

be a finite place of

over



, a rational prime. Recall that

→ E

is an algebraic homomorphism, i.e.





= f(x), f ∈ E(X

, . . . , X

We can extend this to K



= (K ⊗



)

v|

to get a homomorphism

: K



→ E

This is still algebraic, so it is certainly continuous for the -adic topology.

Now consider the character ψ

: J

→ E

, where now

((x

)) = ˜χ(x)ϕ

((x

)

v|

This is then continuous for the



-adic topology on

, and moreover, we see

that

(

) =

{

}

˜χ|

−1

while

. Since

˜χ

(

×,0

∞

) =

{

}

, we

know that ψ

it is in fact defined on C

∼

Obviously, ψ

determines ˜χ and hence χ.

Fact.



-adic character

→ E

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(

) for the primes v | .

Example. Let K = Q and χ = | · |

, then

˜χ = sgn(x

∞

)



((x

)) = sgn(x

∞

)

p6=

· |x



· x



Note that |x



∈ Z



. We have

∼

Under this isomorphism, the map



is just the projection onto



followed

by the inclusion, and by class field theory,



Gal

(

Q/Q

)

→ Z



is just the

cyclotomic character of the field Q({ζ



}),

σ(ζ



) = ζ

(σ) mod 



Example.

Consider the elliptic curve

− 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



E = lim E[

]

is a Z



[i]-module. If λ | , then we define

E = T



E ⊗

[i]

Then Γ

act by Γ

: Aut

E = K

We now want to study the infinity types of an algebraic Hecke character.

Lemma.

Let

be a number field,

→ E

⊆ C

be an algebraic

homomorphism, and suppose E/Q is Galois. Then ϕ factors as

norm

−→ (K ∩ E)

−→ E

Note that since E is Galois, the intersection K ∩ E makes perfect sense.

Proof. By definition, we can write

ϕ(x) =

σ:K→C

σ(x)

n(σ)

Then since ϕ(x) ∈ E, for all x ∈ K

and τ ∈ Γ

, we have

τσ(x)

n(σ)

σ(x)

n(σ)

In other words, we have

σ(x)

n(τ

−1

σ)

σ(x)

n(σ)

Since the homomorphisms

are independent, we must have

(

τσ

) =

(

) for

all embeddings σ : K →

Q and τ ∈ Γ

. This implies the theorem.

Recall that if

is a modulus, then we defined open subgroups

⊆ J

consisting of the elements (

) such that if a real

v | m

, then

0, and if

v | m

for a finite v, then v(x

− 1) ≥ m

. We can write this as

= U

m,∞

× U

∞

Proposition.

Let

→ C

be an algebraic homomorphism. Then

the infinity type of an algebraic Hecke character χ iff ϕ(O

) is finite.

Proof.

To prove the (

⇒

) direction, suppose

∞

is an algebraic Hecke

character with infinity type

. Then

∞

(

∞

) = 1 for some

. Let

∩U

⊆ O

, a subgroup of finite index. As

∞

(

) = 1 =

(

), we know

∞

) = 1. So ϕ(O

) is finite.

To prove (

⇐

), given

with

(

) finite, we can find some

such that

(

) = 1. Then (

ϕ,

1) :

∞

× U

∞

→ C

is trivial on

. So we can extend

this to a homomorphism

∞

∼

∞

→ C

since

∩ U

. But the LHS is a finite index subgroup of

. So the

map extends to some χ.

Here are some non-standard terminology:

Definition

(Serre type)

A homomorphism

→ C

is of Serre type if it

is algebraic and ϕ(O

) is finite.

These are precisely homomorphisms that occur as infinity types of algebraic

Hecke characters.

Note that the unit theorem implies that

→ K

×,1

∞

= {x ∈ K

∞

: |x|

= 1}

has compact cokernel. If

(

) is finite, then

(

×,1

∞

) is compact. So it maps

into U(1).

Example. Suppose K is totally real. Then

∞

= (R

)

{σ:K→R}

Then we have

×,1

∞

= {(x

) :

= ±1}.

Then

((

)) =

n(σ)

, so

|ϕ

(

×,1

∞

)

= 1. In other words, all the

are equal.

Thus, ϕ is just a power of the norm map.

Thus, algebraic Hecke characters are all of the form

| · |

· (finite order character).

Another class of examples comes from CM fields.

Definition

(CM field)

. K

is a CM field if

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

) =

{

, c}

Gal

(

) for every

w | v | ∞

. So

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 ⊆ C

given by the fixed field of the subgroup

hcσcσ

−1

: σ ∈ Γ

i ⊆ Γ

For example, we see that the compositum of two CM fields is another CM field.

Exercise.

Let

be a totally complex

-extension over

. Then

is not CM,

but the quadratic subfields is complex and is equal to K ∩ Q

Example.

Let

be a CM field of degree 2

. Then Dirichlet’s unit theorem

tells us

rk O

= r − 1 = rk O

is a finite index subgroup of

. So

→ C

is of Serre type iff

it is algebraic and its restriction to

+,×

is of Serre type. In other words, we

need n(σ) + n(¯σ) to be independent of σ.

Theorem.

Suppose

is arbitrary, and

→ E

⊆ C

is algebraic, and

we assume

E/Q

is Galois, containing the normal closure of

. Thus, we can

write

ϕ(x) =

σ:K→E

σ(x)

n(σ)

Then the following are equivalent:

(i) ϕ is of Serre type.

(ii) ϕ

ψ ◦N

K/F

, where

is the maximal CM subfield and

is of Serre type.

(iii)

For all

∈ Gal

(

E/Q

) conjugate to complex conjugation

, the map

σ 7→ n(σ) + n(c

σ) is constant.

(iv)

(in the case

K ⊆ C

and

K/Q

is Galois with Galois group

) Let

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

[K : Q]

n(σ).

So in particular, it is independent of c

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

is of Serre type,

then

|ϕ(K

×,1

∞

)| = 1.

Now if (x

) ∈ K

∞

, we have

|ϕ((x

))| =

real v

n(σ

)

complex v

n(σ

)+n(¯σ

)

(n(σ

)+n(¯σ

))

Here the modulus without the subscript is the usual modulus. Then

|ϕ

(

×,1

∞

)

= 1 implies

(

) +

(

¯σ

) is constant. In other words,

(

) +

n(cσ) = m is constant.

But if

τ ∈ Gal

(

E/Q

), and

τ ◦ϕ

(

) =

(

−1

), then this is also of

Serre type. So

m = n

(σ) + n

(cσ) = n(τ

−1

σ) + n(τ

−1

cσ) = n(τ

−1

σ) + n((τ

−1

cτ)τ

−1

σ).

–

(iii)

⇒

(ii): Suppose

(

) +

(

) =

for all

and all

τcτ

−1

. Then

we must have

n(c

σ) = n(cσ)

for all σ. So

n(σ) = n(cτcτ

−1

σ)

is invariant under

= [

c, Gal

(

E/Q

)]

≤ Gal

(

E/Q

), noting that

has order 2. So

takes values in the fixed field

E ∩ Q

. By

the proposition last time, this implies

factors through

K/F

, where

F = E

∩ K = K ∩ Q

Recall that a homomorphism

→ C

is algebraic iff it is a character

of the commutative algebraic group

K/Q

, so that

(

) =

, i.e.

there is an algebraic character

/C → G

such that

restricted to

(Q) is ϕ.

Then

is of Serre type iff

is a character of

, where

is the

Zariski closure of

and

is the identity component, which is the same

as the Zariski closure of ∆

⊆ O

, where ∆ is a sufficiently small finite-index

subgroup.

The group

is called the connected Serre group. We have a commutative

diagram (with exact rows)

1 K

0 K

S Γ

This

S is a projective limit of algebraic groups over Q. We have

Hom(

S, C

) = Hom(

S, G

/C) = {algebraic Hecke characters of K}

The infinity type is just the restriction to

Langlands created a larger group, the Tamiyama group, an extension of

Gal

(

Q/Q

) by

, which is useful for abelian varieties with CM and conjugations

and Shimura varieties.