IV Topics in Number Theory (Full)

Part IV — Topics in Number Theory

Based on lectures by A. J. Scholl

Notes taken by Dexter Chua

Lent 2018

These notes are not endorsed by the lecturers, and I have modified them (often

significantly) after lectures. They are nowhere near accurate representations of what

was actually lectured, and in particular, all errors are almost surely mine.

The “Langlands programme” is a far-ranging series of conjectures describing the

connections between automorphic forms on the one hand, and algebraic number theory

and arithmetic algebraic geometry on the other. In these lectures we will give an

introduction to some aspects of this programme.

Pre-requisites

The course will follow on naturally from the Michaelmas term courses Algebraic Number

Theory and Modular Forms and L-Functions, and knowledge of them will be assumed.

Some k nowledge of algebraic geometry will be required in places.

Contents

0 Introduction

1 Class field theory

1.1 Preliminaries

1.2 Local class field theory

1.3 Global class field theory

1.4 Ideal-theoretic description of global class field theory

2 L-functions

2.1 Hecke characters

2.2 Abelian L-functions

2.3 Non-abelian L-functions

3 -adic representations

4 The Langlands correspondence

4.1 Representations of groups

4.2 Hecke algebras

4.3 The Langlands classification

4.4 Local Langlands correspondence

5 Modular forms and representation theory

0 Introduction

In this course, we shall first give an outline of class field theory. We then look at

abelian

-functions (Hecke, Tate). We then talk about non-abelian

-functions,

and in particular the Weil–Deligne group and local L- and ε-factors.

We then talk about local Langlands for

a bit, and do a bit of global

theory and automorphic forms at the end.

The aim is not to prove everything, because that will take 3 courses instead of

one, but we are going to make precise definitions and statements of everything.

1 Class field theory

1.1 Preliminaries

Class field theory is the study of abelian extensions of local or global fields.

Before we can do class field theory, we must first know Galois theory.

Notation.

Let

be a field. We will write

for a separable closure of

, and

= Gal(

K/K). We have

= lim

L/K finite separable

Gal(L/K),

which is a profinite group. The associated topology is the Krull topology.

Galois theory tells us

Theorem (Galois theory). There are bijections



closed subgroups of



←→



subfields

K ⊆ L ⊆





open subgroups of



←→



finite subfields

K ⊆ L ⊆



Notation.

We write

for the maximal abelian subextension of

, and then

Gal(K

/K) = Γ

[Γ

, Γ

]

It is crucial to note that while

is unique, it is only unique up to non-

canonical isomorphism. Indeed, it has many automorphisms, given by elements

Gal

(

K/K

). Thus, Γ

is well-defined up to conjugation only. On the other

hand, the abelianization Γ

is well-defined. This will be important in later

naturality statements.

Definition

(Non-Archimedean local field)

A non-Archimedean local field is a

finite extension of Q

or F

((t)).

We can also define Archimedean local fields, but they are slightly less inter-

esting.

Definition

(Archimedean local field)

An Archimedean local field is a field that

is R or C.

is a non-Archimedean local field, then it has a canonical normalized

valuation

v = v

: F

 Z.

Definition

(Valuation ring)

The valuation ring of a non-Archimedean local

field F is

O = O

= {x ∈ F : v(x) ≥ 0}.

Any element

∈ O

with

(

) = 1 is called a uniformizer . This generates

the maximal ideal

m = m

= {x ∈ O

: v(x) ≥ 1}.

Definition

(Residue field)

The residue field of a non-Archimedean local field

F is

k = k

= O

This is a finite field of order q = p

A particularly well-understood subfield of

is the maximal unramified

extension F

. We have

Gal(F

/F ) = Gal(

k/k) =

Z = lim

n≥1

Z/nZ.

and this is completely determined by the behaviour of the residue field. The rest

of Γ

is called the inertia group.

Definition (Inertia group). The inertia group I

is defined to be

= Gal(

F /F

) ⊆ Γ

We also define

Definition

(Wild inertia group)

The wild inertia group

is the maximal

pro-p-subgroup of I

Returning to the maximal unramified extension, note that saying

Gal

(

k/k

)

∼

requires picking an isomorphism, and this is equivalent to picking an element

Z to be the “1”. Naively, we might pick the following:

Definition

(Arithmetic Frobenius)

The arithmetic Frobenius

∈ Gal

(

k/k

)

(where |k| = q) is defined to be

(x) = x

Identifying this with 1

∈

leads to infinite confusion, and we shall not do so.

Instead, we define

Definition (Geometric Frobenius). The geometric Frobenius is

Frob

= ϕ

−1

∈ Gal(

k/k).

We shall identify Gal(

k/k)

∼

Z by setting the geometric Frobenius to be 1.

The reason this is called the geometric Frobenius is that if we have a scheme

over a finite field

, then there are two ways the Frobenius can act on it — either

as a Galois action, or as a pullback along the morphism (

−

)

k → k

. The latter

corresponds to the geometric Frobenius.

We now turn to understand the inertia groups. The point of introducing the

wild inertia group is to single out the “

-phenomena”, which we would like to

avoid. To understand

better, let

be a natural number prime to

. As usual,

we write

(

k) = {ζ ∈

k : ζ

= 1}.

We also pick an nth root of π in

F , say π

. By definition, this has π

= π.

Definition

(Tame mod

character)

The tame mod

character is the map

t(n) : I

= Gal(

F /F

) → µ

(

k) given by

γ 7→ γ(π

)/π

(mod π).

Note that since γ fixes π = π

, we indeed have



γ(π

)



γ(π

)

= 1.

Moreover, this doesn’t depend on the choice of

. Any other choice differs by

th root of unity, but the

th root of unity lies in

since

is prime to

. So

fixes it and so it cancels out in the fraction. For the same reason, if

moves

at all, then this is visible down in

, since

would have multiplied

by an nth root of unity, and these nth roots are present in

Now that everything is canonically well-defined, we can take the limit over

all n to obtain a map

t : I

→ lim

(n,p)=1

(

k) =

6=p

lim

m≥1



(

k) ≡

6=p



(1)(

k).

This



(1)(

) is the Tate module of

. This is isomorphic to



, but not

canonically.

Theorem. ker

t = P

Thus, it follows that maximal tamely ramified extension of

, i.e. the fixed

field of P

[

(n,p)=1

(

√

π).

Note that

(

) extends to a map Γ

→ µ

given by the same formula, but

this now depends on the choice of

, and further, it is not a homomorphism,

because

t(n)(γδ) =

γδ(π

)

γ(π

)



δ(π

)



= t(n)(γ) · γ(t(n)(δ)).

So this formula just says that

(

) is a 1-cocycle. Of course, picking another

will modify t(n) by a coboundary.

1.2 Local class field theory

Local class field theory is a (collection of) theorems that describe abelian exten-

sions of a local field. The key takeaway is that finite abelian extensions of

correspond to open finite index subgroups of

, but the theorem says a bit

more than that:

Theorem (Local class field theory).

(i)

Let

be a local field. Then there is a continuous homomorphism, the

local Artin map

Art

: F

→ Γ

with dense image characterized by the properties

(a) The following diagram commutes:

Art

∼

(b) If F

/F is finite, then the following diagram commutes:

)

= Gal(F

0ab

)

= Gal(F

/F )

Art

restriction

Art

(ii) Moreover, the existence theorem says Art

−1

induces a bijection



open finite index

subgroups of F



←→



open subgroups of Γ



Of course, open subgroups of Γ

further corresponds to finite abelian

extensions of F .

(iii) Further, Art

induces an isomorphism

∼

→ im(I

→ Γ

)

and this maps (1 +

πO

)

to the image of

. Of course, the quotient

/(1 + πO

)

∼

= µ

∞

(k).

(iv)

Finally, this is functorial, namely if we have an isomorphism

∼

→ F

and

extend it to

¯α

∼

→

, then this induces isomorphisms between the Galois

groups

∗

: Γ

∼

→

(up to conjugacy), and

∗

◦Art

Art

◦α

∗

On the level of finite Galois extensions

E/F

, we can rephrase the first part

of the theorem as giving a map

Art

E/F

)

→ Gal(E/F )

which is now an isomorphism (since a dense subgroup of a discrete group is the

whole thing!).

We can write down these maps explicitly in certain special cases. We will

not justify the following example:

Example. If F = Q

, then

= Q

(µ

∞

) =

[

(µ

) = Q

(µ

∞

Moreover, if we write x ∈ Q

as p

y with y ∈ Z

, then

Art

(x)|

= Frob

, Art

(x)|

(µ

∞

)

= (ζ

7→ ζ

y mod p

If we had the arithmetic Frobenius instead, then we would have a

−y

in the

power there, which is less pleasant.

The cases of the Archimedean local fields are easy to write down and prove

directly!

Example.

, then Γ

= Γ

= 1 is trivial, and the Artin map is similarly

trivial. There are no non-trivial open finite index subgroups of

, just as there

are no non-trivial open subgroups of the trivial group.

Example.

, then

and Γ

= Γ

{±

}

. The Artin

map is given by the sign map. The unique open finite index subgroup of R

, and this corresponds to the finite Galois extension C/R.

As stated in the theorem, the Artin map has dense image, but is not surjective

(in general). To fix this problem, it is convenient to introduce the Weil group.

Definition

(Weil group)

Let

be a non-Archimedean local field. Then the

Weil group of F is the topological group W

defined as follows:

– As a group, it is

= {γ ∈ Γ

| γ|

= Frob

for some n ∈ Z}.

Recall that

Gal

(

) =

, and we are requiring

γ|

to be in

. In

particular, I

⊆ W

–

The topology is defined by the property that

is an open subgroup with

the profinite topology. Equivalently,

is a fiber product of topological

groups

where Z has the discrete topology.

Note that W

is not profinite. It is totally disconnected but not compact.

This seems like a slightly artificial definition, but this is cooked up precisely

so that

Proposition. Art

induces an isomorphism of topological groups

Art

: F

→ W

This maps O

isomorphically onto the inertia subgroup of Γ

In the case of Archimedean local fields, we make the following definitions.

They will seem rather ad hoc, but we will provide some justification later.

–

The Weil group of

is defined to be

, and the Artin map

Art

is defined to be the identity.

– The Weil group of R is defined the non-abelian group

= hC

, σ | σ

= −1 ∈ C

, σzσ

−1

= ¯z for all z ∈ C

This is a (non)-split extension of C

by Γ

1 → C

→ W

→ Γ

→ 1,

where the last map sends

z 7→

0 and

σ 7→

1. This is in fact the unique

non-split extension of Γ

by C

where Γ

acts on C

in a natural way.

The map Art

is better described by its inverse, which maps

(Art

)

−1

: W

∼

−→ R

z 7−→ z¯z

σ 7−→ −1

To understand these definitions, we need the notion of the relative Weil

group.

Definition

(Relative Weil group)

Let

be a non-Archimedean local field, and

E/F Galois but not necessarily finite. We define

E/F

= {γ ∈ Gal(E

/F ) : γ|

= Frob

, n ∈ Z} =

, W

]

with the quotient topology.

The W

F /F

= W

, while W

F/F

= W

= F

by local class field theory.

Now if

E/F

is a finite extension, then we have an exact sequence of Galois

groups

1 Gal(E

/E) Gal(E

/F ) Gal(E/F ) 1

1 W

E/F

Gal(E/F ) 1.

By the Artin map,

. So the relative Weil group is an extension of

Gal(E/F ) by E

. In the case of non-Archimedean fields, we have

lim

= {1},

where the field extensions are joined by the norm map. So

is invisible in

lim W

E/F

. The weirdness above comes from the fact that the separable

closures of R and C are finite extensions.

We are, of course, not going to prove local class field theory in this course.

However, we can say something about the proofs. There are a few ways of

proving it:

–

The cohomological method (see Artin–Tate, Cassels–Fr¨ohlich), which only

treats the first part, namely the existence of

Art

. We start off with a

finite Galois extension E/F , and we want to construct an isomorphism

Art

E/F

: F

E/F

) → Gal(E/F )

Writing G = Gal(E/F ), this uses the cohomological interpretation

E/F

) =

(G, E

where

is the Tate cohomology of finite groups. On the other hand, we

have

= H

(G, Z) =

−2

(G, Z).

The main step is to compute

(G, E

) =

(G, E

)

∼

Z/Z ⊆ Q/Z = H

(Γ

where

= [

]. The final group

(Γ

) is the Brauer group

(

and the subgroup is just the kernel of Br(F ) → Br(E).

Once we have done this, we then define

Art

E/F

to be the cup product

with the generator of

(

G, E

), and this maps

−2

(

G, Z

)

→

(

G, E

The fact that this map is an isomorphism is rather formal.

The advantage of this method is that it generalizes to duality theorems

about

∗

(

G, M

) for arbitrary

, but this map is not at all explicit, and

is very much tied to abelian extensions.

–

Formal group methods: We know that the maximal abelian extension of

is obtained in two steps — we can write

= Q

(µ

∞

) = Q

(torsion points in

where

is the formal multiplication group, which we can think of as

(1 + m

)

. This generalizes to any F/Q

— we have

= F

(torsion points in

where

is the “Lubin–Tate formal group”. This is described in Iwasawa’s

book, and also in a paper of Yoshida’s. The original paper by Lubin and

Tate is also very readable.

The advantage of this is that it is very explicit, and when done correctly,

gives both the existence of the Artin map and the existence theorem. This

also has a natural generalization to non-abelian extensions. However, it

does not give duality theorems.

–

Neukrich’s method: Suppose

E/F

is abelian and finite. If

g ∈ Gal

(

E/F

we want to construct

Art

−1

E/F

(

)

∈ F

E/F

(

). The point is that there

is only one possibility, because

hgi

is a cyclic subgroup of

Gal

(

E/F

), and

corresponds to some cyclic extension

Gal

(

E/F

). We have the following

lemma:

Lemma.

There is a finite

K/F

such that

K ∩E

, so

Gal

(

KE/K

)

∼

Gal(E/F

) = hgi. Moreover, KE/K is unramified.

Let

, and suppose

Frob

KE/K

. If local class field theory is

true, then we have to have

Art

−1

KE/K

) = π

(mod N

KE/K

(KE

)).

Then by our compatibility conditions, this implies

Art

−1

E/F

(g) = N

K/F

(π

) (mod N

E/F

)).

The problem is then to show that this does not depend on the choices,

and then show that it is a homomorphism. These are in fact extremely

complicated. Note that everything so far is just Galois theory. Solving

these two problems is then where all the number theory goes in.

–

When class field theory was first done, we first did global class field theory,

and deduced the local case from that. No one does that anymore nowadays,

since we now have purely local proofs. However, when we try to generalize

to the Langlands programme, what we have so far all start with global

theorems and then proceed to deduce local results.

1.3 Global class field theory

We now proceed to discuss global class field theory.

Definition

(Global field)

A global field is a number field or

(

) for a smooth

projective absolutely irreducible curve C/F

, i.e. a finite extension of F

(t).

A lot of what we can do can be simultaneously done for both types of global

fields, but we are mostly only interested in the case of number fields, and our

discussions will mostly focus on those.

Definition

(Place)

Let

be a global field. Then a place is a valuation on

is a number field, we say a valuation

is a finite place if it is the valuation

at a prime

p C O

. A valuation

is an infinite place if comes from a complex

or real embedding of

. We write Σ

for the set of places of

, and Σ

∞

and

K,∞

for the sets of finite and infinite places respectively. We also write

v - ∞

if v is a finite place, and v | ∞ otherwise.

is a function field, then all places are are finite, and these correspond to

closed points of the curve.

v ∈

is a place, then there is a completion

K → K

. If

is infinite,

then

, i.e. an Archimedean local field. Otherwise,

is a non-

Archimedean local field.

Example.

, then there is one infinite prime, which we write as

∞

given by the embedding

Q → R

. If

, then we get the embedding

Q → Q

into the p-adic completion.

Notation.

is a finite place, we write

⊆ K

for the valuation ring of the

completion.

Any local field has a canonically normalized valuation, but there is no

canonical absolute value. It is useful to fix the absolute value of our local fields.

For doing class field theory, the right way to put an absolute value on

(and

hence K) is by

|x|

= q

−v(x)

where



is the cardinality of the residue field at

. For example, if

and

then q

= p, and |p|

In the Archimedean case, if

is real, then we set

|x|

to be the usual absolute

value; if K

is complex, then we take |x|

= x¯x = |x|

The reason for choosing these normalizations is that we have the following

product formula:

Proposition (Product formula). If x ∈ K

, then

v∈Σ

|x|

= 1.

The proof is not difficult. First observe that it is true for

, and then show

that this formula is “stable under finite extensions”, which extends the result to

all finite extensions of Q.

Global class field theory is best stated in terms of adeles and ideles. We

make the following definition:

Definition (Adele). The adeles is defined to be the restricted product

)

v∈K

: x

∈ O

for all but finitely many v ∈ Σ

∞

We can write this as K

∞

K or A

K,∞

× A

∞

, where

∞

= A

K,∞

= K ⊗

R = R

× C

consists of the product over the infinite places, and

K = A

∞

v-∞

[

S⊆Σ

∞

finite

v∈S

v∈Σ

∞

This contains

v-∞

. In the case of a number field,

is the profinite

completion of O

. More precisely, if K is a number field then

= lim

/a = lim O

/N O

= O

⊗

where the last equality follows from the fact that O

is a finite Z-module.

Definition (Idele). The ideles is the restricted product

= A

)

∈

: x

∈ O

for almost all v

These objects come with natural topologies. On

, we take

∞

to be an open subgroup with the product topology. Once we have done this,

there is a unique structure of a topological ring for which this holds. On

, we

take

∞

to be open with the product topology. Note that this is not the

subspace topology under the inclusion

→ A

. Instead, it is induced by the

inclusion

→ A

× A

x 7→ (x, x

−1

It is a basic fact that K

⊆ J

is a discrete subgroup.

Definition (Idele class group). The idele class group is then

= J

The idele class group plays an important role in global class field theory.

Note that J

comes with a natural absolute value

| · |

: J

→ R

) 7→

v∈Σ

The product formula implies that

{

}

. So this in fact a map

→ R

Moreover, we have

Theorem. The map | · |

: C

→ R

has compact kernel.

This seemingly innocent theorem is actually quite powerful. For example,

we will later construct a continuous surjection

→ Cl

(

) to the ideal class

group of

. In particular, this implies

(

) is finite! In fact, the theorem is

equivalent to the finiteness of the class group and Dirichlet’s unit theorem.

Example. In the case K = Q, we have, by definition,

= R

Suppose we have an idele

= (

). It is easy to see that there exists a unique

rational number

y ∈ Q

such that

sgn

(

) =

sgn

(

∞

) and

(

) =

(

). So we

have

= Q

Here we think of

as being embedded diagonally into

, and as we have

previously mentioned,

is discrete. From this description, we can read out a

description of the idele class group

= R

and

is the kernel | · |

From the decomposition above, we see that

has a maximal connected

subgroup

. In fact, this is the intersection of all open subgroups containing

1. For a general

, we write

for the maximal connected subgroup, and then

) = C

In the case of K = Q, we can naturally identify

) =

= lim

(Z/nZ)

= lim

Gal(Q(ζ

)/Q) = Gal(Q(ζ

∞

)/Q).

The field

(

∞

) is not just any other field. The Kronecker–Weber theorem says

this is in fact the maximal abelian extension of

. Global class field theory is a

generalization of this isomorphism to all fields.

Just like in local class field theory, global class field theory involves a certain

Artin map. In the local case, we just pulled them out of a hat. To construct the

global Artin map, we simply have to put these maps together to form a global

Artin map.

Let

L/K

be a finite Galois extension,

a place of

, and

w | v

a place of

extending

. For finite places, this means the corresponding primes divide; for

infinite places, this means the embedding

is an extension of

. We then have

the decomposition group

Gal(L

) ⊆ Gal(L/K).

L/K

is abelian, since any two places lying above

are conjugate, this depends

only on v. In this case, we can now define the global Artin map

Art

L/K

: J

−→ Gal(L/K)

)

7−→

Art

where we pick one

for each

. To see this is well-defined, note that if

∈ O

and

L/K

is unramified at

v - ∞

, then

Art

(

) = 1. So the product is in

fact finite.

By the compatibility of the Artin maps, we can passing on to the limit over

all extensions L/K, and get a continuous map

Art

: J

→ Γ

Theorem

(Artin reciprocity law)

. Art

(

) =

{

}

, so induces a map

→

. Moreover,

(i)

char

(

) =

p >

0, then

Art

is injective, and induces an isomorphism

Art

∼

→ W

, where

is defined as follows: since

is a finite

extension of

(

), and wlog assume

∩ K

≡ k

. Then

defined as the pullback

= Gal(

K/K)

∼

Gal(

k/k)

restr.

(ii) If char(K) = 0, we have an isomorphism

Art

: π

) =

∼

→ Γ

Moreover, if L/K is finite, then we have a commutative diagram

L/K

Art

restr.

Art

If this is in fact Galois, then this induces an isomorphism

Art

L/K

)

∼

→ Gal(L/K)

Finally, this is functorial, namely if

∼

→ K

is an isomorphism, then we

have a commutative square

Art

Observe that naturality and functoriality are immediate consequences of the

corresponding results for local class field theory.

As a consequence of the isomorphism, we have a correspondence



finite abelian extensions

L/K



←→



finite index open subgroups

of J

containing K



L 7−→ ker(Art

L/K

: J

→ Gal(L/K))

Note that there exists finite index subgroups that are not open!

Recall that in local class field theory, if we decompose

hπi ×O

, then

the local Artin map sends

to (the image of) the inertia group. Thus an

extension

is unramified iff the local Artin map kills of

. Globally,

this tells us

Proposition.

L/K

is an abelian extension of global fields, which corresponds

to the open subgroup

U ⊆ J

under the Artin map, then

L/K

is unramified at

a finite v - ∞ iff O

⊆ U.

We can extend this to the infinite places if we make the appropriate definitions.

Since

cannot be further extended, there is nothing to say for complex places.

Definition

(Ramification)

v | ∞

is a real place of

, and

L/K

is a finite

abelian extension, then we say

is ramified if for some (hence all) places

above v, w is complex.

The terminology is not completely standard. In this case, Neukrich would

say v is inert instead.

With this definition,

L/K

is unramified at a real place

iff

⊆ U

Note that since U is open, it automatically contains R

We can similarly read off splitting information.

Proposition.

is finite and unramified, then

splits completely iff

⊆ U

Proof. v

splits completely iff

for all

w | v

, iff

Art

(

) =

{

}

Example.

We will use global class field theory to compute all

extensions

L/Q which are unramified outside 5 and 7.

If we didn’t have global class field theory, then to solve this problem we have

to find all cubics whose discriminant are divisible by 5 and 7 only, and there is a

cubic diophantine problem to solve.

While

is not an abelian group, it is solvable. So we can break our potential

extension as a chain

2 = hσi

Since

L/Q

is unramified outside 5 and 7, we know that

must be one of

(

√

(

√

−7

) and

(

√

−35

). We then consider each case in turn, and then see what

are the possibilities for

. We shall only do the case

(

√

−7

) here. If we

perform similar computations for the other cases, we find that the other choices

of K do not work.

So fix

(

√

−7

). We want

L/K

to be cyclic of degree 3, and

must act

non-trivially on L (otherwise we get an abelian extension).

Thus, by global class field theory, we want to find a subgroup

U ≤ J

index 3 such that

⊆ U

for all

v -

35. We also need

(

) =

, or else the

composite extension would not even be Galois, and

has to acts as

−

1 on

∼

Z/3Z to get a non-abelian extension.

We know

(

√

−7

) has has class number 1, and the units are

1. So we

know

{±1}

By assumption, we know

contains

v-35

. So we have to look at the places

that divide 35. In O

√

−7)

, the prime 5 is inert and 7 is ramified.

Since 5 is inert, we know

is an unramified quadratic extension. So

we can write

(5)

= F

× (1 + 5O

(5)

)

The second factor is a pro-5 group, and so it must be contained in

for the

quotient to have order 3. On

acts as the Frobenius

(

) =

. Since

is cyclic of order 24, there is a unique index 3 subgroup, cyclic of order

6. This gives an index 3 subgroup

⊆ O

(5)

. Moreover, on here,

acts by

x 7→ x

= x

−1

. Thus, we can take

U =

v6=(5)

× U

and this gives an

extension of

that is unramified outside 5 and 7. It is an

exercise to explicitly identify this extension.

We turn to the prime 7 = −

√

−7

. Since this is ramified, we have

(

√

−7)

= F



1 + (

√

−7)O

√

−7



and again the second factor is a pro-7 group. Moreover

acts trivially on

. So

U must contain O

(

√

−7)

. So what we found above is the unique such extension.

We previously explicitly described

. It would be nice to have

a similar description of

for an arbitrary

. The connected component will

come from the infinite places

∞

v|∞

. The connected component is given

×,0

∞

= (R

)

× (C

)

where there are r

real places and r

complex ones. Thus, we find that

Proposition.

{±1}

There is a natural map from the ideles to a more familiar group, called the

content homomorphism.

Definition

(Content homomorphism)

The content homomorphism is the map

c : J

→ fractional ideals of K

)

7→

v-∞

v(x

)

where

is the prime ideal corresponding to

. We ignore the infinite places

completely.

Observe that

(

) is the set of all principal ideals by definition. Moreover,

the kernel of the content map is

∞

, by definition. So we have a short

exact sequence

1 →

{±1}

→ C

→ Cl(K) → 1.

(

√

−D

), then

is finite, and in particular is closed. But

in general, it will not be closed, and taking the closure is indeed needed.

Returning to the case

, our favorite abelian extensions are those of

the form L = Q(ζ

) with N > 1. This comes with an Artin map

∼

→ Gal(L/Q)

∼

(Z/N Z)

By local class field theory for

, we see that with our normalizations, this is

just the quotient map, whose kernel is

(1 + N

p-N

p|N

(1 + NZ

)

⊆

Note that if we used the arithmetic Frobenius, then we would get the inverse of

the quotient map.

These subgroups of

are rather special ones. First of all (1 +

)

form

a neighbourhood of the identity in

. Thus, any open subgroup contains a

subgroup of this form. Equivalently, every abelian extension of

is contained

in Q(ζ

) for some N . This is the Kronecker–Weber theorem.

For a general number field

, we want to write down an explicit basis for

open subgroups of 1 in π

Definition (Modulus). A modulus is a finite formal sum

m =

v∈Σ

· (v)

of places of K, where m

≥ 0 are integers.

Given a modulus m, we define the subgroup

v|∞,m

×,0

v|∞,m

v-∞,m

(1+p

)

v-∞,m

⊆ J

Then essentially by definition of the topology of

, any open subgroup of

containing K

×,0

∞

contains some U

In our previous example, our moduli are all of the form

Definition

(

∞

))

a C O

is an ideal, we write

(

∞

) for the modulus with

= v(a) for all v - ∞, and m

= 1 for all v | ∞.

and

= (

)(

∞

), then we simply get

(1 +

)

, and

= (Z/nZ)

corresponding to the abelian extension Q(ζ

In general, we define

Definition

(Ray class field)

L/K

is abelian with

Gal

(

L/K

)

∼

under the Artin map, we call L the ray class field of K modulo m.

Definition

(Conductor)

corresponds to

U ⊆ J

, then

U ⊇ K

for

some m. The minimal such m is the conductor of L/K.

1.4 Ideal-theoretic description of global class field theory

Originally, class field theory was discovered using ideals, and the ideal-theoretic

formulation is at times more convenient.

Let

be a modulus, and let

be the set of finite

such that

0. Let

be the group of fractional ideals prime to S. Consider

= {(x) ∈ I

: x ≡ 1 mod m}.

To be precise, we require that for all v ∈ S, we have v(x − 1) ≥ m

, and for all

infinite

real with

0, then

(

)

0 for

K → R

the corresponding to

In other words, x ∈ K

∩ U

Note that if

is trivial, then

is the ideal class group. Thus, it makes

sense to define

Definition

(Ray class group)

Let

be a modulus. The generalized ideal class

group, or ray class group modulo m is

(K) = I

One can show that this is always a finite group.

Proposition. There is a canonical isomorphism

∼

→ Cl

(K)

such that for v 6∈ S ∪ Σ

K,∞

, the composition

→ J

→ Cl

(K)

sends x 7→ p

−v(x)

Thus, in particular, the Galois group

Gal

(

L/K

) of the ray class field modulo

(

). Concretely, if

p 6∈ S

is an ideal, then [

]

∈ Cl

(

) corresponds to

∈ Gal

(

L/K

), the arithmetic Frobenius. This was Artin’s original reciprocity

law.

When

= 0, then this map is the inverse of the map given by content.

However, in general, it is not simply (the inverse of) the prime-to-

content map,

even for ideles whose content is prime to

. According to Fr¨olich, this is the

“fundamental mistake of class field theory”.

Proof sketch. Let J

(S) ⊆ J

be given by

(S) =

v6∈S∪Σ

K,∞

Here we do have the inverse of the content map

−1

: J

(S)  I

) 7→

−v(x

)

We want to extend it to an isomorphism. Observe that

(S) ∩ U

v6∈S∪Σ

K,∞

which is precisely the kernel of the map

−1

. So

−1

extends uniquely to a

homomorphism

(S)U

∼

(S)

(S) ∩ U

→ I

We then use that K

(S)U

= J

(weak approximation), and

∩ V

= {x ≡ 1 mod m, x ∈ K

∗

where

= J

(S)U

= {(x

) ∈ J

| for all v with m

> 0, x

∈ U

2 L-functions

Recall that Dirichlet characters

: (

Z/N Z

)

→ C

give rise to Dirichlet

functions. Explicitly, we extend

to a function on

by setting

(

) = 0 if

[a] 6∈ (Z/N Z)

, and then the Dirichlet L-function is defined by

L(χ, s) =

∞

n=1

χ(n)

(1 − χ(p)p

−s

)

−1

As one can see from the

and

appearing, Dirichlet

-functions are “about

”, and we would like to have a version of

-functions that are for arbitrary

number fields

. If

= 1, we already know what this should be, namely the

ζ-function

(s) =

aCO

N(a)

pCO

1 − N(p)

−s

where the first sum is over all ideals of

, and the second product is over all

prime ideals of O

In general, the replacement of Dirichlet characters is a Hecke character.

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/N Z)

→ 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.

2.2 Abelian L-functions

We are now going to define

-functions for Hecke characters. Recall that amongst

all other things, an

-function is a function in a complex variable. Here we are

going to do things slightly differently. For any Hecke character

, we will define

L(χ), which will be a number. We then define

L(χ, s) = L(| · |

χ).

We shall define

(

) as an Euler product, and then later show it can be written

as a sum.

Definition

(Hecke

-function)

Let

→ C

be a Hecke character. For

v ∈ Σ

, we define local L-factors L(χ

) as follows:

– If v is non-Archimedean and χ

unramified, i.e. χ

= 1, we set

L(χ

) =

1 − χ

(π

)

– If v is non-Archimedean and χ

is ramified, then we set

L(χ

) = 1.

– If v is a real place, then χ

is of the form

(x) = x

−N

|x|

where N = 0, 1. We write

L(χ

) = Γ

(s) = π

−s/2

Γ(s/2).

– If v is a complex place, then χ

is of the form

(x) = σ(x)

−N

|x|

where σ is an embedding of K

into C and N ≥ 0. Then

L(χ

) = Γ

(s) = 2(2π)

−s

Γ(s)

We then define

L(χ

, s) = L(χ

· | · |

So for finite unramified v, we have

L(χ

, s) =

1 − χ

(π

−s

where q

= |O

/(π

)|.

Finally, we define

L(χ, s) =

v-∞

L(χ

, s)

Λ(χ, s) =

L(χ

, s).

Recall that the kernel of the idelic norm

| · |

→ R

is compact. It is

then not hard to see that for every

, there is some

t ∈ R

such that

χ · | · |

unitary. Thus,

(

χ, s

) converges absolutely on some right half-plane. Observe

that

Λ(χ| · |

, s) = Λ(χ, t + s).

Theorem (Hecke–Tate).

(i)

Λ(

χ, s

) has a meromorphic continuation to

, entire unless

| · |

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

for some A ∈ C

and B ∈ R

such that

Λ(χ, s) = ε(χ, s)Λ(χ

−1

, 1 − s).

(iii) There is a factorization

ε(χ, s) =

(χ

, µ

, ψ

, s),

where

= 1 for almost all

, and

depends only on

and certain

auxiliary data ψ

, µ

. These are the local ε-factors.

Traditionally, we write

L(χ, s) =

finite v

L(χ

, s),

and then

Λ(χ, s) = L(χ, s)L

∞

(χ, s).

However, Tate (and others, especially the automorphic people) use

(

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

and

. This generalizes considerably, e.g. to

automorphic representations.

We can explain some ideas of Hecke’s method. We have a decomposition

∞

= K ⊗

∼

× C

∼

and this has a norm

k · k

induces by the Euclidean metric on

. Let ∆

⊆ O

K,+

be a subgroup of totally positive units of finite index, which is

∼

−1

. This

has an embedding ∆ =

×,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

∆⊗R

kuxk

du =

stuff

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 =

∞

−∞

|ε

x + ε

−t

RHS =

stuff

|xx

The consequence of this is that if a ⊆ K is a fractional ideal, then

06=x∈a mod ∆

K/Q

(x)|

= stuff ·

06=x∈a mod ∆

∆⊗R

kuxk

= stuff ·

∆⊗R/∆





06=x∈a

kuxk





The integrand has a name, and is called the Epstein

-function of the lattice

(

a, ku · k

). By the Poisson summation formula, we get an analytic continuity

and functional equation for the epsilon

function. On the other hand, taking

linear combinations of the left gives

(

χ, s

) for

(

)

→ C

. For more

general

, we modify this with some extra factors. When the infinity type is

non-trivial, this is actually quite subtle.

Note that if

is unramified outside

and ramified at

, recall we had a

homomorphism Θ : I

→ C

sending p

7→ χ

(π

)

−1

. So

L(χ, s) =

finite v6∈S



1 − Θ(p

)

−1

(Np

)

−s



a∈O

prime to S

Θ(a)

−1

(Na)

This was Hecke’s original definition of the Hecke character.

and

→ C

is of finite order, then it factors through

→ C

∼

→

(

Z/N Z

)

, and so

is just some Dirichlet character

: (

Z/nZ

)

→ C

. The associated

-functions are just Dirichlet

-functions.

Indeed, if p - N, then

(p) = χ(1, . . . , 1, p, 1, . . .) = χ(p

−1

, . . . , p

−1

, 1, p

−1

, . . .) = ϕ(p mod N )

−1

In other words,

(

χ, s

) is the Dirichlet

-series of

−1

(assuming

is chosen so

that χ ramifies exactly at v | N).

Tate’s method uses local

-factors

(

, µ

, ψ

, s

), where

→

U(1) is

a non-trivial additive character, e.g. for v finite,

∼

Z[v

]/Z C

tr e

2πix

which we needed because Fourier transforms take in additive measures, and

is a Haar measure on K

. The condition for (iii) to hold is

: A

→ U(1)

is well-defined and trivial on

K ⊆ A

, and

is a well-defined measure

on A

, i.e. µ

) = 1 for all v and

= 1.

There exists explicit formulae for these

’s. If

is unramified, then it is just

, and is usually 1; for ramified finite v, they are given by Gauss sums.

2.3 Non-abelian L-functions

Let K be a number field. Then we have a reciprocity isomorphism

Art

: C

∼

→ Γ

→ C

is a Hecke character of finite order, then we can view it

as a map ψ = χ ◦ Art

−1

: Γ

→ C

. Then

L(χ, s) =

finite v unramified

1 − χ

(π

−s

−1

1 − ψ(Frob

−s

where

Frob

∈

is the geometric Frobenius, using that

(

) = 1.

Artin generalized this to arbitrary complex representations of Γ

Let ρ : Γ

→ GL

L(ρ, s) =

finite v

L(ρ

, s),

where

is the restriction to the decomposition group at

, and depends only

on the isomorphism class of

. We first define these local factors for non-

Archimedean fields:

Definition.

Let

be local and non-Archimedean. Let

→ GL

(

) be a

representation. Then we define

L(ρ, s) = det(1 − q

−s

ρ(Frob

)

−1

where V

is the invariants under I

Note that in this section, all representations will be finite-dimensional and

continuous for the complex topology (so in the case of

, we require

ker σ

be open).

Proposition.

(i) If

0 → (ρ

, V

) → (ρ, V ) → (ρ

, V

) → 0

is exact, then

L(ρ, s) = L(ρ

, s) · L(ρ

, s).

(ii)

E/F

is finite separable,

→ GL

(

) and

Ind

→

(U), then

L(ρ, s) = L(σ, s).

Proof.

(i) Since ρ has open kernel, we know ρ(I

) is finite. So

0 → (V

)

→ V

→ (V

)

→ 0

is exact. Then the result follows from the multiplicativity of det.

(ii) We can write

U = {ϕ : W

→ V : ϕ(gx) = ρ(g)ϕ(x) for all g ∈ W

, x ∈ W

where W

acts by

σ(g)ϕ(x) = ϕ(xg).

Then we have

= {ϕ : W

→ V : ···}.

Then whenever ϕ ∈ U

and g ∈ I

, then

σ(g)ϕ(x) = ϕ(xg) = ϕ((xgx

−1

)x) = ϕ(x).

So in fact ϕ takes values in V

. Therefore

= Ind

Of course, W

∼

Z, which contains W

as a subgroup. Moreover,

Frob

= Frob

where d = [k

: k

]. We note the following lemma:

Lemma.

Let

hgi ⊇ H

H → GL

(

) and

Ind

Then

det(1 − t

ρ(h)) = det(1 − tσ(g)).

Proof.

Both sides are multiplicative for exact sequences of representations

. So we can reduce to the case of

dim V

= 1, where

(

) =

λ ∈ C

We then check it explicitly.

To complete the proof of (ii), take

Frob

and

−s

so that

= q

−s

For Archimedean

, we define

(

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

→ C

, then

Is the corresponding character of

under the

Artin map.

F ' C

, then this is rather easy, since every irreducible representation of

∼

is one-dimensional. We then just define for

1-dimensional using

∼

and extend to all

by (i). The Jordan–H¨older theorem tells us this

is well-defined.

If F ' R, then recall that

= hC

, s : s

= −1 ∈ C

, szs

−1

= ¯zi.

Contained in here is W

(1)

= hU(1), si. Then

= W

(1)

× R

It is then easy to see that the irreducible representations of W

are

(i) 1-dimensional ρ

; or

(ii) 2-dimensional, σ = Ind

ρ, where ρ 6= ρ

: 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

, i.e.

ρ(z) = (z¯z)

Then

Ind

is reducible, and is a sum of characters of

∼

, namely

x 7→ |x|

and x 7→ sgn(x)|x|

= x

−1

|x|

t+1

. Then (ii) follows from the identity

(s)Γ

(s + 1) = Γ

(s) = 2(2π)

−s

Γ(s).

Now let

be global, and let

: Γ

→ GL

(

). For each

v ∈

, choose

over

. Let Γ

∼

be the decomposition group at

¯v

. These contain

and we have the geometric Frobenius

Frob

∈

. We define

ρ|

, and

then set

L(ρ, s) =

v-∞

L(ρ

, s) =

v-∞

det(1 − q

−s

Frob

)

−1

Λ(ρ, s) = LL

∞

v|∞

L(ρ

, s).

This is well-defined as the decomposition groups

¯v | v

are conjugate. If

dim V

= 1,

then ρ = χ ◦ Art

−1

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

, s) = L(ρ, s)L(ρ

, s).

(ii) If L/K is finite separable and ρ : Γ

→ GL

(V ) and σ = Ind

(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 ∈

over

v ∈

and consider the

local extension L

, then

L(ρ

, s) = L(Ind

In the global world, we have to take care of the splitting of primes. This boils

down to the fact that



Ind





w|v

Ind

(ρ|

). (∗)

We fix a valuation

¯v

over

. Write Γ

¯v/v

for the decomposition group in Γ

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

∼

/Γ

¯v/v

We then have

∼

\Γ

/Γ

¯v/v

which is compatible with the obvious map

S → S.

For ¯w = g¯v, we have

¯w/v

= gΓ

¯v/v

−1

Conjugating by

−1

, we can identify this with Γ

¯v/v

. Similarly, if

¯w|

, then

this contains

¯w/w

= gΓ

¯v/v

−1

∩ Γ

and we can identify this with Γ

¯v/v

∩ g

−1

There is a theorem, usually called Mackey’s formula, which says if

H, K ⊆ G

are two subgroups of finite index, and

H → GL

(

) is a representation of

Then

(Ind

V )|

∼

g∈H\G/K

Ind

K∩g

−1

(

−1

V ),

where

−1

is the

K ∩ g

−1

-representation where

−1

acts by

(

). We

then apply this to G = Γ

, H = Γ

, K = Γ

¯v/v

Example. If ρ is trivial, then

L(ρ, s) =

(1 − q

−s

)

−1

aCO

= ζ

(s).

This is just the Dedekind ζ-function of K.

Example.

Let

L/K

be a finite Galois extension with Galois group

. Consider

the regular representation

L/K

[

]. This decomposes as

, where

{ρ

}

run over the irreducible representations of G of dimension d

. We also have

L/K

= Ind

(1).

So by the induction formula, we have

(s) = L(r

L/K

, s) =

L(ρ

, s)

Example. For example, if L/K = Q(ζ

)/Q, then

Q(ζ

)

(s) =

L(χ, s),

where the product runs over all primitive Dirichlet characters mod

M | N

. Since

Q(ζ

)

, ζ

have simple poles at s = 1, we know that L(χ, 1) 6= 0 if χ 6= χ

Theorem

(Brauer induction theorem)

Suppose

G → GL

(

) is a represen-

tation of a finite group. Then there exists subgroups

⊆ G

and homomorphisms

: H

→ C

and integers m

∈ Z such that

tr ρ =

tr Ind

Note that the

need not be non-negative. So we cannot quite state this as a

statement about representations.

Corollary.

Let

: Γ

→ GL

(

). Then there exists finite separable

and χ

: Γ

→ C

of finite order and m

∈ Z such that

L(ρ, s) =

L(χ

, s)

In particular,

(

ρ, s

) has meromorphic continuation to

and has a functional

equation

Λ(ρ, s) = L · L

∞

= ε(ρ, s)L(˜ρ, 1 − s)

where

ε(ρ, s) = AB

ε(χ

, s)

and ˜ρ(g) =

ρ(g

−1

Conjecture

(Artin conjecture)

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

as an sum of virtual

induced characters. But when we take the product of the

factors, it is always

well-defined. We also know that

ε(χ

, s) =

(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) =

(ρ

, s),

we cannot just take

(

, s

) =

(

j,v

, s

), as this is not well-defined. However,

Langlands proved that there exists a unique factorization of

(

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

is a Haar measure on

and

F →

U(1) is a non-trivial character.

Let n(ψ) be the least integer such that ψ(π

) = 1. Then

ε(χ, ψ, µ) =

(

µ(O

) χ unramified, n(ψ) = 0

−1

· ψ dµ χ ramified

Since

and

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

→ GL

(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)

E/F

is finite separable, and

is a virtual representation of

of degree

0 and σ = Ind

ρ, then

ε(σ, ψ, µ) = ε(ρ, ψ ◦tr

E/F

, µ

Note that this is independent of the choice of

and

, since “

dim V

= 0”.

(iv) If dim ρ = 1, then ε(ρ) is the usual abelian ε(χ).

3 `-adic representations

In this section, we shall discuss



-adic representations of the Galois group, which

often naturally arise from geometric situations. At the end of the section, we

will relate these to complex representations of the Weil–Langlands group, which

will be what enters the Langlands correspondence.

Definition

(



-adic representation)

Let

be a topological group. An



-adic

representation consists of the following data:

– A finite extension E/Q



;

– An E-vector space V ; and

– A continuous homomorphism ρ : G → GL

(V )

∼

(E).

In this section,we will always take

= Γ

, where

F/Q

is a finite

extension with p 6= .

Example.

The cyclotomic character

cycl

: Γ

→ Z



⊆ Q



is defined by the

relation

cycl

(γ)

= γ(ζ)

for all

ζ ∈

with



= 1 and

γ ∈

. This is a one-dimensional



-adic

representation.

Example. Let E/K be an elliptic curve. We define the Tate module by



E = lim

←−

E[

](

K), V



E = T



E ⊗



Then V



E is is a 2-dimensional -adic representation of Γ

over Q



Example. More generally, if X/K is any algebraic variety, then

V = H

´et

(X ⊗

K, Q



)

is an -adic representation of Γ

We will actually focus on the representations of the Weil group

instead of

the full Galois group

. The reason is that every representation of the Galois

group restricts to one of the Weil group, and since the Weil group is dense, no

information is lost when doing so. On the other hand, the Weil group can have

more representations, and we seek to be slightly more general.

Another reason to talk about the Weil group is that local class field theory

says there is an isomorphism

Art

: W

∼

So one-dimensional representations of

are the same as one-dimensional

representations of F

For example, there is an absolute value map

→ Q

, inducing a represen-

tation

→ Q

. Under the Artin map, this sends the geometric Frobenius

. In fact, ω is the restriction of the cyclotomic character to W

Recall that we previously defined the tame character. Pick a sequence

∈

by π

= π and π



n+1

= π

. We defined, for any γ ∈ Γ



(γ) =



γ(π

)



∈ lim

←−



(

F ) = Z



(1).

When we restrict to the inertia group, this is a homomorphism, independent

of the choice of (

), which we call the tame character. In fact, this map is

-equivariant, where Γ

acts on

by conjugation. In general, this still defines

a function Γ

→ Z



(1), which depends on the choice of π

Example.

Continuing the previous notation, where

is a uniformizer of

, we

let T

be the 

-torsion subgroup of

/hπi. Then

= hζ

, π

i/hπ



∼

(Z/

The



th power map

→ T

n−1

is surjective, and we can form the inverse limit

, which is then isomorphic to



. This then gives a 2-dimensional



-adic

representation of Γ

In terms of the basis (ζ



), (π

), the representation is given by

γ 7→



cycl

(γ) t



(γ)

0 1



Notice that the image of



1 Z



0 1



. In particular, it is infinite. This cannot

happen for one-dimensional representations.

Perhaps surprisingly, the category of



-adic representations of

over

does not depend on the topology of

, but only the

as an abstract field. In

particular, if we take



, then after taking care of the slight annoyance that

it is infinite over



, the category of representations over



does not depend

on !

To prove this, we make use of the following understanding of



-adic represen-

tations.

Theorem

(Grothendieck’s monodromy theorem)

Fix an isomorphism



(1)

∼



. In other words, fix a system (



) such that



n−1

. We then view



a homomorphism I

→ Z



via this identification.

Let

→ GL

(

) be an



-adic representation over

. Then there exists

an open subgroup

⊆ I

and a nilpotent

N ∈ End

such that for all

γ ∈ I

ρ(γ) = exp(t



(γ)N) =

∞

j=0



(γ)N)

In particular, ρ(I

) unipotent and abelian.

In our previous example, N =



0 1

0 0



Proof. If ρ(I

) is finite, let I

= ker ρ ∩ I

and N = 0, and we are done.

Otherwise, first observe that

is any compact group and

G → GL

(

)

is an



-adic representation, then

contains a

-invariant lattice, i.e. a finitely-

generated

-submodule of maximal rank. To see this, pick any lattice

⊆ V

Then ρ(G)L

is compact, so generates a lattice which is G-invariant.

Thus, pick a basis of an

-invariant lattice. Then

→ GL

(

)

restricts to a map I

→ GL

We seek to understand this group

(

) better. We define a filtration on

) by

= {g ∈ GL

) : g ≡ I mod 

which is an open subgroup of

(

). Note that for

k ≥

1, there is an

isomorphism

k+1

→ M

/O

sending 1 +



. Since the latter is an



-group, we know

is a pro-



group.

Also, by definition, (G

)



⊆ G

k+1

Since

−1

(

) is open, we can pick an open subgroup

⊆ I

such that

(

)

⊆ G

. Recall that



(

) is the maximal pro-



quotient of

, because the

tame characters give an isomorphism

∼

-p



(1).

So ρ|

: I

→ G

factors as



) = 



` ν

using the assumption that ρ(I

) is infinite.

Now for r ≥ s, let T

= ν(

) = T

r−s

∈ G

r+2−s

. For r sufficiently large,

= log(T

) =

m≥1

(−1)

m−1

− 1)

converges -locally, and then T

= exp N

We claim that

is nilpotent. To see this, if we enlarge

, we may assume

that all the eigenvalues of N

are in E. For δ ∈ W

and γ ∈ I

, we know



(δγδ

−1

) = ω(δ)t



(γ).

ρ(δγδ

−1

) = ρ(γ)

w(σ)

for all γ ∈ I

. So

ρ(σ)N

ρ(δ

−1

) = ω(δ)N

Choose

lifting

(

) =

. Then if

is an eigenvector for

with eigenvalue

, then

(

)

is an eigenvector of eigenvalue

−1

. Since

has finitely many

eigenvalues, but we can do this as many times as we like, it must be the case

that λ = 0.

Then take

N =



for r sufficiently large, and this works.

There is a slight unpleasantness in this theorem that we fixed a choice of



roots of unity. To avoid this, we can say there exists an

(1) =

V ⊗



(1)

→

V nilpotent such that for all γ ∈ I

, we have

ρ(γ) = exp(t



(γ)N).

Grothendieck’s monodromy theorem motivates the definition of the Weil–

Deligne groups, whose category of representations are isomorphic to the category

of -adic representations. It is actually easier to state what the representations

of the Weil–Deligne group are. One can then write down a definition of the

Weil–Deligne group as a semi-direct product if they wish.

Definition

(Weil–Deligne representation)

A Weil–Deligne representation of

over a field E of characteristic 0 is a pair (ρ, N), where

– ρ

→ GL

(

) is a finite-dimensional representation of

over

with open kernel; and

– N ∈ End

(V ) is nilpotent such that for all γ ∈ W

, we have

ρ(γ)Nρ(γ)

−1

= ω(γ)N,

Note that giving

is the same as giving a unipotent

exp N

, which

is the same as giving an algebraic representation of

. So a Weil–Deligne

representation is a representation of a suitable semi-direct product W

n G

Weil–Deligne representations form a symmetric monoidal category in the

obvious way, with

(ρ, N) ⊗ (ρ

, N

) = (ρ ⊗ ρ

, N ⊗ 1 + 1 ⊗ N).

There are similarly duals.

Theorem.

Let

E/Q



be finite (and

 6

). Then there exists an equivalence of

(symmetric monoidal) categories



-adic representations

of W

over E



←→



Weil–Deligne representations

of W

over E



Note that the left-hand side is pretty topological, while the right-hand side

is almost purely algebraic, apart from the requirement that

has open kernel.

In particular, the topology of E is not used.

Proof.

We have already fixed an isomorphism



(1)

∼



. We also pick a lift

Φ ∈ W

of the geometric Frobenius. In other words, we are picking a splitting

= hΦi n I

The equivalence will take an



-adic representation



to the Weil–Deligne repre-

sentation (ρ, N ) on the same vector space such that



(Φ

γ) = ρ(Φ

γ) exp t



(γ)N (∗)

for all m ∈ Z and γ ∈ I

To check that this “works”, we first look at the right-to-left direction. Suppose

we have a Weil–Deligne representation (

ρ, N

) on

. We then define



→

Aut

(

) by (

∗

). Since

has open kernel, it is continuous. Since



is also

continuous, we know



is continuous. To see that



is a homomorphism,

suppose

γ · Φ

δ = Φ

m+n

where γ, δ ∈ I

and

= Φ

−n

γΦ

Then

exp t



(γ)N · ρ(Φ

δ) =

j≥0



(γ)

ρ(Φ

δ)

j≥0



(γ)q

ρ(Φ

δ)N

= ρ(Φ

δ) exp(q



(γ)).

But



(γ

) = t



(Φ

−n

γΦ

) = ω(Φ

−n



(γ) = q



(γ).

So we know that



(Φ

γ)ρ



(Φ

δ) = ρ



(Φ

m+n

δ).

Notice that if

γ ∈ I

∩ ker ρ

, then



(

) =

exp t



(

)

. So

is the nilpotent

endomorphism occurring in the Grothendieck theorem.

Conversely, given an



-adic representation



, let

N ∈ End

be given by

the monodromy theorem. We then define

by (

∗

). Then the same calculation

shows that (

ρ, N

) is a Weil–Deligne representation, and if

⊆ I

is the open

subgroup occurring in the theorem, then



(

) =

exp t



(

)

for all

γ ∈ I

. So

by (∗), we know ρ(I

) = {1}, and so ρ has open kernel.

This equivalence depends on two choices — the isomorphism



(1)

∼



and

also on the choice of Φ. It is not hard to check that up to natural isomorphisms,

the equivalence does not depend on the choices.

We can similarly talk about representations over



, instead of some finite

extension

. Note that if we have a continuous homomorphism

→

(



), then there exists a finite E/Q



such that ρ factors through GL

(E).

Indeed,

(

)

⊆ GL

(



) is compact, since it is a continuous image of

compact group. So it is a complete metric space. Moreover, the set of finite

extensions of

E/Q



is countable (Krasner’s lemma). So by the Baire category

theorem,

(

) is contained in some

(

), and of course,

(Φ) is contained

in some GL

(E).

Recalling that a Weil–Deligne representation over

only depends on

as a

field, and



∼



for any , 

, we know that

Theorem.

Let

, 

. Then the category of



representations of

equivalent to the category of



representations of W

Conjecturally,



-adic representations coming from algebraic geometry have

semi-simple Frobenius. This notion is captured by the following proposi-

tion/definition.

Proposition.

Suppose



is an



-adic representation corresponding to a Weil–

Deligne representation (ρ, N ). Then the following are equivalent:

(i) ρ



(Φ) is semi-simple (where Φ is a lift of Frob

(ii) ρ



(γ) is semi-simple for all γ ∈ W

\ I

(iii) ρ is semi-simple.

(iv) ρ(Φ) is semi-simple.

In this case, we say



and (

ρ, N

) are

-semisimple (where

refers to Frobenius).

Proof.

Recall that

∼

Z o I

, and

(

) is finite. So that part is always

semisimple, and thus (iii) and (iv) are equivalent.

Moreover, since



(Φ) =

(Φ), we know (i) and (iii) are equivalent. Finally,



(Φ) is semi-simple iff



(Φ

) is semi-simple for all Φ. Then this is equivalent

to (ii) since the equivalence before does not depend on the choice of Φ.

Example.

The Tate module of an elliptic curve over

is not semi-simple, since

it has matrix



(γ) =



ω(γ) t



(γ)

0 1



However, it is F -semisimple, since

ρ(γ) =



ω(γ) 0

0 1



, N =



0 1

0 0



It turns out we can classify all the indecomposable and

-semisimple Weil–

Deligne representations. In the case of vector spaces, if a matrix

acts nilpo-

tently on a vector space

, then the Jordan normal form theorem says there is a

basis of

in which

takes a particularly nice form, namely the only non-zero

entries are the entries of the form (

i, i

+ 1), and the entries are all either 0 or 1.

In general, we have the following result:

Theorem

(Jordan normal form)

is semi-simple,

N ∈ End

(

) is nilpotent

with

m+1

= 0, then there exists subobjects

, . . . , P

⊆ V

(not unique as

subobjects, but unique up to isomorphism), such that

→ N

is an

isomorphism, and N

r+1

= 0, and

V =

r=0

⊕ N P

⊕ ··· ⊕ N

r=0

⊗

Z[N]

r+1

)

For vector spaces, this is just the Jordan normal form for nilpotent matrices.

Proof.

If we had the desired decomposition, then heuristically, we want to set

to be the things killed by

but not in the image of

. Thus, using semisimplicity,

we pick P

to be a splitting

ker N = (ker N ∩ im N) ⊕ P

Similarly, we can pick P

ker N

= (ker N + (im N ∩ ker N

)) ⊕ P

One then checks that this works.

We will apply this when

is a representation of

→ GL

(

) and

is the

nilpotent endomorphism of a Weil–Deligne representation. Recall that we had

ρ(γ)Nρ(γ)

−1

= ω(γ)N,

is a map

V → V ⊗ ω

−1

, rather than an endomorphism of

. Thankfully,

the above result still holds (note that

V ⊗ω

−1

is still the same vector space, but

with a different action of the Weil–Deligne group).

Proposition. Let (ρ, N ) be a Weil–Deligne representation.

(i) (ρ, N) is irreducible iff ρ is irreducible and N = 0.

(ii) (ρ, N) is indecomposable and F -semisimple iff

(ρ, N) = (σ, 0) ⊗ sp(n),

where

is an irreducible representation of

and

(

)

∼

is the

representation

ρ = diag(ω

n−1

, . . . , ω, 1), N =







0 1







Example. If

ρ =





ω ω





, N =





0 0 0

0 0 1

0 0 0





then this is an indecomposable Weil–Deligne representation not of the above

form.

Proof. (i) is obvious.

For (ii), we first prove (

⇐

). If (

ρ, N

) = (

σ,

⊗ sp

(

), then

-semisimplicity

is clear, and we have to check that it is indecomposable. Observe that the kernel

is still a representation of

. Writing

N=0

for the kernel of

, we

note that V

N=0

= σ ⊗ ω

n−1

, which is irreducible. Suppose that

(ρ, N) = U

⊕ U

Then for each

, we must have

N=0

= 0 or

N=0

. We may wlog assume

N=0

= 0. Then this forces U

= 0. So we are done.

Conversely, if (

ρ, N, V

) is

-semisimple and indecomposable, then

is a

representation of

which is semi-simple and

V → V ⊗ ω

−1

. By Jordan

normal form, we must have

V = U ⊕ N U ⊕ ··· ⊕ N

with N

r+1

= 0, and U is irreducible. So V = (σ, 0) ⊗ sp(r + 1).

Given this classification result, when working over complex representations,

the representation theory of

(2) lets us capture the

bit of

-semisimple

Weil–Deligne representation via the following group:

Definition

(Weil–Langlands group)

We define the (Weil–)Langlands group to

= W

× SU(2).

A representation of

is a continuous action on a finite-dimensional vector

space (thus, the restriction to W

has open kernel).

Theorem.

There exists a bijection between

-semisimple Weil–Deligne repre-

sentations over

and semi-simple representations of

, compatible with tensor

products, duals, dimension, etc. In this correspondence:

–

The representations

that factor through

correspond to the

Weil–Deligne representations (ρ, 0).

–

More generally, simple

representations

σ ⊗

(

Sym

n−1

) correspond to

the Weil–Deligne representation (σ ⊗ω

(−1+n)/2

, 0) ⊗ sp(n).

If one sits down and checks the theorem, then one sees that the twist in the

second part is required to ensure compatibility with tensor products.

Of course, the (

-semisimple) Weil–Deligne representations over

are in

bijection those over



, using an isomorphism



∼

4 The Langlands correspondence

Local class field theory says we have an isomorphism

∼

If we want to state this in terms of the full Weil group, we can talk about the

one-dimensional representations of

, and write local class field theory as a

correspondence



characters of GL

(F )



←→



1-dimensional representations

of W



The Langlands correspondence aims to understand the representations of

(

), and it turns out this corresponds to

-dimensional representations of

. That is, if we put enough adjectives in front of these words.

4.1 Representations of groups

The adjectives we need are fairly general. The group

(

) contains a profinite

open subgroup

(

). The general theory applies to any topological group

with a profinite open subgroup K, with G/K countable.

Definition

(Smooth representation)

A smooth representation of

is a contin-

uous representation of

over

, where

is given the discrete topology. That

is, it is a pair (

π, V

) where

is a complex vector space and

G → GL

(

) a

homomorphism such that for every v ∈ V , the stabilizer of v in G is open.

Note that we can replace

with any field, but we like

. Typically,

is an

infinite-dimensional vector space. To retain some sanity, we often desire the

following property:

Definition

(Admissible representation)

We say (

π, V

) is admissible if for every

open compact subgroup

K ⊆ G

the fixed set

{v ∈ V

(

)

v ∀g ∈ K}

is finite-dimensional.

Example.

Take

(

). Then

(

) has a right action of

by linear

transformations. In fact, we can write P

(F ) as

(F ) =



∗ ∗

0 ∗



\G.

Let

be the space of all locally constant functions

(

)

→ C

. There are

lots of such functions, because

(

) is totally disconnected. However, since

(F ) is compact, each such function can only take finitely many values.

We let

π(g)f = (x 7→ f(xg)).

It is not difficult to see that this is an infinite-dimensional admissible representa-

tion.

Of course, any finite-dimensional representation is an example, but

(

)

does not have very interesting finite-dimensional representations.

Proposition.

Let

(

). If (

π, V

) is a smooth representation with

dim V < ∞, then

π = σ ◦det

for some σ : F

→ GL

(V ).

So these are pretty boring.

Proof. If V =

i=1

, then

ker π =

i=1

(stabilizers of e

)

is open. It is also a normal subgroup, so

ker π ⊇ K

= {g ∈ GL

(O) : g ≡ I mod 

}

for some

, where



is a uniformizer of

. In particular,

ker π

contains

(

)

for some

i 6

and

, which is the matrix that is the identity except at entry

(i, j), where it is x.

But since

ker π

is normal, conjugation by diagonal matrices shows that it

contains all

(

) for all

x ∈ F

and

i 6

. For any field, these matrices generate

(F ). So we are done.

So the interesting representations are all infinite-dimensional. Fortunately, a

lot of things true for finite-dimensional representations also hold for these. For

example,

Lemma

(Schur’s lemma)

Let (

π, V

) be an irreducible representation. Then

every endomorphism of V commuting with π is a scalar.

In particular, there exists ω

: Z(G) → C

such that

π(zg) = ω

(z)π(g)

for all z ∈ Z(G) and g ∈ G. This is called the central character.

At this point, we are already well-equipped to state a high-level description

of the local Langlands correspondence.

Theorem (Harris–Taylor, Henniart). There is a bijection



irreducible, admissible

representations of GL

(F )



←→



semi-simple n-dimensional

representations of L



In the next section, we will introduce the Hecke algebra, which allows us to

capture these scary infinite dimensional representations of

(

) in terms of

something finite-dimensional.

Afterwards, we are going to state the Langlands classification of irreducible

admissible representations of

(

). We can then state a detailed version of

the local Langlands correspondence in terms of this classification.

4.2 Hecke algebras

Let G, K be as before.

Notation.

We write

∞

(

) for the vector space of locally constant functions

f : G → C of compact support.

Definition (Hecke algebra). The Hecke algebra is defined to be

H(G, K) = {ϕ ∈ C

∞

(G) : ϕ(kgk

) = ϕ(g) for all k, k

∈ K}.

This is spanned by the characteristic functions of double cosets KgK.

This algebra comes with a product called the convolution product. To define

this, we need the Haar measure on

. This is a functional

∞

(

)

→ C

, written

f 7→

f(g) dµ(g),

that is invariant under left translation, i.e. for all h ∈ G,we have

f(hg) dµ(g) =

f(g) dµ(g).

To construct the Haar measure, we take

) = 1. Then if

⊆ K

is an open

subgroup, then it is of finite index, and since we want

) =

), we

must have

µ(1

) =

(K : K

)

We then set

) =

) for any

x ∈ G

, and since these form a basis of the

topology, this defines µ.

Definition (Convolution product). The convolution product on H(G, K) is

(ϕ ∗ ϕ

)(g) =

ϕ(x)ϕ

−1

g) dµ(x).

Observe that this integral is actually a finite sum.

It is an exercise to check that this is a C-algebra with unit

µ(K)

Now if (

π, V

) is a smooth representation, then for all

v ∈ V

and

ϕ ∈ H

(

G, K

consider the expression

π(ϕ)v =

ϕ(g)π(g)v dµ(g).

Note that since the stabilizer of

is open, the integral is actually a finite sum,

so we can make sense of it. One then sees that

π(ϕ)π(ϕ

) = π(ϕ ∗ ϕ

This would imply

is a

(

G, K

)-module, if the unit acted appropriately. It

doesn’t, however, since in fact

(

) always maps into

. Indeed, if

k ∈ K

then

π(k)π(ϕ)v =

ϕ(g)π(kg)v dµ(g) =

ϕ(g)π(g) dµ(g) = π(ϕ)v,

using that ϕ(g) = ϕ(k

−1

g) and dµ(g) = dµ(k

−1

g).

So our best hope is that

is an

(

G, K

)-module, and one easily checks

that

(

) indeed acts as the identity. We also have a canonical projection

π(e

) : V  V

In good situations, this Hecke module determines V .

Proposition.

There is a bijection between isomorphism classes of irreducible ad-

missible (

π, V

) with

= 0 and isomorphism classes of simple finite-dimensional

H(G, K)-modules, which sends (π, V ) to V

with the action we described.

If we replace K by a smaller subgroup K

⊆ K, then we have an inclusion

H(G, K) → H(G, K

which does not take

. We can then form the union of all of these, and

let

H(G) = lim

−→

H(G, K)

which is an algebra without unit. Heuristically, the unit should be the delta

function concentrated at the identity, but that is not a function.

This

(

) acts on any smooth representation, and we get an equivalence of