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
L
-functions (Hecke, Tate). We then talk about non-abelian
L
-functions,
and in particular the Weil–Deligne group and local L- and ε-factors.
We then talk about local Langlands for
GL
n
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
K
be a field. We will write
¯
K
for a separable closure of
K
, and
Γ
K
= Gal(
¯
K/K). We have
Γ
K
= 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
Γ
K
subfields
K L
¯
K
open subgroups of
Γ
K
finite subfields
K L
¯
K
Notation.
We write
K
ab
for the maximal abelian subextension of
¯
K
, and then
Gal(K
ab
/K) = Γ
ab
K
=
Γ
K
K
, Γ
K
]
.
It is crucial to note that while
¯
K
is unique, it is only unique up to non-
canonical isomorphism. Indeed, it has many automorphisms, given by elements
of
Gal
(
¯
K/K
). Thus, Γ
K
is well-defined up to conjugation only. On the other
hand, the abelianization Γ
ab
K
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
p
or F
p
((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.
If
F
is a non-Archimedean local field, then it has a canonical normalized
valuation
v = v
F
: F
×
Z.
Definition
(Valuation ring)
.
The valuation ring of a non-Archimedean local
field F is
O = O
F
= {x F : v(x) 0}.
Any element
π
=
π
f
O
F
with
v
(
π
) = 1 is called a uniformizer . This generates
the maximal ideal
m = m
F
= {x O
F
: v(x) 1}.
Definition
(Residue field)
.
The residue field of a non-Archimedean local field
F is
k = k
F
= O
F
/m
F
.
This is a finite field of order q = p
r
.
A particularly well-understood subfield of
F
ab
is the maximal unramified
extension F
ur
. We have
Gal(F
ur
/F ) = Gal(
¯
k/k) =
ˆ
Z = lim
n1
Z/nZ.
and this is completely determined by the behaviour of the residue field. The rest
of Γ
F
is called the inertia group.
Definition (Inertia group). The inertia group I
F
is defined to be
I
F
= Gal(
¯
F /F
ur
) Γ
F
.
We also define
Definition
(Wild inertia group)
.
The wild inertia group
P
F
is the maximal
pro-p-subgroup of I
F
.
Returning to the maximal unramified extension, note that saying
Gal
(
¯
k/k
)
=
ˆ
Z
requires picking an isomorphism, and this is equivalent to picking an element
of
ˆ
Z to be the “1”. Naively, we might pick the following:
Definition
(Arithmetic Frobenius)
.
The arithmetic Frobenius
ϕ
q
Gal
(
¯
k/k
)
(where |k| = q) is defined to be
ϕ
q
(x) = x
q
.
Identifying this with 1
ˆ
Z
leads to infinite confusion, and we shall not do so.
Instead, we define
Definition (Geometric Frobenius). The geometric Frobenius is
Frob
q
= ϕ
1
q
Gal(
¯
k/k).
We shall identify Gal(
¯
k/k)
=
ˆ
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
k
, then there are two ways the Frobenius can act on it either
as a Galois action, or as a pullback along the morphism (
)
q
:
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
p
-phenomena”, which we would like to
avoid. To understand
I
F
better, let
n
be a natural number prime to
p
. As usual,
we write
µ
n
(
¯
k) = {ζ
¯
k : ζ
n
= 1}.
We also pick an nth root of π in
¯
F , say π
n
. By definition, this has π
n
n
= π.
Definition
(Tame mod
n
character)
.
The tame mod
n
character is the map
t(n) : I
F
= Gal(
¯
F /F
ur
) µ
n
(
¯
k) given by
γ 7→ γ(π
n
)
n
(mod π).
Note that since γ fixes π = π
n
n
, we indeed have
γ(π
n
)
π
n
n
=
γ(π
n
n
)
π
n
n
= 1.
Moreover, this doesn’t depend on the choice of
π
n
. Any other choice differs by
an
n
th root of unity, but the
n
th root of unity lies in
F
ur
since
n
is prime to
p
. So
γ
fixes it and so it cancels out in the fraction. For the same reason, if
γ
moves
π
n
at all, then this is visible down in
¯
k
, since
γ
would have multiplied
π
n
by an nth root of unity, and these nth roots are present in
¯
k.
Now that everything is canonically well-defined, we can take the limit over
all n to obtain a map
ˆ
t : I
F
lim
(n,p)=1
µ
n
(
¯
k) =
Y
6=p
lim
m1
µ
m
(
¯
k)
Y
6=p
Z
(1)(
¯
k).
This
Z
(1)(
¯
k
) is the Tate module of
¯
k
×
. This is isomorphic to
Z
, but not
canonically.
Theorem. ker
ˆ
t = P
F
.
Thus, it follows that maximal tamely ramified extension of
F
, i.e. the fixed
field of P
F
is
[
(n,p)=1
F
ur
(
n
π).
Note that
t
(
n
) extends to a map Γ
F
µ
n
given by the same formula, but
this now depends on the choice of
π
n
, and further, it is not a homomorphism,
because
t(n)(γδ) =
γδ(π
n
)
π
n
=
γ(π
n
)
π
n
γ
δ(π
n
)
π
n
= t(n)(γ) · γ(t(n)(δ)).
So this formula just says that
t
(
n
) is a 1-cocycle. Of course, picking another
π
n
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
F
correspond to open finite index subgroups of
F
×
, but the theorem says a bit
more than that:
Theorem (Local class field theory).
(i)
Let
F
be a local field. Then there is a continuous homomorphism, the
local Artin map
Art
F
: F
×
Γ
ab
F
with dense image characterized by the properties
(a) The following diagram commutes:
F
×
Γ
ab
F
Γ
F
/I
F
Z
ˆ
Z
v
F
Art
F
(b) If F
0
/F is finite, then the following diagram commutes:
(F
0
)
×
Γ
ab
F
0
= Gal(F
0ab
/F
0
)
F
×
Γ
ab
F
= Gal(F
ab
/F )
Art
F
0
N
F
0
/F
restriction
Art
F
(ii) Moreover, the existence theorem says Art
1
F
induces a bijection
open finite index
subgroups of F
×
open subgroups of Γ
ab
F
Of course, open subgroups of Γ
ab
F
further corresponds to finite abelian
extensions of F .
(iii) Further, Art
F
induces an isomorphism
O
×
F
im(I
F
Γ
ab
F
)
and this maps (1 +
πO
F
)
×
to the image of
P
F
. Of course, the quotient
O
×
F
/(1 + πO
F
)
×
=
k
×
= µ
(k).
(iv)
Finally, this is functorial, namely if we have an isomorphism
α
:
F
F
0
and
extend it to
¯α
:
¯
F
¯
F
0
, then this induces isomorphisms between the Galois
groups
α
: Γ
F
Γ
F
0
(up to conjugacy), and
α
ab
Art
F
=
Art
F
0
α
ab
.
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
:
F
×
N
E/F
(E
×
)
Gal(E/F )
ab
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
p
, then
F
ab
= Q
p
(µ
) =
[
Q
p
(µ
n
) = Q
ur
p
(µ
p
).
Moreover, if we write x Q
×
p
as p
n
y with y Z
×
p
, then
Art
Q
(x)|
Q
ur
p
= Frob
n
p
, Art
Q
(x)|
Q
p
(µ
p
)
= (ζ
p
n
7→ ζ
y mod p
n
p
n
).
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.
If
F
=
C
, then Γ
F
= Γ
ab
F
= 1 is trivial, and the Artin map is similarly
trivial. There are no non-trivial open finite index subgroups of
C
×
, just as there
are no non-trivial open subgroups of the trivial group.
Example.
If
F
=
R
, then
¯
R
=
C
and Γ
F
= Γ
ab
F
=
Z/
2
Z
=
1
}
. The Artin
map is given by the sign map. The unique open finite index subgroup of R
×
is
R
×
>0
, 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
F
be a non-Archimedean local field. Then the
Weil group of F is the topological group W
F
defined as follows:
As a group, it is
W
F
= {γ Γ
F
| γ|
F
ur
= Frob
n
q
for some n Z}.
Recall that
Gal
(
F
ur
/F
) =
ˆ
Z
, and we are requiring
γ|
F
ur
to be in
Z
. In
particular, I
F
W
F
.
The topology is defined by the property that
I
F
is an open subgroup with
the profinite topology. Equivalently,
W
F
is a fiber product of topological
groups
W
F
Γ
F
Z
ˆ
Z
where Z has the discrete topology.
Note that W
F
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
F
induces an isomorphism of topological groups
Art
W
F
: F
×
W
ab
F
.
This maps O
×
F
isomorphically onto the inertia subgroup of Γ
ab
F
.
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
C
is defined to be
W
C
=
C
×
, and the Artin map
Art
W
R
is defined to be the identity.
The Weil group of R is defined the non-abelian group
W
R
= hC
×
, σ | σ
2
= 1 C
×
, σzσ
1
= ¯z for all z C
×
i.
This is a (non)-split extension of C
×
by Γ
R
,
1 C
×
W
R
Γ
R
1,
where the last map sends
z 7→
0 and
σ 7→
1. This is in fact the unique
non-split extension of Γ
R
by C
×
where Γ
R
acts on C
×
in a natural way.
The map Art
W
R
is better described by its inverse, which maps
(Art
W
R
)
1
: W
ab
R
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
F
be a non-Archimedean local field, and
E/F Galois but not necessarily finite. We define
W
E/F
= {γ Gal(E
ab
/F ) : γ|
F
ur
= Frob
n
q
, n Z} =
W
F
[W
E
, W
E
]
.
with the quotient topology.
The W
¯
F /F
= W
F
, while W
F/F
= W
ab
F
= 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
ab
/E) Gal(E
ab
/F ) Gal(E/F ) 1
1 W
ab
E
W
E/F
Gal(E/F ) 1.
By the Artin map,
W
ab
E
=
E
×
. So the relative Weil group is an extension of
Gal(E/F ) by E
×
. In the case of non-Archimedean fields, we have
lim
E
E
×
= {1},
where the field extensions are joined by the norm map. So
¯
F
×
is invisible in
W
F
=
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–Fohlich), which only
treats the first part, namely the existence of
Art
K
. We start off with a
finite Galois extension E/F , and we want to construct an isomorphism
Art
E/F
: F
×
/N
E/F
(E
×
) Gal(E/F )
ab
.
Writing G = Gal(E/F ), this uses the cohomological interpretation
F
×
/N
E/F
(E
×
) =
ˆ
H
0
(G, E
×
),
where
ˆ
H
is the Tate cohomology of finite groups. On the other hand, we
have
G
ab
= H
1
(G, Z) =
ˆ
H
2
(G, Z).
The main step is to compute
H
2
(G, E
×
) =
ˆ
H
2
(G, E
×
)
=
1
n
Z/Z Q/Z = H
2
F
,
¯
F
×
).
where
n
= [
E
:
F
]. The final group
H
2
F
,
¯
F
×
) is the Brauer group
Br
(
F
),
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
ˆ
H
2
(
G, E
×
), and this maps
ˆ
H
2
(
G, Z
)
ˆ
H
0
(
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
H
(
G, M
) for arbitrary
M
, 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
Q
p
is obtained in two steps we can write
Q
ab
p
= Q
ur
p
(µ
p
) = Q
ur
p
(torsion points in
ˆ
G
m
),
where
ˆ
G
m
is the formal multiplication group, which we can think of as
(1 + m
¯
Q
p
)
×
. This generalizes to any F/Q
p
we have
F
ab
= F
ur
(torsion points in
ˆ
G
π
),
where
ˆ
G
π
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
(
g
)
F
×
/N
E/F
(
E
×
). 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
0
). We have the following
lemma:
Lemma.
There is a finite
K/F
0
such that
K E
=
F
0
, so
Gal
(
KE/K
)
=
Gal(E/F
0
) = hgi. Moreover, KE/K is unramified.
Let
g
0
|
E
=
g
, and suppose
g
0
=
Frob
a
KE/K
. If local class field theory is
true, then we have to have
Art
1
KE/K
(g
0
) = π
a
K
(mod N
KE/K
(KE
×
)).
Then by our compatibility conditions, this implies
Art
1
E/F
(g) = N
K/F
(π
a
K
) (mod N
E/F
(E
×
)).
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
k
(
C
) for a smooth
projective absolutely irreducible curve C/F
q
, i.e. a finite extension of F
q
(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
K
be a global field. Then a place is a valuation on
K
.
If
K
is a number field, we say a valuation
v
is a finite place if it is the valuation
at a prime
p C O
K
. A valuation
v
is an infinite place if comes from a complex
or real embedding of
K
. We write Σ
K
for the set of places of
K
, and Σ
K
and
Σ
K,
for the sets of finite and infinite places respectively. We also write
v -
if v is a finite place, and v | otherwise.
If
K
is a function field, then all places are are finite, and these correspond to
closed points of the curve.
If
v
Σ
K
is a place, then there is a completion
K K
v
. If
v
is infinite,
then
K
v
is
R
or
C
, i.e. an Archimedean local field. Otherwise,
K
v
is a non-
Archimedean local field.
Example.
If
Q
=
K
, then there is one infinite prime, which we write as
,
given by the embedding
Q R
. If
v
=
p
, then we get the embedding
Q Q
p
into the p-adic completion.
Notation.
If
v
is a finite place, we write
O
v
K
v
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
K
v
(and
hence K) is by
|x|
v
= q
v(x)
v
,
where
q
v
=
O
v
π
v
O
v
is the cardinality of the residue field at
v
. For example, if
K
=
Q
and
v
=
p
,
then q
v
= p, and |p|
v
=
1
p
.
In the Archimedean case, if
K
v
is real, then we set
|x|
to be the usual absolute
value; if K
v
is complex, then we take |x|
v
= x¯x = |x|
2
.
The reason for choosing these normalizations is that we have the following
product formula:
Proposition (Product formula). If x K
×
, then
Y
vΣ
K
|x|
v
= 1.
The proof is not difficult. First observe that it is true for
Q
, 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
A
K
=
Y
v
0
K
V
=
n
(x
v
)
vK
v
: x
v
O
v
for all but finitely many v Σ
K
o
.
We can write this as K
×
ˆ
K or A
K,
× A
K
, where
K
= A
K,
= K
Q
R = R
r
1
× C
r
2
.
consists of the product over the infinite places, and
ˆ
K = A
K
=
Y
v-
0
K
V
=
[
SΣ
K
finite
Y
vS
K
v
×
Y
vΣ
K
\S
O
v
.
This contains
ˆ
O
K
=
Q
v-
O
K
. In the case of a number field,
ˆ
O
K
is the profinite
completion of O
K
. More precisely, if K is a number field then
ˆ
O
K
= lim
a
O
K
/a = lim O
K
/N O
K
= O
K
Z
ˆ
Z,
where the last equality follows from the fact that O
K
is a finite Z-module.
Definition (Idele). The ideles is the restricted product
J
K
= A
×
K
=
Y
v
0
K
×
v
=
n
(x
v
)
v
Y
K
×
v
: x
v
O
×
v
for almost all v
o
.
These objects come with natural topologies. On
A
K
, we take
K
×
ˆ
O
K
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
J
K
, we
take
K
×
×
ˆ
O
×
K
to be open with the product topology. Note that this is not the
subspace topology under the inclusion
J
K
A
K
. Instead, it is induced by the
inclusion
J
K
A
K
× A
K
x 7→ (x, x
1
).
It is a basic fact that K
×
J
K
is a discrete subgroup.
Definition (Idele class group). The idele class group is then
C
K
= J
K
/K
×
.
The idele class group plays an important role in global class field theory.
Note that J
K
comes with a natural absolute value
| · |
A
: J
K
R
×
>0
(x
v
) 7→
Y
vΣ
K
|x
v
|
v
.
The product formula implies that
|K
×
|
A
=
{
1
}
. So this in fact a map
C
K
R
×
>0
.
Moreover, we have
Theorem. The map | · |
A
: C
K
R
×
>0
has compact kernel.
This seemingly innocent theorem is actually quite powerful. For example,
we will later construct a continuous surjection
C
K
Cl
(
K
) to the ideal class
group of
K
. In particular, this implies
Cl
(
K
) 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,
J
Q
= R
×
×
Y
p
0
Q
×
p
.
Suppose we have an idele
x
= (
x
v
). It is easy to see that there exists a unique
rational number
y Q
×
such that
sgn
(
y
) =
sgn
(
x
) and
v
p
(
y
) =
v
p
(
x
p
). So we
have
J
Q
= Q
×
×
R
×
>0
×
Y
p
Z
×
p
!
.
Here we think of
Q
×
as being embedded diagonally into
J
Q
, and as we have
previously mentioned,
Q
×
is discrete. From this description, we can read out a
description of the idele class group
C
Q
= R
×
>0
×
ˆ
Z
×
,
and
ˆ
Z
×
is the kernel | · |
A
.
From the decomposition above, we see that
C
Q
has a maximal connected
subgroup
R
×
>0
. In fact, this is the intersection of all open subgroups containing
1. For a general
K
, we write
C
0
K
for the maximal connected subgroup, and then
π
0
(C
K
) = C
K
/C
0
K
,
In the case of K = Q, we can naturally identify
π
0
(C
Q
) =
ˆ
Z
×
= lim
n
(Z/nZ)
×
= lim
n
Gal(Q(ζ
n
)/Q) = Gal(Q(ζ
)/Q).
The field
Q
(
ζ
) is not just any other field. The Kronecker–Weber theorem says
this is in fact the maximal abelian extension of
Q
. 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,
v
a place of
K
, and
w | v
a place of
L
extending
v
. For finite places, this means the corresponding primes divide; for
infinite places, this means the embedding
w
is an extension of
v
. We then have
the decomposition group
Gal(L
w
/K
v
) Gal(L/K).
If
L/K
is abelian, since any two places lying above
v
are conjugate, this depends
only on v. In this case, we can now define the global Artin map
Art
L/K
: J
K
Gal(L/K)
(x
v
)
v
7−
Y
v
Art
L
w
/K
v
(x
v
),
where we pick one
w
for each
v
. To see this is well-defined, note that if
x
v
O
×
v
and
L/K
is unramified at
v -
, then
Art
K
w
/K
v
(
x
v
) = 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
K
: J
K
Γ
ab
K
.
Theorem
(Artin reciprocity law)
. Art
K
(
K
×
) =
{
1
}
, so induces a map
C
K
Γ
ab
K
. Moreover,
(i)
If
char
(
K
) =
p >
0, then
Art
K
is injective, and induces an isomorphism
Art
K
:
C
k
W
ab
K
, where
W
K
is defined as follows: since
K
is a finite
extension of
F
q
(
T
), and wlog assume
¯
F
q
K
=
F
q
k
. Then
W
K
is
defined as the pullback
W
K
Γ
K
= Gal(
¯
K/K)
Z
ˆ
Z
=
Gal(
¯
k/k)
restr.
(ii) If char(K) = 0, we have an isomorphism
Art
K
: π
0
(C
K
) =
C
K
C
0
K
Γ
ab
K
.
Moreover, if L/K is finite, then we have a commutative diagram
C
L
Γ
ab
L
C
K
Γ
ab
K
N
L/K
Art
L
restr.
Art
K
If this is in fact Galois, then this induces an isomorphism
Art
L/K
:
J
K
K
×
N
L/K
(J
L
)
Gal(L/K)
ab
.
Finally, this is functorial, namely if
σ
:
K
K
0
is an isomorphism, then we
have a commutative square
C
K
Γ
ab
K
C
K
0
Γ
ab
K
0
Art
K
σ
Art
K
0
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
K
containing K
×
L 7− ker(Art
L/K
: J
K
Gal(L/K))
Note that there exists finite index subgroups that are not open!
Recall that in local class field theory, if we decompose
K
v
=
hπi ×O
×
v
, then
the local Artin map sends
O
×
v
to (the image of) the inertia group. Thus an
extension
L
w
/K
v
is unramified iff the local Artin map kills of
O
×
v
. Globally,
this tells us
Proposition.
If
L/K
is an abelian extension of global fields, which corresponds
to the open subgroup
U J
K
under the Artin map, then
L/K
is unramified at
a finite v - iff O
×
v
U.
We can extend this to the infinite places if we make the appropriate definitions.
Since
C
cannot be further extended, there is nothing to say for complex places.
Definition
(Ramification)
.
If
v |
is a real place of
K
, and
L/K
is a finite
abelian extension, then we say
v
is ramified if for some (hence all) places
w
of
L
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
v
iff
K
×
v
=
R
×
U
.
Note that since U is open, it automatically contains R
×
>0
.
We can similarly read off splitting information.
Proposition.
If
v
is finite and unramified, then
v
splits completely iff
K
×
v
U
.
Proof. v
splits completely iff
L
w
=
K
v
for all
w | v
, iff
Art
L
w
/K
v
(
K
×
v
) =
{
1
}
.
Example.
We will use global class field theory to compute all
S
3
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
S
3
is not an abelian group, it is solvable. So we can break our potential
extension as a chain
Q
K
L
2 = hσi
3
Since
L/Q
is unramified outside 5 and 7, we know that
K
must be one of
Q
(
5
),
Q
(
7
) and
Q
(
35
). We then consider each case in turn, and then see what
are the possibilities for
L
. We shall only do the case
K
=
Q
(
7
) here. If we
perform similar computations for the other cases, we find that the other choices
of K do not work.
So fix
K
=
Q
(
7
). We want
L/K
to be cyclic of degree 3, and
σ
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
K
of
index 3 such that
O
×
v
U
for all
v -
35. We also need
σ
(
U
) =
U
, or else the
composite extension would not even be Galois, and
σ
has to acts as
1 on
J
K
/U
=
Z/3Z to get a non-abelian extension.
We know
K
=
Q
(
7
) has has class number 1, and the units are
±
1. So we
know
C
K
C
0
K
=
ˆ
O
×
K
1}
.
By assumption, we know
U
contains
Q
v-35
O
×
v
. So we have to look at the places
that divide 35. In O
Q(
7)
, the prime 5 is inert and 7 is ramified.
Since 5 is inert, we know
K
5
/Q
5
is an unramified quadratic extension. So
we can write
O
×
(5)
= F
×
25
× (1 + 5O
(5)
)
×
.
The second factor is a pro-5 group, and so it must be contained in
U
for the
quotient to have order 3. On
F
×
25
,
σ
acts as the Frobenius
σ
(
x
) =
x
5
. Since
F
×
25
is cyclic of order 24, there is a unique index 3 subgroup, cyclic of order
6. This gives an index 3 subgroup
U
5
O
×
(5)
. Moreover, on here,
σ
acts by
x 7→ x
5
= x
1
. Thus, we can take
U =
Y
v6=(5)
O
×
v
× U
5
,
and this gives an
S
3
extension of
Q
that is unramified outside 5 and 7. It is an
exercise to explicitly identify this extension.
We turn to the prime 7 =
7
2
. Since this is ramified, we have
O
×
(
7)
= F
×
7
×
1 + (
7)O
7
×
,
and again the second factor is a pro-7 group. Moreover
σ
acts trivially on
F
×
7
. So
U must contain O
×
(
7)
. So what we found above is the unique such extension.
We previously explicitly described
C
Q
as
R
×
>0
×
ˆ
Z
. It would be nice to have
a similar description of
C
K
for an arbitrary
K
. The connected component will
come from the infinite places
K
×
Q
v|∞
K
×
v
. The connected component is given
by
K
×,0
= (R
×
>0
)
r
1
× (C
×
)
r
2
,
where there are r
1
real places and r
2
complex ones. Thus, we find that
Proposition.
C
K
/C
0
K
=
1}
r
1
×
ˆ
K
×
K
×
.
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
K
fractional ideals of K
(x
v
)
v
7→
Y
v-
p
v(x
v
)
v
,
where
p
v
is the prime ideal corresponding to
v
. We ignore the infinite places
completely.
Observe that
c
(
K
×
) is the set of all principal ideals by definition. Moreover,
the kernel of the content map is
K
×
×
ˆ
O
×
K
, by definition. So we have a short
exact sequence
1
1}
r
1
×
ˆ
O
×
K
O
×
K
C
K
/C
0
K
Cl(K) 1.
If
K
=
Q
or
Q
(
D
), then
O
×
K
=
O
×
K
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
K
=
Q
, our favorite abelian extensions are those of
the form L = Q(ζ
N
) with N > 1. This comes with an Artin map
ˆ
Z
×
=
C
Q
/C
0
Q
Gal(L/Q)
=
(Z/N Z)
×
.
By local class field theory for
Q
p
, we see that with our normalizations, this is
just the quotient map, whose kernel is
(1 + N
ˆ
Z)
×
=
Y
p-N
Z
×
p
×
Y
p|N
(1 + NZ
p
)
×
Y
Z
×
p
=
ˆ
Z
×
.
Note that if we used the arithmetic Frobenius, then we would get the inverse of
the quotient map.
These subgroups of
ˆ
Z
×
are rather special ones. First of all (1 +
N
ˆ
Z
)
×
form
a neighbourhood of the identity in
ˆ
Z
×
. Thus, any open subgroup contains a
subgroup of this form. Equivalently, every abelian extension of
Q
is contained
in Q(ζ
N
) for some N . This is the Kronecker–Weber theorem.
For a general number field
K
, we want to write down an explicit basis for
open subgroups of 1 in π
0
(C
k
).
Definition (Modulus). A modulus is a finite formal sum
m =
X
vΣ
k
m
v
· (v)
of places of K, where m
v
0 are integers.
Given a modulus m, we define the subgroup
U
m
=
Y
v|∞,m
v
>0
K
×,0
v
×
Y
v|∞,m
v
=0
K
×
v
×
Y
v-,m
v
>0
(1+p
m
v
v
O
v
)
×
×
Y
v-,m
v
=0
O
×
v
J
K
.
Then essentially by definition of the topology of
J
K
, any open subgroup of
J
K
containing K
×,0
contains some U
m
.
In our previous example, our moduli are all of the form
Definition
(
a
(
))
.
If
a C O
K
is an ideal, we write
a
(
) for the modulus with
m
v
= v(a) for all v - , and m
v
= 1 for all v | .
If
k
=
Q
and
m
= (
N
)(
), then we simply get
U
m
=
R
×
>0
×
(1 +
N
ˆ
Z
)
×
, and
so
J
Q
Q
×
U
m
= (Z/nZ)
×
,
corresponding to the abelian extension Q(ζ
N
).
In general, we define
Definition
(Ray class field)
.
If
L/K
is abelian with
Gal
(
L/K
)
=
J
K
/K
×
U
m
under the Artin map, we call L the ray class field of K modulo m.
Definition
(Conductor)
.
If
L
corresponds to
U J
K
, then
U K
×
U
m
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
m
be a modulus, and let
S
be the set of finite
v
such that
m
v
>
0. Let
I
S
be the group of fractional ideals prime to S. Consider
P
m
= {(x) I
S
: x 1 mod m}.
To be precise, we require that for all v S, we have v(x 1) m
v
, and for all
infinite
v
real with
m
v
>
0, then
τ
(
x
)
>
0 for
τ
:
K R
the corresponding to
v
.
In other words, x K
×
U
m
.
Note that if
m
is trivial, then
I
S
/P
m
is the ideal class group. Thus, it makes
sense to define
Definition
(Ray class group)
.
Let
m
be a modulus. The generalized ideal class
group, or ray class group modulo m is
Cl
m
(K) = I
S
/P
m
.
One can show that this is always a finite group.
Proposition. There is a canonical isomorphism
J
K
K
×
U
m
Cl
m
(K)
such that for v 6∈ S Σ
K,
, the composition
K
×
v
J
K
Cl
m
(K)
sends x 7→ p
v(x)
v
.
Thus, in particular, the Galois group
Gal
(
L/K
) of the ray class field modulo
m
is
Cl
m
(
K
). Concretely, if
p 6∈ S
is an ideal, then [
p
]
Cl
m
(
K
) corresponds to
σ
p
Gal
(
L/K
), the arithmetic Frobenius. This was Artin’s original reciprocity
law.
When
m
= 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-
S
content map,
even for ideles whose content is prime to
S
. According to Folich, this is the
fundamental mistake of class field theory”.
Proof sketch. Let J
K
(S) J
K
be given by
J
K
(S) =
Y
v6∈SΣ
K,
K
×
v
.
Here we do have the inverse of the content map
c
1
: J
K
(S) I
S
(x
v
) 7→
Y
p
v(x
v
)
v
We want to extend it to an isomorphism. Observe that
J
K
(S) U
m
=
Y
v6∈SΣ
K,
O
×
v
,
which is precisely the kernel of the map
c
1
. So
c
1
extends uniquely to a
homomorphism
J
K
(S)U
m
U
m
=
J
K
(S)
J
K
(S) U
m
I
S
.
We then use that K
×
J
K
(S)U
m
= J
K
(weak approximation), and
K
×
V
m
= {x 1 mod m, x K
},
where
V
m
= J
K
(S)U
m
= {(x
v
) J
K
| for all v with m
v
> 0, x
v
U
m
}.
2 L-functions
Recall that Dirichlet characters
χ
: (
Z/N Z
)
×
C
×
give rise to Dirichlet
L
-
functions. Explicitly, we extend
χ
to a function on
Z
by setting
χ
(
a
) = 0 if
[a] 6∈ (Z/N Z)
×
, and then the Dirichlet L-function is defined by
L(χ, s) =
X
n=1
χ(n)
n
s
=
Y
p
(1 χ(p)p
s
)
1
.
As one can see from the
n
and
p
appearing, Dirichlet
L
-functions are “about
Q
”, and we would like to have a version of
L
-functions that are for arbitrary
number fields
K
. If
χ
= 1, we already know what this should be, namely the
ζ-function
ζ
K
(s) =
X
aCO
K
1
N(a)
s
=
Y
pCO
K
1
1 N(p)
s
,
where the first sum is over all ideals of
O
K
, and the second product is over all
prime ideals of O
K
.
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
K
/K
×
C
×
.
These are also known as quasi-characters in some places, where character
means unitary. However, we shall adopt the convention that characters need not
be unitary. The German term Gr¨oßencharakter (or suitable variations) are also
used.
In this section, we will seek to understand Hecke characters better, and see
how Dirichlet characters arise as a special case where
K
=
Q
. Doing so is useful
if we want to write down actual Hecke characters. The theory of
L
-functions
will be deferred to the next chapter.
We begin with the following result:
Proposition.
Let
G
be a profinite group, and
ρ
:
G GL
n
(
C
) continuous.
Then ker ρ is open.
Of course, the kernel is always closed.
Proof.
It suffices to show that
ker ρ
contains an open subgroup. We use the fact
that
GL
n
(
C
) has “no small subgroups”, i.e. there is an open neighbourhood
U
of
1
GL
n
(
C
) such that
U
contains no non-trivial subgroup of
GL
n
(
C
) (exercise!).
For example, if n = 1, then we can take U to be the right half plane.
Then for such
U
, we know
ρ
1
(
U
) is open. So it contains an open subgroup
V
. Then
ρ
(
V
) is a subgroup of
GL
n
(
C
) contained in
U
, hence is trivial. So
V ker(ρ).
While the multiplicative group of a local field is not profinite, it is close
enough, and we similarly have
Exercise.
Let
F
be a local field. Then any continuous homomorphism
F
×
C
×
has an open kernel, i.e. χ(1 + p
N
F
) = 1 for some N 0.
Definition
(Unramified character)
.
If
F
is a local field, a character
χ
:
F C
×
is unramified if
χ|
O
×
F
= 1.
If F
=
R, we say χ : F
×
C
×
is unramified if χ(1) = 1.
Using the decomposition F
=
O
F
× hπ
F
i (for the local case), we see that
Proposition. χ is unramified iff χ(x) = |x|
s
F
for some s C.
We now return to global fields. We will think of Hecke characters as continuous
maps
J
K
C
×
that factor through
J
K
/K
×
, since it is easier to reason about
J
K
than the quotient. We can begin by discussing arbitrary characters
χ
:
J
K
C
×
.
Proposition.
The set of continuous homomorphisms
χ
:
J
K
=
Q
0
v
K
×
v
C
×
bijects with the set of all families (
χ
v
)
vΣ
k
,
χ
v
:
K
×
v
C
×
such that
χ
v
is
unramified for almost all (i.e. all but finitely many)
v
, with the bijection given
by χ 7→ (χ
v
), χ
v
= χ|
K
×
v
.
Proof.
Let
χ
:
J
K
C
×
be a character. Since
ˆ
O
K
J
K
is profinite, we know
ker χ|
ˆ
O
×
K
is an open subgroup. Thus, it contains
O
×
v
for all but finitely many
v
.
So we have a map from the LHS to the RHS.
In the other direction, suppose we are given a family (
χ
v
)
v
. We attempt to
define a character χ : J
K
C
×
by
χ(x
v
) =
Y
χ
v
(x
v
).
By assumption,
χ
v
(
x
v
) = 1 for all but finitely many
v
. So this is well-defined.
These two operations are clearly inverses to each other.
In general, we can write χ as
χ = χ
χ
, χ
=
Y
v-
χ
v
: K
×,
C
×
, χ
=
Y
v|∞
χ
v
: K
×
C
×
.
Lemma. Let χ be a Hecke character. Then the following are equivalent:
(i) χ has finite image.
(ii) χ
(K
×,0
) = 1.
(iii) χ
2
= 1.
(iv) χ(C
0
K
) = 1.
(v) χ factors through Cl
m
(K) for some modulus m.
In this case, we say χ is a ray class character.
Proof.
Since
χ
(
K
×,0
) is either 1 or infinite, we know (i)
(ii). It is clear that
(ii) (iii), and these easily imply (iv). Since C
K
/C
0
K
is profinite, if (iii) holds,
then
χ
factors through
C
K
/C
0
K
and has open kernel, hence the kernel contains
U
m
for some modulus
m
. So
χ
factors through
Cl
m
(
K
). Finally, since
Cl
m
(
K
) is
finite, (v) (i) is clear.
Using this, we are able to classify all Hecke characters when K = Q.
Example.
The idele norm
| · |
A
:
C
K
R
×
>0
is a character not of finite order.
In the case
K
=
Q
, we have
C
Q
=
R
×
>0
×
ˆ
Z
×
. The idele norm is then the
projection onto R
×
>0
.
Thus, if
χ
;
C
Q
C
×
is a Hecke character, then the restriction to
R
×
>0
is of
the form x 7→ x
s
for some s. If we write
χ(x) = |x|
s
A
· χ
0
(x)
for some
χ
0
, then
χ
0
vanishes on
R
×
>0
, hence factors through
ˆ
Z
×
and has finite
order. Thus, it factors as
χ
0
: C
Q
ˆ
Z
×
(Z/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
K
, we have finite order characters as we just saw. They
correspond to characters on
I
S
which are trivial on
P
m
. In fact, we can describe
all Hecke characters in terms of ideals.
There is an alternative way to think about Hecke characters. We can
think of Dirichlet characters as (partial) functions
Z C
×
that satisfy certain
multiplicativity properties. In general, a Hecke character can be thought of as a
function on a set of ideals of O
K
.
Pick a modulus
m
such that
χ
is trivial on
ˆ
K
×
U
m
. Let
S
be the set of
finite
v
such that
m
v
is positive, and let
I
S
be the set of fractional ideals prime
to S. We then define a homomorphism
Θ : I
S
C
×
p
v
7→ χ
v
(π
v
)
1
One would not expect Θ to remember much information about the infinite
part
χ
. However, once we know Θ and
χ
(and
m
), it is not difficult to see
that we can recover all of χ.
On the other hand, an arbitrary pair
, χ
) doesn’t necessarily come from
a Hecke character. Indeed, the fact that
χ
vanishes on
K
×
implies there is some
compatibility condition between Θ and χ
.
Suppose x K
×
is such that x 1 mod m. Then (x) I
S
, and we have
1 = χ(x) = χ
(x)
Y
v6∈S finite
χ
v
(x) = χ
(x)
Y
finite v6∈S
χ
v
(π
v
)
v(x)
.
Writing
P
m
for the set of principal ideals generated by these
x
, as we previously
did, we see that for all x P
m
,
χ
(x) = Θ(x).
One can check that given (Θ
, χ
) (and
m
) satisfying this compatibility condition,
there is a unique Hecke character that gives rise to this pair. This was Hecke’s
original definition of a Hecke character, which is more similar to the definition
of a Dirichlet character.
Example.
Take
K
=
Q
(
i
), and fix an embedding into
C
. Since
Cl
(
K
) = 1, we
have
C
K
=
C
×
×
ˆ
O
×
K
µ
4
= 1, ±i}
.
Let
v
2
be the place over 2, corresponding to the prime (1 +
i
)
O
K
. Then
K
v
2
is
a ramified extension over Q
2
of degree 2. Moreover,
O
K
(1 + i)
3
O
K
×
= µ
4
= 1, ±i}.
So we have a decomposition
O
×
v
2
= (1 + (1 + i)
3
O
v
2
) × µ
4
.
Thus, there is a natural projection
C
K
C
×
× O
×
v
2
µ
4
=
C
×
× (1 + (1 + i)
3
O
v
2
) C
×
.
This gives a Hecke character with χ
(z) = z, and is trivial on
Y
v6∈{v
2
,∞}
O
×
v
× (1 + (1 + i)
3
O
v
2
),
This has modulus
m = 3(v
2
).
In ideal-theoretic terms, if
p 6
= (1 +
i
) is a prime ideal of
K
, then
p
= (
π
p
)
for a unique π
p
O
K
with π
p
1 mod (1 + i)
3
. Then Θ sends p to π
p
.
This is an example of an algebraic Hecke character.
Definition
(Algebraic homomorphism)
.
A homomorphism
K
×
C
×
is alge-
braic if there exists integers n(σ) (for all σ : K C) such that
ϕ(x) =
Y
σ(x)
n(σ)
.
The first thing to note is that if
ϕ
is algebraic, then
ϕ
(
K
×
) is contained
in the Galois closure of
K
in
C
. In particular, it takes values in the number
field. Another equivalent definition is that it is algebraic in the sense of algebraic
geometry, i.e. if
K
=
L
Qe
i
for
i
= 1
, . . . , n
as a vector space, then we can view
K
as the
Q
-points of an
n
-dimensional affine group scheme. We can then define
R
K/Q
G
m
A
to be the set on which
X
is invertible, and then an algebraic
Hecke character is a homomorphism of algebraic groups (
T
K
)
/C G
m
/C
, where
T
K
= Res
K/Q
(G
m
).
If we have a real place
v
of
K
, then this corresponds to a real embedding
σ
v
:
K K
v
=
R
, and if
v
is a complex place, we have a pair of embedding
σ
v
, ¯σ
v
:
K K
v
' C
, picking one of the pair to be
σ
v
. So
ϕ
extends to a
homomorphism
ϕ : K
×
C
×
given by
ϕ(x
v
) =
Y
v real
x
n(σ
v
)
v
Y
v complex
x
n(σ
v
)
v
¯x
n(¯σ
v
)
v
Definition
(Algebraic Hecke character)
.
A Hecke character
χ
=
χ
χ
:
J
K
/K
×
C
×
is algebraic if there exists an algebraic homomorphism
ϕ
:
K
×
C
×
such that
ϕ
(
x
) =
χ
(
x
) for all
x K
×,0
, i.e.
χ
=
ϕ
Q
v real
sgn
e
v
v
for e
v
{0, 1}.
We say ϕ (or the tuple (n(σ))
σ
) is the infinite type of χ.
Example. The adelic norm | · |
A
: J
K
C
×
has
χ
=
Y
| · |
v
,
and so
χ
is algebraic, and the associated
ϕ
is just
N
K/Q
:
K
×
Q
×
C
×
, with
(n
σ
) = (1, . . . , 1).
Exercise.
Let
K
=
Q
(
i
), and
χ
from the previous example, whose associated
character of ideals was Θ :
p 7→ π
p
, where
π
p
1
mod
(2 + 2
i
). The infinity type
is the inclusion K
×
C
×
, i.e. it has type (1, 0).
Observe that the image of an algebraic homomorphism
ϕ
:
K
×
C
×
lies in
the normal closure of K. More generally,
Proposition.
If
χ
is an algebraic Hecke character, then
χ
takes values in
some number field. We write E = E(χ) for the smallest such field.
Of course, we cannot expect
χ
to take algebraic values, since
J
K
contains
copies of R and C.
Proof.
Observe that
χ
(
ˆ
O
×
K
) is finite subgroup, so is
µ
n
for some
n
. Let
x K
×
,
totally positive. Then
χ
(x) = χ
(x)
1
= ϕ(x)
1
K
cl
,
where
K
cl
is the Galois closure. Then since
K
×
>0
×
ˆ
O
×
K
ˆ
K
×
has finite cokernel
(by the finiteness of the class group), so
χ
(
ˆ
K
×
) =
d
a
i=1
z
i
χ
(K
×
>0
ˆ
O
×
K
),
where
z
d
i
χ
(
K
×
>0
ˆ
O
×
K
), and is therefore contained inside a finite extension of
the image of K
×
>0
×
ˆ
O
×
K
.
Hecke characters of finite order (i.e. algebraic Hecke characters with infinity
type (0
, . . . ,
0)) are in bijection with continuous homomorphisms Γ
K
C
×
,
necessarily of finite order. What we show now is how to associate to a general
algebraic Hecke character
χ
a continuous homomorphism
ψ
: Γ
K
E
(
χ
)
×
λ
Q
×
, where
λ
is a place of
E
(
χ
) over
. This is continuous for the
-adic topology
on
E
λ
. In general, this will not be of finite order. Thus, algebraic Hecke
characters correspond to -adic Galois representations.
The construction works as follows: since
χ
(
x
) =
ϕ
(
x
)
1
, we can restrict
the infinity type
ϕ
to a homomorphism
ϕ
:
K
×
E
×
. We define
˜χ
:
J
K
E
×
as follows: if x = x
x
K
×
K
,×
J
K
, then we set
˜χ(x) = χ(x)ϕ(x
)
1
.
Notice that this is not trivial on
K
×
in general. Then
˜χ
takes values in
1
}
.
Thus,
˜χ
takes values in
E
×
. Thus, we know that
˜χ
has open kernel, i.e. it is
continuous for the discrete topology on E
×
, and ˜χ|
K
×
= ϕ
1
.
Conversely, if
˜χ
:
K
×
E
×
is a continuous homomorphism for the discrete
topology on
E
×
, and
˜χ|
K
×
is an algebraic homomorphism, then it comes from
an algebraic Hecke character in this way.
Let
λ
be a finite place of
E
over
, a rational prime. Recall that
ϕ
:
K
×
E
×
is an algebraic homomorphism, i.e.
ϕ
X
x
i
e
i
= f(x), f E(X
1
, . . . , X
n
).
We can extend this to K
×
= (K
Q
Q
)
×
=
Q
v|
K
×
v
to get a homomorphism
ϕ
λ
: K
×
E
×
λ
This is still algebraic, so it is certainly continuous for the -adic topology.
Now consider the character ψ
λ
: J
K
E
×
λ
, where now
ψ
λ
((x
v
)) = ˜χ(x)ϕ
λ
((x
v
)
v|
).
This is then continuous for the
-adic topology on
E
×
λ
, and moreover, we see
that
ψ
λ
(
K
×
) =
{
1
}
as
˜χ|
K
×
=
ϕ
1
while
ϕ
λ
|
K
×
=
ϕ
. Since
˜χ
(
K
×,0
) =
{
1
}
, we
know that ψ
λ
it is in fact defined on C
K
/C
0
K
=
Γ
ab
K
.
Obviously, ψ
λ
determines ˜χ and hence χ.
Fact.
An
-adic character
ψ
:
C
K
/C
0
K
E
×
λ
comes from an algebraic Hecke
character in this way if and only if the associated Galois representation is
Hodge–Tate, which is a condition on the restriction to the decomposition groups
Gal(
¯
K
v
/K
v
) for the primes v | .
Example. Let K = Q and χ = | · |
A
, then
˜χ = sgn(x
)
Y
p
|x
p
|
p
.
So
ψ
((x
v
)) = sgn(x
)
Y
p6=
|x
p
|
p
· |x
|
· x
.
Note that |x
|
x
Z
×
. We have
C
Q
/C
0
Q
=
ˆ
Z
×
.
Under this isomorphism, the map
ˆ
Z
×
Q
×
is just the projection onto
Z
×
followed
by the inclusion, and by class field theory,
ψ
:
Gal
(
¯
Q/Q
)
Z
×
is just the
cyclotomic character of the field Q({ζ
n
}),
σ(ζ
n
) = ζ
ψ
`
(σ) mod
n
n
.
Example.
Consider the elliptic curve
y
2
=
x
3
x
with complex multiplication
over Q(i). In other words, End(E/Q(i)) = Z[i], where we let i act by
i · (x, y) 7→ (x, iy).
Its Tate module
T
E = lim E[
n
]
is a Z
[i]-module. If λ | , then we define
V
λ
E = T
E
Z
`
[i]
K
λ
.
Then Γ
K
act by Γ
K
: Aut
K
λ
V
λ
E = K
×
λ
.
We now want to study the infinity types of an algebraic Hecke character.
Lemma.
Let
K
be a number field,
ϕ
:
K
×
E
×
C
×
be an algebraic
homomorphism, and suppose E/Q is Galois. Then ϕ factors as
K
×
norm
(K E)
×
φ
0
E
×
.
Note that since E is Galois, the intersection K E makes perfect sense.
Proof. By definition, we can write
ϕ(x) =
Y
σ:KC
σ(x)
n(σ)
.
Then since ϕ(x) E, for all x K
×
and τ Γ
E
, we have
Y
τσ(x)
n(σ)
=
Y
σ(x)
n(σ)
.
In other words, we have
Y
σ
σ(x)
n(τ
1
σ)
=
Y
σ
σ(x)
n(σ)
.
Since the homomorphisms
σ
are independent, we must have
n
(
τσ
) =
n
(
σ
) for
all embeddings σ : K
¯
Q and τ Γ
E
. This implies the theorem.
Recall that if
m
is a modulus, then we defined open subgroups
U
m
J
K
,
consisting of the elements (
x
v
) such that if a real
v | m
, then
x
v
>
0, and if
v | m
for a finite v, then v(x
v
1) m
v
. We can write this as
U
m
= U
m,
× U
m
.
Proposition.
Let
ϕ
:
K
×
C
×
be an algebraic homomorphism. Then
ϕ
is
the infinity type of an algebraic Hecke character χ iff ϕ(O
×
K
) is finite.
Proof.
To prove the (
) direction, suppose
χ
=
χ
χ
is an algebraic Hecke
character with infinity type
ϕ
. Then
χ
(
U
m
) = 1 for some
m
. Let
E
m
=
K
×
U
m
O
×
K
, a subgroup of finite index. As
χ
(
E
m
) = 1 =
χ
(
E
m
), we know
χ
(E
m
) = 1. So ϕ(O
×
K
) is finite.
To prove (
), given
ϕ
with
ϕ
(
O
×
K
) finite, we can find some
m
such that
ϕ
(
E
m
) = 1. Then (
ϕ,
1) :
K
×
× U
m
C
×
is trivial on
E
m
. So we can extend
this to a homomorphism
K
×
U
m
K
×
K
×
=
K
×
U
m
E
m
C
×
,
since
E
m
=
K
×
U
m
. But the LHS is a finite index subgroup of
C
K
. So the
map extends to some χ.
Here are some non-standard terminology:
Definition
(Serre type)
.
A homomorphism
ϕ
:
K
×
C
×
is of Serre type if it
is algebraic and ϕ(O
×
K
) is finite.
These are precisely homomorphisms that occur as infinity types of algebraic
Hecke characters.
Note that the unit theorem implies that
O
×
K
K
×,1
= {x K
×
: |x|
A
= 1}
has compact cokernel. If
ϕ
(
O
×
K
) is finite, then
ϕ
(
K
×,1
) is compact. So it maps
into U(1).
Example. Suppose K is totally real. Then
K
×
= (R
×
)
{σ:KR}
.
Then we have
K
×,1
= {(x
σ
) :
Y
x
σ
= ±1}.
Then
ϕ
((
x
σ
)) =
Q
x
n(σ)
σ
, so
|ϕ
(
K
×,1
)
|
= 1. In other words, all the
n
σ
are equal.
Thus, ϕ is just a power of the norm map.
Thus, algebraic Hecke characters are all of the form
| · |
m
A
· (finite order character).
Another class of examples comes from CM fields.
Definition
(CM field)
. K
is a CM field if
K
is a totally complex quadratic
extension of a totally real number field K
+
.
This CM refers to complex multiplication.
This is a rather restrictive condition, since this implies
Gal
(
K/K
+
) =
{
1
, c}
=
Gal
(
K
w
/K
+
v
) for every
w | v |
. So
c
is equal to complex conjugation for every
embedding K C.
From this, it is easy to see that CM fields are all contained in
Q
CM
¯
Q C
,
given by the fixed field of the subgroup
h
1
: σ Γ
Q
i Γ
Q
.
For example, we see that the compositum of two CM fields is another CM field.
Exercise.
Let
K
be a totally complex
S
3
-extension over
Q
. Then
K
is not CM,
but the quadratic subfields is complex and is equal to K Q
CM
.
Example.
Let
K
be a CM field of degree 2
r
. Then Dirichlet’s unit theorem
tells us
rk O
×
K
= r 1 = rk O
×
K
+
.
So
O
×
K
is a finite index subgroup of
O
×
K
+
. So
ϕ
:
K
×
C
×
is of Serre type iff
it is algebraic and its restriction to
K
+,×
is of Serre type. In other words, we
need n(σ) + n(¯σ) to be independent of σ.
Theorem.
Suppose
K
is arbitrary, and
ϕ
:
K
×
E
×
C
×
is algebraic, and
we assume
E/Q
is Galois, containing the normal closure of
K
. Thus, we can
write
ϕ(x) =
Y
σ:KE
σ(x)
n(σ)
.
Then the following are equivalent:
(i) ϕ is of Serre type.
(ii) ϕ
=
ψ N
K/F
, where
F
is the maximal CM subfield and
ψ
is of Serre type.
(iii)
For all
c
0
Gal
(
E/Q
) conjugate to complex conjugation
c
, the map
σ 7→ n(σ) + n(c
0
σ) is constant.
(iv)
(in the case
K C
and
K/Q
is Galois with Galois group
G
) Let
λ
=
P
n(σ)σ Z[G]. Then for all τ G, we have
(τ 1)(c + 1)λ = 0 = (c + 1)(τ 1)λ.
Note that in (iii), the constant is necessarily
2
[K : Q]
X
σ
n(σ).
So in particular, it is independent of c
0
.
Proof.
(iii) (iv): This is just some formal symbol manipulation.
(ii) (i): The norm takes units to units.
(i)
(iii): By the previous lecture, we know that if
ϕ
is of Serre type,
then
|ϕ(K
×,1
)| = 1.
Now if (x
v
) K
×
, we have
|ϕ((x
v
))| =
Y
real v
|x
v
|
n(σ
v
)
Y
complex v
|x
v
|
n(σ
v
)+n(¯σ
v
)
=
Y
v
|x
v
|
1
2
(n(σ
v
)+n(¯σ
v
))
v
.
Here the modulus without the subscript is the usual modulus. Then
|ϕ
(
K
×,1
)
|
= 1 implies
n
(
σ
v
) +
n
(
¯σ
v
) is constant. In other words,
n
(
σ
) +
n() = m is constant.
But if
τ Gal
(
E/Q
), and
ϕ
0
=
τ ϕ
,
n
0
(
σ
) =
n
(
τ
1
σ
), then this is also of
Serre type. So
m = n
0
(σ) + n
0
() = n(τ
1
σ) + n(τ
1
) = n(τ
1
σ) + n((τ
1
)τ
1
σ).
(iii)
(ii): Suppose
n
(
σ
) +
n
(
c
0
σ
) =
m
for all
σ
and all
c
0
=
τ
1
. Then
we must have
n(c
0
σ) = n()
for all σ. So
n(σ) = n(
1
σ)
So
n
is invariant under
H
= [
c, Gal
(
E/Q
)]
Gal
(
E/Q
), noting that
c
has order 2. So
ϕ
takes values in the fixed field
E
H
=
E Q
CM
. By
the proposition last time, this implies
ϕ
factors through
N
K/F
, where
F = E
H
K = K Q
CM
.
Recall that a homomorphism
ϕ
:
K
×
C
×
is algebraic iff it is a character
of the commutative algebraic group
T
K
=
R
K/Q
G
m
, so that
T
K
(
Q
) =
K
×
, i.e.
there is an algebraic character
ϕ
0
:
T
K
/C G
m
/C
such that
ϕ
0
restricted to
T
K
(Q) is ϕ.
Then
ϕ
is of Serre type iff
ϕ
is a character of
K
S
0
=
T
K
/E
0
K
, where
E
K
is the
Zariski closure of
O
×
K
in
T
K
and
E
0
K
is the identity component, which is the same
as the Zariski closure of
O
×
K
, where is a sufficiently small finite-index
subgroup.
The group
K
S
0
is called the connected Serre group. We have a commutative
diagram (with exact rows)
1 K
×
J
K
J
K
/K
×
1
1
K
S
0 K
S Γ
ab
K
1
π
0
This
K
S is a projective limit of algebraic groups over Q. We have
Hom(
K
S, C
×
) = Hom(
K
S, G
m
/C) = {algebraic Hecke characters of K}
The infinity type is just the restriction to
K
S
0
.
Langlands created a larger group, the Tamiyama group, an extension of
Gal
(
¯
Q/Q
) by
K
S
0
, which is useful for abelian varieties with CM and conjugations
and Shimura varieties.
2.2 Abelian L-functions
We are now going to define
L
-functions for Hecke characters. Recall that amongst
all other things, an
L
-function is a function in a complex variable. Here we are
going to do things slightly differently. For any Hecke character
χ
, we will define
L(χ), which will be a number. We then define
L(χ, s) = L(| · |
s
A
χ).
We shall define
L
(
χ
) as an Euler product, and then later show it can be written
as a sum.
Definition
(Hecke
L
-function)
.
Let
χ
:
C
K
C
×
be a Hecke character. For
v Σ
K
, we define local L-factors L(χ
v
) as follows:
If v is non-Archimedean and χ
v
unramified, i.e. χ
v
|
O
×
K
v
= 1, we set
L(χ
v
) =
1
1 χ
v
(π
v
)
.
If v is non-Archimedean and χ
v
is ramified, then we set
L(χ
v
) = 1.
If v is a real place, then χ
v
is of the form
χ
v
(x) = x
N
|x|
s
v
,
where N = 0, 1. We write
L(χ
v
) = Γ
R
(s) = π
s/2
Γ(s/2).
If v is a complex place, then χ
v
is of the form
χ
v
(x) = σ(x)
N
|x|
s
v
,
where σ is an embedding of K
v
into C and N 0. Then
L(χ
v
) = Γ
C
(s) = 2(2π)
s
Γ(s)
We then define
L(χ
v
, s) = L(χ
v
· | · |
s
v
).
So for finite unramified v, we have
L(χ
v
, s) =
1
1 χ
v
(π
v
)q
s
v
,
where q
v
= |O
K
v
/(π
v
)|.
Finally, we define
L(χ, s) =
Y
v-
L(χ
v
, s)
Λ(χ, s) =
Y
v
L(χ
v
, s).
Recall that the kernel of the idelic norm
| · |
A
:
C
K
R
×
>0
is compact. It is
then not hard to see that for every
χ
, there is some
t R
such that
χ · | · |
t
A
is
unitary. Thus,
L
(
χ, s
) converges absolutely on some right half-plane. Observe
that
Λ(χ| · |
t
A
, s) = Λ(χ, t + s).
Theorem (Hecke–Tate).
(i)
Λ(
χ, s
) has a meromorphic continuation to
C
, entire unless
χ
=
| · |
t
A
for
some t C, in which case there are simple poles at s = 1 t, t.
(ii) There is some function, the global ε-factor ,
ε(χ, s) = AB
s
for some A C
×
and B R
>0
such that
Λ(χ, s) = ε(χ, s)Λ(χ
1
, 1 s).
(iii) There is a factorization
ε(χ, s) =
Y
v
ε
v
(χ
v
, µ
v
, ψ
v
, s),
where
ε
v
= 1 for almost all
v
, and
ε
v
depends only on
χ
v
and certain
auxiliary data ψ
v
, µ
v
. These are the local ε-factors.
Traditionally, we write
L(χ, s) =
Y
finite v
L(χ
v
, s),
and then
Λ(χ, s) = L(χ, s)L
(χ, s).
However, Tate (and others, especially the automorphic people) use
L
(
χ, s
) for
the product over all v.
At first, Hecke proved (i) and (ii) using global methods, using certain Θ
functions. Later, Tate proved (i) to (iii) using local-global methods and espe-
cially Fourier analysis on
K
v
and
A
K
. This generalizes considerably, e.g. to
automorphic representations.
We can explain some ideas of Hecke’s method. We have a decomposition
K
= K
Q
R
=
R
r
1
× C
r
2
=
C
n
,
and this has a norm
k · k
induces by the Euclidean metric on
R
n
. Let
O
×
K,+
be a subgroup of totally positive units of finite index, which is
=
Z
r
1
+r
2
1
. This
has an embedding =
K
×,1
, which extends to a continuous homomorphism
R K
×,1
. The key fact is
Proposition. Let x K
×
. Pick some invariant measure du on R. Then
Z
R
1
kuxk
2s
du =
stuff
|N
K/Q
(x)|
2s/n
,
where the stuff is some ratio of Γ factors and powers of π (and depends on s).
Exercise. Prove this when K = Q[
d] for d > 0. Then = hεi, and then
LHS =
Z
−∞
1
|ε
t
x + ε
t
x
0
|
2s
dt
RHS =
stuff
|xx
0
|
s
.
The consequence of this is that if a K is a fractional ideal, then
X
06=xa mod ∆
1
|N
K/Q
(x)|
s
= stuff ·
X
06=xa mod ∆
Z
R
1
kuxk
ns
du
= stuff ·
Z
R/
X
06=xa
1
kuxk
ns
du
The integrand has a name, and is called the Epstein
ζ
-function of the lattice
(
a, ku · k
2
). By the Poisson summation formula, we get an analytic continuity
and functional equation for the epsilon
ζ
function. On the other hand, taking
linear combinations of the left gives
L
(
χ, s
) for
χ
:
Cl
(
K
)
C
×
. For more
general
χ
, we modify this with some extra factors. When the infinity type is
non-trivial, this is actually quite subtle.
Note that if
χ
is unramified outside
S
and ramified at
S
, recall we had a
homomorphism Θ : I
S
C
×
sending p
v
7→ χ
v
(π
v
)
1
. So
L(χ, s) =
Y
finite v6∈S
1
1 Θ(p
v
)
1
(Np
v
)
s
=
X
a∈O
K
prime to S
Θ(a)
1
(Na)
s
.
This was Hecke’s original definition of the Hecke character.
If
K
=
Q
and
χ
:
C
Q
C
×
is of finite order, then it factors through
C
Q
C
Q
/C
0
Q
=
ˆ
Z
×
(
Z/N Z
)
×
, and so
χ
is just some Dirichlet character
ϕ
: (
Z/nZ
)
×
C
×
. The associated
L
-functions are just Dirichlet
L
-functions.
Indeed, if p - N, then
χ
p
(p) = χ(1, . . . , 1, p, 1, . . .) = χ(p
1
, . . . , p
1
, 1, p
1
, . . .) = ϕ(p mod N )
1
.
In other words,
L
(
χ, s
) is the Dirichlet
L
-series of
ϕ
1
(assuming
N
is chosen so
that χ ramifies exactly at v | N).
Tate’s method uses local
ε
-factors
ε
(
χ
v
, µ
v
, ψ
v
, s
), where
ψ
v
:
K
v
U(1) is
a non-trivial additive character, e.g. for v finite,
K
V
Q
p
Q
p
/Z
p
=
Z[v
p
]/Z C
×
,
tr e
2πix
which we needed because Fourier transforms take in additive measures, and
µ
v
is a Haar measure on K
v
. The condition for (iii) to hold is
Y
ψ
v
: A
K
U(1)
is well-defined and trivial on
K A
K
, and
µ
A
=
Q
µ
v
is a well-defined measure
on A
K
, i.e. µ
v
(O
v
) = 1 for all v and
Z
A
K
/K
µ
A
= 1.
There exists explicit formulae for these
ε
v
’s. If
χ
v
is unramified, then it is just
A
v
B
s
v
, and is usually 1; for ramified finite v, they are given by Gauss sums.
2.3 Non-abelian L-functions
Let K be a number field. Then we have a reciprocity isomorphism
Art
K
: C
K
/C
0
K
Γ
ab
K
.
If
χ
:
C
K
C
0
K
C
×
is a Hecke character of finite order, then we can view it
as a map ψ = χ Art
1
K
: Γ
K
C
×
. Then
L(χ, s) =
Y
finite v unramified
1
1 χ
v
(π
v
)q
s
v
1
=
Y
1
1 ψ(Frob
v
)q
s
v
,
where
Frob
v
Γ
K
v
/I
K
v
is the geometric Frobenius, using that
ψ
(
I
K
v
) = 1.
Artin generalized this to arbitrary complex representations of Γ
K
.
Let ρ : Γ
K
GL
n
(C) be a representation. Define
L(ρ, s) =
Y
finite v
L(ρ
v
, s),
where
ρ
v
is the restriction to the decomposition group at
v
, and depends only
on the isomorphism class of
ρ
. We first define these local factors for non-
Archimedean fields:
Definition.
Let
F
be local and non-Archimedean. Let
ρ
:
W
F
GL
C
(
V
) be a
representation. Then we define
L(ρ, s) = det(1 q
s
ρ(Frob
F
)|
V
I
F
)
1
,
where V
I
F
is the invariants under I
F
.
Note that in this section, all representations will be finite-dimensional and
continuous for the complex topology (so in the case of
W
F
, we require
ker σ
to
be open).
Proposition.
(i) If
0 (ρ
0
, V
0
) (ρ, V ) (ρ
00
, V
0
) 0
is exact, then
L(ρ, s) = L(ρ
0
, s) · L(ρ
00
, s).
(ii)
If
E/F
is finite separable,
ρ
:
W
E
GL
C
(
V
) and
σ
=
Ind
W
F
W
E
ρ
:
W
F
GL
C
(U), then
L(ρ, s) = L(σ, s).
Proof.
(i) Since ρ has open kernel, we know ρ(I
F
) is finite. So
0 (V
0
)
I
F
V
I
F
(V
0
)
I
F
0
is exact. Then the result follows from the multiplicativity of det.
(ii) We can write
U = {ϕ : W
F
V : ϕ(gx) = ρ(g)ϕ(x) for all g W
E
, x W
F
}.
where W
F
acts by
σ(g)ϕ(x) = ϕ(xg).
Then we have
U
I
F
= {ϕ : W
F
/I
F
V : ···}.
Then whenever ϕ U
I
F
and g I
E
, then
σ(g)ϕ(x) = ϕ(xg) = ϕ((xgx
1
)x) = ϕ(x).
So in fact ϕ takes values in V
I
E
. Therefore
U
I
F
= Ind
W
F
/I
F
W
E
/I
E
V
I
E
.
Of course, W
F
/I
F
=
Z, which contains W
E
/I
E
as a subgroup. Moreover,
Frob
d
F
= Frob
E
,
where d = [k
E
: k
F
]. We note the following lemma:
Lemma.
Let
G
=
hgi H
=
hh
=
g
d
i
,
ρ
:
H GL
C
(
V
) and
σ
=
Ind
G
H
ρ
.
Then
det(1 t
d
ρ(h)) = det(1 (g)).
Proof.
Both sides are multiplicative for exact sequences of representations
of
H
. So we can reduce to the case of
dim V
= 1, where
ρ
(
h
) =
λ C
×
.
We then check it explicitly.
To complete the proof of (ii), take
g
=
Frob
F
and
t
=
q
s
F
so that
t
d
= q
s
E
.
For Archimedean
F
, we define
L
(
ρ, s
) in such a way to ensure that (i) and
(ii) hold, and if dim V = 1, then
L(ρ, s) = L(χ, s),
where if
ρ
:
W
ab
F
C
×
, then
χ
Is the corresponding character of
F
×
under the
Artin map.
If
F ' C
, then this is rather easy, since every irreducible representation of
W
F
=
C
×
is one-dimensional. We then just define for
ρ
1-dimensional using
W
ab
F
=
F
×
and extend to all
ρ
by (i). The Jordan–H¨older theorem tells us this
is well-defined.
If F ' R, then recall that
W
R
= hC
×
, s : s
2
= 1 C
×
, szs
1
= ¯zi.
Contained in here is W
(1)
R
= hU(1), si. Then
W
R
= W
(1)
R
× R
×
>0
.
It is then easy to see that the irreducible representations of W
R
are
(i) 1-dimensional ρ
W
R
; or
(ii) 2-dimensional, σ = Ind
W
R
C
ρ, where ρ 6= ρ
s
: C
×
C
×
.
In the first case, we define
L(ρ, s) = L(χ, s)
using the Artin map, and in the second case, we define
L(σ, s) = L(ρ, s)
using (ii).
To see that the properties are satisfied, note that (i) is true by construction,
and there is only one case to check for (ii), which is if ρ = ρ
s
, i.e.
ρ(z) = (z¯z)
t
.
Then
Ind
W
R
C
×
ρ
is reducible, and is a sum of characters of
W
ab
R
=
R
×
, namely
x 7→ |x|
t
and x 7→ sgn(x)|x|
t
= x
1
|x|
t+1
. Then (ii) follows from the identity
Γ
R
(s
R
(s + 1) = Γ
C
(s) = 2(2π)
s
Γ(s).
Now let
K
be global, and let
ρ
: Γ
K
GL
C
(
V
). For each
v
Σ
K
, choose
¯
k
of
¯
K
over
v
. Let Γ
v
=
Γ
K
v
be the decomposition group at
¯v
. These contain
I
V
,
and we have the geometric Frobenius
Frob
v
Γ
v
/I
V
. We define
ρ
v
=
ρ|
Γ
v
, and
then set
L(ρ, s) =
Y
v-
L(ρ
v
, s) =
Y
v-
det(1 q
s
v
Frob
v
|
V
I
v
)
1
Λ(ρ, s) = LL
L
=
Y
v|∞
L(ρ
v
, s).
This is well-defined as the decomposition groups
¯v | v
are conjugate. If
dim V
= 1,
then ρ = χ Art
1
K
for a finite-order Hecke character χ, and then
L(ρ, s) = L(χ, s).
The facts we had for local factors extend to global statements
Proposition.
(i) L(ρ ρ
0
, s) = L(ρ, s)L(ρ
0
, s).
(ii) If L/K is finite separable and ρ : Γ
L
GL
C
(V ) and σ = Ind
Γ
K
Γ
L
(ρ), then
L(ρ, s) = L(σ, s).
The same are true for Λ(ρ, s).
Proof.
(i) is clear. For (ii), we saw that if
w
Σ
L
over
v
Σ
K
and consider the
local extension L
w
/K
v
, then
L(ρ
w
, s) = L(Ind
Γ
K
v
Γ
L
w
ρ
w
).
In the global world, we have to take care of the splitting of primes. This boils
down to the fact that
Ind
Γ
K
Γ
L
ρ
Γ
K
v
=
M
w|v
Ind
Γ
K
v
Γ
L
w
(ρ|
Γ
L
w
). ()
We fix a valuation
¯v
of
¯
K
over
v
. Write Γ
¯v/v
for the decomposition group in Γ
K
.
Write
¯
S for the places of
¯
K over v, and S the places of L over v.
The Galois group acts transitively on
¯
S, and we have
¯
S
=
Γ
K
/Γ
¯v/v
.
We then have
S
=
Γ
L
\Γ
K
/Γ
¯v/v
,
which is compatible with the obvious map
¯
S S.
For ¯w = g¯v, we have
Γ
¯w/v
= gΓ
¯v/v
g
1
.
Conjugating by
g
1
, we can identify this with Γ
¯v/v
. Similarly, if
w
=
¯w|
L
, then
this contains
Γ
¯w/w
= gΓ
¯v/v
g
1
Γ
L
,
and we can identify this with Γ
¯v/v
g
1
Γ
L
g.
There is a theorem, usually called Mackey’s formula, which says if
H, K G
are two subgroups of finite index, and
ρ
:
H GL
C
(
V
) is a representation of
H
.
Then
(Ind
G
H
V )|
K
=
M
gH\G/K
Ind
K
Kg
1
Hg
(
g
1
V ),
where
g
1
V
is the
K g
1
Hg
-representation where
g
1
xg
acts by
ρ
(
x
). We
then apply this to G = Γ
K
, H = Γ
L
, K = Γ
¯v/v
.
Example. If ρ is trivial, then
L(ρ, s) =
Y
v
(1 q
s
v
)
1
=
X
aCO
K
1
Na
s
= ζ
K
(s).
This is just the Dedekind ζ-function of K.
Example.
Let
L/K
be a finite Galois extension with Galois group
G
. Consider
the regular representation
r
L/K
on
C
[
G
]. This decomposes as
L
ρ
d
i
i
, where
{ρ
i
}
run over the irreducible representations of G of dimension d
i
. We also have
r
L/K
= Ind
Γ
K
Γ
L
(1).
So by the induction formula, we have
ζ
L
(s) = L(r
L/K
, s) =
Y
i
L(ρ
i
, s)
d
i
.
Example. For example, if L/K = Q(ζ
N
)/Q, then
ζ
Q(ζ
N
)
(s) =
Y
χ
L(χ, s),
where the product runs over all primitive Dirichlet characters mod
M | N
. Since
ζ
Q(ζ
N
)
, ζ
Q
have simple poles at s = 1, we know that L(χ, 1) 6= 0 if χ 6= χ
0
.
Theorem
(Brauer induction theorem)
.
Suppose
ρ
:
G GL
N
(
C
) is a represen-
tation of a finite group. Then there exists subgroups
H
j
G
and homomorphisms
χ
j
: H
j
C
×
and integers m
i
Z such that
tr ρ =
X
j
m
j
tr Ind
G
H
j
χ
j
.
Note that the
m
j
need not be non-negative. So we cannot quite state this as a
statement about representations.
Corollary.
Let
ρ
: Γ
K
GL
N
(
C
). Then there exists finite separable
L
j
/K
and χ
j
: Γ
L
j
C
×
of finite order and m
j
Z such that
L(ρ, s) =
Y
j
L(χ
j
, s)
m
j
.
In particular,
L
(
ρ, s
) has meromorphic continuation to
C
and has a functional
equation
Λ(ρ, s) = L · L
= ε(ρ, s)L(˜ρ, 1 s)
where
ε(ρ, s) = AB
s
=
Y
ε(χ
j
, s)
m
j
,
and ˜ρ(g) =
t
ρ(g
1
).
Conjecture
(Artin conjecture)
.
If
ρ
does not contain the trivial representation,
then Λ(ρ, s) is entire.
This is closely related to the global Langlands conjecture.
In general, there is more than one way to write
ρ
as an sum of virtual
induced characters. But when we take the product of the
ε
factors, it is always
well-defined. We also know that
ε(χ
j
, s) =
Y
ε
v
(χ
j,v
, s)
is a product of local factors. It turns out the local decomposition is not indepen-
dent of the decomposition, so if we want to write
ε(ρ, s) =
Y
v
ε
v
(ρ
v
, s),
we cannot just take
ε
v
(
ρ
v
, s
) =
Q
ε
v
(
χ
j,v
, s
), as this is not well-defined. However,
Langlands proved that there exists a unique factorization of
ε
(
ρ, s
) satisfying
certain conditions.
We fix F a non-Archimedean local field, χ : F
×
C
×
and local ε factors
ε(χ, ψ, µ),
where
µ
is a Haar measure on
F
and
ψ
:
F
U(1) is a non-trivial character.
Let n(ψ) be the least integer such that ψ(π
n
F
O
F
) = 1. Then
ε(χ, ψ, µ) =
(
µ(O
F
) χ unramified, n(ψ) = 0
R
F
×
χ
1
· ψ dµ χ ramified
Since
χ
and
ψ
are locally constant, the integral is actually sum, which turns out
to be finite (this uses the fact that χ is ramified).
For a F
×
and b > 0, we have
ε(χ, ψ(ax), ) = χ(a)|a|
1
(χ, ψ, µ).
Theorem
(Langlands–Deligne)
.
There exists a unique system of local constants
ε(ρ, ψ, µ) for ρ : W
F
GL
C
(V ) such that
(i) ε
is multiplicative in exact sequences, so it is well-defined for virtual
representations.
(ii) ε(ρ, ψ, ) = b
dim V
ε(ρ, ψ, µ).
(iii)
If
E/F
is finite separable, and
ρ
is a virtual representation of
W
F
of degree
0 and σ = Ind
W
F
W
E
ρ, then
ε(σ, ψ, µ) = ε(ρ, ψ tr
E/F
, µ
0
).
Note that this is independent of the choice of
µ
and
µ
0
, since
dim V
= 0”.
(iv) If dim ρ = 1, then ε(ρ) is the usual abelian ε(χ).
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
G
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
E
(V )
=
GL
n
(E).
In this section,we will always take
G
= Γ
F
or
W
F
, where
F/Q
p
is a finite
extension with p 6= .
Example.
The cyclotomic character
χ
cycl
: Γ
K
Z
×
Q
×
is defined by the
relation
ζ
χ
cycl
(γ)
= γ(ζ)
for all
ζ
¯
K
with
ζ
n
= 1 and
γ
Γ
K
. This is a one-dimensional
-adic
representation.
Example. Let E/K be an elliptic curve. We define the Tate module by
T
E = lim
n
E[
n
](
¯
K), V
E = T
E
Z
`
Q
.
Then V
E is is a 2-dimensional -adic representation of Γ
K
over Q
.
Example. More generally, if X/K is any algebraic variety, then
V = H
i
´et
(X
K
¯
K, Q
)
is an -adic representation of Γ
K
.
We will actually focus on the representations of the Weil group
W
F
instead of
the full Galois group
G
F
. 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
F
: W
ab
F
=
F
×
.
So one-dimensional representations of
W
F
are the same as one-dimensional
representations of F
×
.
For example, there is an absolute value map
F
×
Q
×
, inducing a represen-
tation
ω
:
W
F
Q
×
. Under the Artin map, this sends the geometric Frobenius
to
1
q
. In fact, ω is the restriction of the cyclotomic character to W
F
.
Recall that we previously defined the tame character. Pick a sequence
π
n
¯
F
by π
0
= π and π
n+1
= π
n
. We defined, for any γ Γ
F
,
t
(γ) =
γ(π
n
)
π
n
n
lim
µ
n
(
¯
F ) = Z
(1).
When we restrict to the inertia group, this is a homomorphism, independent
of the choice of (
π
n
), which we call the tame character. In fact, this map is
Γ
F
-equivariant, where Γ
F
acts on
I
F
by conjugation. In general, this still defines
a function Γ
F
Z
(1), which depends on the choice of π
n
.
Example.
Continuing the previous notation, where
π
is a uniformizer of
F
, we
let T
n
be the
n
-torsion subgroup of
¯
F
×
/hπi. Then
T
n
= hζ
n
, π
n
i/hπ
n
n
i
=
(Z/
n
Z)
2
.
The
th power map
T
n
T
n1
is surjective, and we can form the inverse limit
T
, which is then isomorphic to
Z
2
. This then gives a 2-dimensional
-adic
representation of Γ
F
.
In terms of the basis (ζ
n
), (π
n
), the representation is given by
γ 7→
χ
cycl
(γ) t
(γ)
0 1
.
Notice that the image of
I
F
is
1 Z
0 1
. In particular, it is infinite. This cannot
happen for one-dimensional representations.
Perhaps surprisingly, the category of
-adic representations of
W
F
over
E
does not depend on the topology of
E
, but only the
E
as an abstract field. In
particular, if we take
E
=
¯
Q
, then after taking care of the slight annoyance that
it is infinite over
¯
Q
, the category of representations over
¯
Q
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
Z
(1)
=
Z
. In other words, fix a system (
ζ
n
) such that
ζ
n
=
ζ
n1
. We then view
t
as
a homomorphism I
F
Z
via this identification.
Let
ρ
:
W
F
GL
(
V
) be an
-adic representation over
E
. Then there exists
an open subgroup
I
0
I
F
and a nilpotent
N End
E
V
such that for all
γ I
0
,
ρ(γ) = exp(t
(γ)N) =
X
j=0
(t
(γ)N)
j
j!
.
In particular, ρ(I
0
) unipotent and abelian.
In our previous example, N =
0 1
0 0
.
Proof. If ρ(I
F
) is finite, let I
0
= ker ρ I
F
and N = 0, and we are done.
Otherwise, first observe that
G
is any compact group and
ρ
:
G GL
(
V
)
is an
-adic representation, then
V
contains a
G
-invariant lattice, i.e. a finitely-
generated
O
E
-submodule of maximal rank. To see this, pick any lattice
L
0
V
.
Then ρ(G)L
0
is compact, so generates a lattice which is G-invariant.
Thus, pick a basis of an
I
F
-invariant lattice. Then
ρ
:
W
F
GL
n
(
E
)
restricts to a map I
F
GL
n
(O
E
).
We seek to understand this group
GL
n
(
O
E
) better. We define a filtration on
GL
n
(O
E
) by
G
k
= {g GL
n
(O
E
) : g I mod
k
},
which is an open subgroup of
GL
n
(
O
E
). Note that for
k
1, there is an
isomorphism
G
k
/G
k+1
M
n
(O
E
/O
E
),
sending 1 +
k
g
to
g
. Since the latter is an
-group, we know
G
1
is a pro-
group.
Also, by definition, (G
k
)
G
k+1
.
Since
ρ
1
(
G
2
) is open, we can pick an open subgroup
I
0
I
F
such that
ρ
(
I
0
)
G
2
. Recall that
t
(
I
F
) is the maximal pro-
quotient of
I
F
, because the
tame characters give an isomorphism
I
F
/P
F
=
Y
-p
Z
(1).
So ρ|
I
0
: I
0
G
2
factors as
I
0
t
(I
0
) =
s
Z
G
2
t
` ν
,
using the assumption that ρ(I
F
) is infinite.
Now for r s, let T
r
= ν(
r
) = T
rs
s
G
r+2s
. For r sufficiently large,
N
r
= log(T
r
) =
X
m1
(1)
m1
(T
r
1)
m
m
converges -locally, and then T
r
= exp N
r
.
We claim that
N
r
is nilpotent. To see this, if we enlarge
E
, we may assume
that all the eigenvalues of N
r
are in E. For δ W
F
and γ I
F
, we know
t
(δγδ
1
) = ω(δ)t
(γ).
So
ρ(δγδ
1
) = ρ(γ)
w(σ)
for all γ I
0
. So
ρ(σ)N
r
ρ(δ
1
) = ω(δ)N
r
.
Choose
δ
lifting
ϕ
q
,
w
(
δ
) =
q
. Then if
v
is an eigenvector for
N
r
with eigenvalue
λ
, then
ρ
(
δ
)
v
is an eigenvector of eigenvalue
q
1
λ
. Since
N
r
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 =
1
r
N
r
for r sufficiently large, and this works.
There is a slight unpleasantness in this theorem that we fixed a choice of
n
roots of unity. To avoid this, we can say there exists an
N
:
v
(1) =
V
Z
`
Z
(1)
V nilpotent such that for all γ I
0
, 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
W
F
over a field E of characteristic 0 is a pair (ρ, N), where
ρ
:
W
F
GL
E
(
V
) is a finite-dimensional representation of
W
F
over
E
with open kernel; and
N End
E
(V ) is nilpotent such that for all γ W
F
, we have
ρ(γ)Nρ(γ)
1
= ω(γ)N,
Note that giving
N
is the same as giving a unipotent
T
=
exp N
, which
is the same as giving an algebraic representation of
G
a
. So a Weil–Deligne
representation is a representation of a suitable semi-direct product W
F
n G
a
.
Weil–Deligne representations form a symmetric monoidal category in the
obvious way, with
(ρ, N) (ρ
0
, N
0
) = (ρ ρ
0
, N 1 + 1 N).
There are similarly duals.
Theorem.
Let
E/Q
be finite (and
6
=
p
). Then there exists an equivalence of
(symmetric monoidal) categories
-adic representations
of W
F
over E
Weil–Deligne representations
of W
F
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
ρ
has open kernel.
In particular, the topology of E is not used.
Proof.
We have already fixed an isomorphism
Z
(1)
=
Z
. We also pick a lift
Φ W
F
of the geometric Frobenius. In other words, we are picking a splitting
W
F
= hΦi n I
F
.
The equivalence will take an
-adic representation
ρ
to the Weil–Deligne repre-
sentation (ρ, N ) on the same vector space such that
ρ
m
γ) = ρ
m
γ) exp t
(γ)N ()
for all m Z and γ I
F
.
To check that this “works”, we first look at the right-to-left direction. Suppose
we have a Weil–Deligne representation (
ρ, N
) on
V
. We then define
ρ
:
W
F
Aut
E
(
V
) by (
). Since
ρ
has open kernel, it is continuous. Since
t
is also
continuous, we know
ρ
is continuous. To see that
ρ
is a homomorphism,
suppose
Φ
m
γ · Φ
m
δ = Φ
m+n
γ
0
δ
where γ, δ I
F
and
γ
0
= Φ
n
γΦ
n
.
Then
exp t
(γ)N · ρ
n
δ) =
X
j0
1
j!
t
(γ)
j
N
j
ρ
n
δ)
=
X
j0
1
j!
t
(γ)q
nj
ρ
n
δ)N
j
= ρ
n
δ) exp(q
n
t
(γ)).
But
t
(γ
0
) = t
n
γΦ
n
) = ω
n
)t
(γ) = q
n
t
(γ).
So we know that
ρ
m
γ)ρ
n
δ) = ρ
m+n
γ
0
δ).
Notice that if
γ I
F
ker ρ
, then
ρ
(
γ
) =
exp t
(
γ
)
N
. So
N
is the nilpotent
endomorphism occurring in the Grothendieck theorem.
Conversely, given an
-adic representation
ρ
, let
N End
E
V
be given by
the monodromy theorem. We then define
ρ
by (
). Then the same calculation
shows that (
ρ, N
) is a Weil–Deligne representation, and if
I
0
I
F
is the open
subgroup occurring in the theorem, then
ρ
(
γ
) =
exp t
(
γ
)
N
for all
γ I
0
. So
by (), we know ρ(I
0
) = {1}, and so ρ has open kernel.
This equivalence depends on two choices the isomorphism
Z
(1)
=
Z
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
¯
Q
, instead of some finite
extension
E
. Note that if we have a continuous homomorphism
ρ
:
W
F
GL
n
(
¯
Q
), then there exists a finite E/Q
such that ρ factors through GL
n
(E).
Indeed,
ρ
(
I
F
)
GL
n
(
¯
Q
) 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,
ρ
(
I
F
) is contained in some
GL
n
(
E
), and of course,
ρ
(Φ) is contained
in some GL
n
(E).
Recalling that a Weil–Deligne representation over
E
only depends on
E
as a
field, and
¯
Q
=
¯
Q
0
for any ,
0
, we know that
Theorem.
Let
,
0
6
=
p
. Then the category of
¯
Q
representations of
W
F
is
equivalent to the category of
¯
Q
0
representations of W
F
.
Conjecturally,
-adic representations coming from algebraic geometry have
semi-simple Frobenius. This notion is captured by the following proposi-
tion/definition.
Proposition.
Suppose
ρ
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
q
).
(ii) ρ
(γ) is semi-simple for all γ W
F
\ I
F
.
(iii) ρ is semi-simple.
(iv) ρ(Φ) is semi-simple.
In this case, we say
ρ
and (
ρ, N
) are
F
-semisimple (where
F
refers to Frobenius).
Proof.
Recall that
W
F
=
Z o I
F
, and
ρ
(
I
F
) is finite. So that part is always
semisimple, and thus (iii) and (iv) are equivalent.
Moreover, since
ρ
(Φ) =
ρ
(Φ), we know (i) and (iii) are equivalent. Finally,
ρ
(Φ) is semi-simple iff
ρ
n
) 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
F
is not semi-simple, since
it has matrix
ρ
(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
F
-semisimple Weil–
Deligne representations. In the case of vector spaces, if a matrix
N
acts nilpo-
tently on a vector space
V
, then the Jordan normal form theorem says there is a
basis of
V
in which
N
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)
.
If
V
is semi-simple,
N End
(
V
) is nilpotent
with
N
m+1
= 0, then there exists subobjects
P
0
, . . . , P
m
V
(not unique as
subobjects, but unique up to isomorphism), such that
N
r
:
P
r
N
r
P
r
is an
isomorphism, and N
r+1
P
r
= 0, and
V =
m
M
r=0
P
r
N P
r
··· N
r
P
r
=
m
M
r=0
P
r
Z
Z[N]
(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
P
0
to be the things killed by
N
but not in the image of
N
. Thus, using semisimplicity,
we pick P
0
to be a splitting
ker N = (ker N im N) P
0
.
Similarly, we can pick P
1
by
ker N
2
= (ker N + (im N ker N
2
)) P
1
.
One then checks that this works.
We will apply this when
V
is a representation of
W
F
GL
(
V
) and
N
is the
nilpotent endomorphism of a Weil–Deligne representation. Recall that we had
ρ(γ)Nρ(γ)
1
= ω(γ)N,
so
N
is a map
V V ω
1
, rather than an endomorphism of
V
. 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
σ
is an irreducible representation of
W
F
and
sp
(
n
)
=
E
n
is the
representation
ρ = diag(ω
n1
, . . . , ω, 1), N =
0 1
.
.
.
.
.
.
0 1
0
Example. If
ρ =
ω
ω ω
1
, 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,
0)
sp
(
n
), then
F
-semisimplicity
is clear, and we have to check that it is indecomposable. Observe that the kernel
of
N
is still a representation of
W
F
. Writing
V
N=0
for the kernel of
N
in
V
, we
note that V
N=0
= σ ω
n1
, which is irreducible. Suppose that
(ρ, N) = U
1
U
2
.
Then for each
i
, we must have
U
N=0
i
= 0 or
V
N=0
. We may wlog assume
U
N=0
1
= 0. Then this forces U
1
= 0. So we are done.
Conversely, if (
ρ, N, V
) is
F
-semisimple and indecomposable, then
V
is a
representation of
W
F
which is semi-simple and
N
:
V V ω
1
. By Jordan
normal form, we must have
V = U N U ··· N
r
U
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
SU
(2) lets us capture the
N
bit of
F
-semisimple
Weil–Deligne representation via the following group:
Definition
(Weil–Langlands group)
.
We define the (Weil–)Langlands group to
be
L
F
= W
F
× SU(2).
A representation of
L
F
is a continuous action on a finite-dimensional vector
space (thus, the restriction to W
F
has open kernel).
Theorem.
There exists a bijection between
F
-semisimple Weil–Deligne repre-
sentations over
C
and semi-simple representations of
L
F
, compatible with tensor
products, duals, dimension, etc. In this correspondence:
The representations
ρ
of
L
F
that factor through
W
F
correspond to the
Weil–Deligne representations (ρ, 0).
More generally, simple
L
F
representations
σ
(
Sym
n1
C
2
) 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 (
F
-semisimple) Weil–Deligne representations over
C
are in
bijection those over
¯
Q
, using an isomorphism
¯
Q
=
C.
4 The Langlands correspondence
Local class field theory says we have an isomorphism
W
ab
F
=
F
×
.
If we want to state this in terms of the full Weil group, we can talk about the
one-dimensional representations of
W
F
, and write local class field theory as a
correspondence
characters of GL
1
(F )
1-dimensional representations
of W
F
The Langlands correspondence aims to understand the representations of
GL
n
(
F
), and it turns out this corresponds to
n
-dimensional representations of
L
F
. 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
GL
n
(
F
) contains a profinite
open subgroup
GL
n
(
O
F
). 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
G
is a contin-
uous representation of
G
over
C
, where
C
is given the discrete topology. That
is, it is a pair (
π, V
) where
V
is a complex vector space and
π
:
G GL
C
(
V
) a
homomorphism such that for every v V , the stabilizer of v in G is open.
Note that we can replace
C
with any field, but we like
C
. Typically,
V
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
K
=
{v V
:
π
(
g
)
v
=
v g K}
is finite-dimensional.
Example.
Take
G
=
GL
2
(
F
). Then
P
1
(
F
) has a right action of
G
by linear
transformations. In fact, we can write P
1
(F ) as
P
1
(F ) =
∗ ∗
0
\G.
Let
V
be the space of all locally constant functions
f
:
P
1
(
F
)
C
. There are
lots of such functions, because
P
1
(
F
) is totally disconnected. However, since
P
1
(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
GL
n
(
F
)
does not have very interesting finite-dimensional representations.
Proposition.
Let
G
=
GL
n
(
F
). If (
π, V
) is a smooth representation with
dim V < , then
π = σ det
for some σ : F
×
GL
C
(V ).
So these are pretty boring.
Proof. If V =
L
d
i=1
Ce
i
, then
ker π =
d
\
i=1
(stabilizers of e
i
)
is open. It is also a normal subgroup, so
ker π K
m
= {g GL
n
(O) : g I mod
m
}
for some
m
, where
is a uniformizer of
F
. In particular,
ker π
contains
E
ij
(
x
)
for some
i 6
=
j
and
x
, 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
E
ij
(
x
) for all
x F
and
i 6
=
j
. For any field, these matrices generate
SL
n
(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
π
: Z(G) C
×
such that
π(zg) = ω
π
(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
n
(F )
semi-simple n-dimensional
representations of L
F
.
In the next section, we will introduce the Hecke algebra, which allows us to
capture these scary infinite dimensional representations of
GL
n
(
F
) in terms of
something finite-dimensional.
Afterwards, we are going to state the Langlands classification of irreducible
admissible representations of
GL
n
(
F
). 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
C
c
(
G
) 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
c
(G) : ϕ(kgk
0
) = ϕ(g) for all k, k
0
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
G
. This is a functional
C
c
(
G
)
C
, written
f 7→
Z
G
f(g) dµ(g),
that is invariant under left translation, i.e. for all h G,we have
Z
f(hg) dµ(g) =
Z
f(g) dµ(g).
To construct the Haar measure, we take
µ
(1
K
) = 1. Then if
K
0
K
is an open
subgroup, then it is of finite index, and since we want
µ
(1
xK
0
) =
µ
(1
K
0
), we
must have
µ(1
K
0
) =
1
(K : K
0
)
.
We then set
µ
(1
xK
0
) =
µ
(1
K
0
) 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
0
)(g) =
Z
G
ϕ(x)ϕ
0
(x
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
e
K
=
1
µ(K)
1
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 =
Z
G
ϕ(g)π(g)v dµ(g).
Note that since the stabilizer of
v
is open, the integral is actually a finite sum,
so we can make sense of it. One then sees that
π(ϕ)π(ϕ
0
) = π(ϕ ϕ
0
).
This would imply
V
is a
H
(
G, K
)-module, if the unit acted appropriately. It
doesn’t, however, since in fact
π
(
ϕ
) always maps into
V
K
. Indeed, if
k K
,
then
π(k)π(ϕ)v =
Z
G
ϕ(g)π(kg)v dµ(g) =
Z
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
V
K
is an
H
(
G, K
)-module, and one easily checks
that
π
(
e
K
) indeed acts as the identity. We also have a canonical projection
π(e
K
) : V V
K
.
In good situations, this Hecke module determines V .
Proposition.
There is a bijection between isomorphism classes of irreducible ad-
missible (
π, V
) with
V
K
6
= 0 and isomorphism classes of simple finite-dimensional
H(G, K)-modules, which sends (π, V ) to V
K
with the action we described.
If we replace K by a smaller subgroup K
0
K, then we have an inclusion
H(G, K) H(G, K
0
),
which does not take
e
K
to
e
K
0
. We can then form the union of all of these, and
let
H(G) = lim
K
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
H
(
G
) acts on any smooth representation, and we get an equivalence of
categories
smooth
G-representations
non-degenerate
H(G)-modules
.
The non-degeneracy condition is V = H(G)V .
Note that if
ϕ H
(
G
) and (
π, V
) is admissible, then the
rank π
(
ϕ
)
<
,
using that
V
K
is finite-dimensional. So the trace is well-defined. The character
of (π, V ) is then the map
ϕ 7→ tr π(ϕ)
In this sense, admissible representations have traces.
4.3 The Langlands classification
Recall that the group algebra
C
[
G
] is an important tool in the representation
theory of finite groups. This decomposes as a direct sum over all irreducible
representations
C[G] =
M
π
π
dim(π)
.
The same result is true for compact groups, if we replace
C
[
G
] by
L
2
(
G
). We
get a decomposition
L
2
(G) =
ˆ
M
π
π
dim(π)
,
where
L
2
is defined with respect to the Haar measure, and the sum is over all
(finite dimensional) irreducible representations of
G
. The hat on the direct sum
says it is a Hilbert space direct sum, which is the completion of the vector space
direct sum. This result is known as the Peter–Weyl theorem. For example,
L
2
(R/Z) =
ˆ
M
nZ
C · e
2πiny
.
However, if G is non-compact, then this is no longer true.
Sometimes, we can salvage this a bit by replacing the discrete direct sum
with a continuous version. For example, the characters of
R
are those of the
form
x 7→ e
2πixy
,
which are not L
2
functions. But we can write any function in L
2
(R) as
x 7→
Z
y
ϕ(y)e
2πixy
dy.
So in a sense,
L
2
(
R
) is the “continuous direct sum” of irreducible representations.
In general,
L
2
(
G
) decomposes as a sum of irreducible representations, and
contains both a discrete sum and a continuous part. However, there are irre-
ducible representations that don’t appear in
L
2
(
G
), discretely or continuously.
These are known as the complementary series representations. This happens,
for example, for G = SL
2
(R) (Bargmann 1947).
We now focus on the case
G
=
GL
n
(
F
), or any reductive
F
-group (it doesn’t
hurt to go for generality if we are not proving anything anyway). It turns out
in this case, we can describe the representations that appear in
L
2
(
G
) pretty
explicitly. These are characterized by the matrix coefficients.
If
π
:
G GL
n
(
C
) is a finite-dimensional representation, then the matrix
coefficients π(g)
ij
can be written as
π(g)
ij
= δ
i
(π(g)e
j
),
where
e
j
C
n
is the
j
th
basis vector and
δ
i
(
C
n
)
is the
i
th
dual basis vector.
More generally, if
π
:
G GL
(
V
) is a finite-dimensional representation, and
v V , V
, then we can think of
π
v,
(g) = (π(g)v)
as a matrix element of π(g), and this defines a function π
v,
: G C.
In the case of the
G
we care about, our representations are fancy infinite-
dimensional representations, and we need a fancy version of the dual known as
the contragredient.
Definition
(Contragredient)
.
Let (
π, V
) be a smooth representation. We define
V
= Hom
C
(V, C), and the representation (π
, V
) of G is defined by
π
(g) = (v 7→ (π(g
1
)v)).
We then define the contragredient (˜π, ˜v) to be the subrepresentation
˜
V = { V
with open stabilizer}.
This contragredient is quite pleasant. Recall that
V =
[
K
V
K
.
We then have
˜
V =
[
(V
)
K
.
Using the projection π(e
K
) : V V
K
, we can identify
(V
)
K
= (V
K
)
.
So in particular, if
π
is admissible, then so is
˜π
, and we have a canonical
isomorphism
V
˜
˜
V.
Definition
(Matrix coefficient)
.
Let (
π, V
) be a smooth representation, and
v V ,
˜
V . The matrix coefficient π
v,
is defined by
π
v,
(g) = (π(g)v).
This is a locally constant function G C.
We can now make the following definition:
Definition
(Square integrable representation)
.
Let (
π, V
) be an irreducible
smooth representation of G. We say it is square integrable if ω
π
is unitary and
|π
v,
| L
2
(G/Z)
for all (v, ).
Note that the fact that
ω
π
is unitary implies
|π
v,
|
is indeed a function on
L
2
(G/Z). In general, it is unlikely that π
v,
is in L
2
(G).
If
Z
is finite, then
ω
π
is automatically unitary and we don’t have to worry
about quotienting about the center. Moreover,
π
is square integrable iff
π
v,
L
2
(
G
). In this case, if we pick
˜
V
non-zero, then
v 7→ π
v,
gives an embedding
of V into L
2
(G). In general, we have
V L
2
(G, ω
π
) = {f : G C : f(zg) = ω
π
(z)f(z), |f| L
2
(G/Z)},
A slight weakening of square integrability is the following weird definition:
Definition
(Tempered representation)
.
Let (
π, V
) be irreducible,
ω
π
unitary.
We say it is tempered if for all (v, ) and ε > 0, we have
|π
v,
| L
2+ε
(G/Z).
The reason for this definition is that
π
is tempered iff it occurs in
L
2
(
G
), not
necessarily discretely.
Weakening in another direction gives the following definition:
Definition
(Essentially square integrable)
.
Let (
π, V
) be irreducible. Then
(π, V ) is essentially square integrable (or essentially tempered) if
π (χ det)
is square integrable (or tempered) for some character χ : F
×
C.
Note that while these definitions seem very analytic, there are in fact purely
algebraic interpretations of these definitions, using Jacquet modules.
A final category of representations is the following:
Definition
(Supercuspidal representation)
.
We say
π
is supercuspidal if for all
(v, ), the support of π
v,
is compact mod Z.
These are important because they are building blocks of all irreducible
representations of GL
n
(F ), in a sense we will make precise.
The key notion is that of parabolic induction, which takes a list of represen-
tations σ
i
of GL
n
i
(F ) to a representation of GL
N
(F ), where N =
P
n
i
.
We first consider a simpler case, where we have an
n
-tuple
χ
= (
χ
1
, . . . , χ
n
) :
(
F
×
)
n
C
×
of characters. The group
G
=
GL
n
(
F
) containing the Borel
subgroup
B
of upper-triangular matrices. Then
B
is the semi-direct product
T N
, where
T
=
(
F
×
)
n
consists of the diagonal matrices and
N
the unipotent
ones. We can then view χ as a character χ : B B/N = T C
×
.
We then induce this up to a representation of
G
. Here the definition of an
induced representation is not the usual one, but has a twist.
Definition
(Induced representation)
.
Let
χ
:
B C
be a character. We define
the induced representation
Ind
G
B
(
χ
) to be the space of locally constant functions
f : g C such that
f(bg) = χ(b)δ
B
(b)
1/2
f(g)
for all b B and g G, where G acts by
π(g)f : x 7→ f(xg).
The function δ
B
(b)
1/2
is defined by
δ
B
(b) = |det ad
B
(b)|.
More explicitly, if the diagonal entries of b B are x
1
, . . . , x
n
, then
δ
B
(b) =
n
Y
i=1
|x
i
|
n+12i
= |x
1
|
n1
|x
2
|
n3
···|x
n
|
n+1
This is a smooth representation since
B\G
is compact. In fact, it is admissible
and of finite length.
When this is irreducible, it is said to be a principle series representation of
G.
Example. Recall that P
1
(F ) = B\GL
2
(F ). In this terminology,
C
(P
1
(F )) = Ind
G
B
(δ
1/2
B
).
This is not irreducible, since it contains the constant functions, but quotienting
by these does give an irreducible representation. This is called the Steinberg
representation.
In general, we start with a parabolic subgroup
P GL
n
(
F
) =
G
, i.e. one
conjugate to block upper diagonal matrices with a partition
n
=
n
1
+
···
+
n
r
.
This then decomposes into MN, where
M
=
Y
i
GL
n
i
(F ), N =
I
n
1
···
.
.
.
.
.
.
I
n
r
.
This is an example of a Levi decomposition, and M is a Levi subgroup.
To perform parabolic induction, we let (
σ, U
) be a smooth representation of
M
, e.g.
σ
1
··· σ
r
, where each
σ
i
is a representation of
GL
n
i
(
F
). This then
defines a representation of
P
via
P P/N
=
M
, and we define
Ind
G
P
(
σ
) to be
the space of all locally constant functions f : G U such that
f(pg) = δ
P
(p)σ(p)f(g)
for all p P, g G and δ
P
is again defined by
δ
P
(p) = |det ad
P
(p)|.
This is again a smooth representation.
Proposition.
(i) σ is admissible implies π = Ind
G
P
σ is admissible.
(ii) σ is unitary implies π is unitary.
(iii) Ind
G
P
(˜σ) = ˜π.
(ii) and (iii) are the reasons for the factor δ
1/2
P
.
Example. Observe
^
C
(P(F )) = Ind(δ
1/2
B
) = {f : G C : f(bg) = δ
B
(b)f(g)}.
There is a linear form to
C
given by integrating
f
over
GL
2
(
O
) (we can’t use
F since GL
2
(F ) is not compact). In fact, this map is G-invariant, and not just
GL
2
(O)-invariant. This is dual to the constant subspace of C
(P
1
(F )).
The rough statement of the classification theorem is that every irreducible
admissible representation of
G
=
GL
n
(
F
), is a subquotient of an induced
representation
Ind
G
P
σ
for some supercuspidal representation of a Levi subgroup
P = GL
n
1
× ··· × GL
n
r
(F ). This holds for any reductive G/F .
For
GL
n
(
F
), we can be more precise. This is called the Langlands classifica-
tion. We first classify all the essentially square integrable representations:
Theorem.
Let
n
=
mr
with
m, r
1. Let
σ
be any supercuspidal representation
of GL
m
(F ). Let
σ(x) = σ |det
m
|
x
.
Write ∆ = (
σ, σ
(1)
, . . . , σ
(
r
1)), a representation of
GL
m
(
F
)
× ··· × GL
m
(
F
).
Then
Ind
G
P
(∆) has a unique irreducible subquotient
Q
(∆), which is essentially
square integrable.
Moreover,
Q
(∆) is square integrable iff the central character is unitary, iff
σ
(
r1
2
) is square-integrable, and every essentially square integrable
π
is a
Q
(∆)
for a unique ∆.
Example. Take n = 2 = r, σ = | · |
1/2
. Take
P = B =
∗ ∗
0
Then
Ind
G
B
(| · |
1/2
, | · |
1/2
) = C
(B\G) C,
where
C
is the constants. Then
C
(
B\G
) is not supercuspidal, but the quo-
tient is the Steinberg representation, which is square integrable. Thus, every
two-dimensional essentially square integrable representation which is not super-
cuspidal is a twist of the Steinberg representation by χ det.
We can next classify tempered representations.
Theorem.
The tempered irreducible admissible representations of
GL
n
(
F
) are
precisely the representations
Ind
G
P
σ
, where
σ
is irreducible square integrable. In
particular, Ind
G
P
σ are always irreducible when σ is square integrable.
Example.
For
GL
2
, we seek a
π
which is tempered but not square integrable.
This must be of the form
π = Ind
G
B
(χ
1
, χ
2
),
where
|χ
1
|
=
|χ
2
|
= 1. If we want it to be essentially tempered, then we only
need |χ
1
| = |χ
2
|.
Finally, we classify all irreducible (admissible) representations.
Theorem.
Let
n
=
n
1
+
···
+
n
r
be a partition, and
σ
i
tempered representation
of
GL
n
i
(
F
). Let
t
i
R
with
t
1
> ··· > t
r
. Then
Ind
G
P
(
σ
1
(
t
1
)
, . . . , σ
r
(
t
r
)) has a
unique irreducible quotient Langlands quotient, and every
π
is (uniquely) of this
form.
Example.
For
GL
2
, the remaining (i.e. not essentially tempered) representations
are the irreducible subquotients of
Ind
G
B
(χ
1
, χ
2
),
where
|χ
i
| = | · |
t
i
F
, t
1
> t
2
.
Note that the one-dimensional representations must occur in this set, because
we haven’t encountered any yet.
For example, if we take χ
1
= | · |
1/2
and χ
2
= | · |
1/2
, then
Ind
G
B
(χ
1
, χ
2
) =
^
C
(B\G),
which has the trivial representation as its irreducible quotient.
4.4 Local Langlands correspondence
Theorem (Harris–Taylor, Henniart). There is a bijection
irreducible, admissible
representations of GL
n
(F )
semi-simple n-dimensional
representations of L
F
.
Moreover,
For n = 1, this is the same as local class field theory.
Under local class field theory, this corresponds between ω
π
and det σ.
The supercuspidals correspond to the irreducible representations of
W
F
itself.
If a supercuspidal
π
0
corresponds to the representation
σ
0
of
W
F
, then the
essentially square integrable representation
π
=
Q
(
π
0
(
r1
2
)
, . . . , π
0
(
r1
2
))
corresponds to σ = σ
0
Sym
r1
C
2
.
If
π
i
correspond to
σ
i
, where
σ
i
are irreducible and unitary, then the
tempered representation
Ind
G
P
(
π
1
··· π
r
) corresponds to
σ
1
··· σ
r
.
For general representations, if π is the Langlands quotient of
Ind(π
1
(t
1
), . . . , π
r
(t
r
))
with each
π
i
tempered, and
π
i
corresponds to unitary representations
σ
i
of L
F
, then π corresponds to
L
σ
i
|Art
1
F
|
t
i
F
.
The hard part of the theorem is the correspondence between the supercuspidal
representations and irreducible representations of
W
F
. This correspondence is
characterized by ε-factors of pairs.
Recall that for an irreducible representation of
W
F
, we had an
ε
factor
ε
(
σ, µ
F
, ψ
). If we have two representations, then we can just take the tensor
product
ε
(
σ
1
σ
2
, µ
F
, ψ
). It turns out for supercuspidals, we can also introduce
ε
-factors
ε
(
π, µ
F
, ψ
). There are also
ε
factors for pairs,
ε
(
π
1
, π
2
, µ
F
, ψ
). Then
the correspondence is such that if π
i
correspond to σ
i
, then
ε(σ
1
σ
2
, µ
F
, ψ) = ε(π
1
, π
2
, µ
F
, ψ).
When
n
= 1, we get local class field theory. Recall that we actually have a
homomorphic correspondence between characters of
F
×
and characters of
W
F
,
and the correspondence is uniquely determined by
(i)
The behaviour on unramified characters, which is saying that the Artin
map sends uniformizers to geometric Frobenii
(ii)
The base change property: the restriction map
W
ab
F
0
W
ab
F
correspond to
the norm map of fields
If we want to extend this to the local Langlands correspondence, the correspond-
ing picture will include
(i)
Multiplication: taking multiplications of
GL
n
and
GL
m
to representations
of GL
mn
(ii)
Base change: sending representations of
GL
n
(
F
) to representations of
GL
n
(F
0
) for a finite extension F
0
/F
These thing exist by virtue of local Langlands correspondence (much earlier,
base change for cyclic extensions was constructed by Arthur–Clozel).
Proposition.
Let
σ
:
W
F
GL
n
(
C
) be an irreducible representation. Then
the following are equivalent:
(i) For some g W
F
\ I
F
, σ(g) has an eigenvalue of absolute value 1.
(ii) im σ is relatively compact, i.e. has compact closure, i.e. is bounded.
(iii) σ is unitary.
Proof. The only non-trivial part is (i) (ii). We know
im σ = hσ(Φ), σ(I
F
) = Hi,
where Φ is some lift of the Frobenius and
H
is a finite group. Moreover,
I
F
is
normal in
W
F
. So for some
n
1,
σ
n
) commutes with
H
. Thus, replacing
g
and Φ
n
with suitable non-zero powers, we can assume
σ
(
g
) =
σ
n
)
h
for some
h H
. Since
H
is finite, and
σ
n
) commutes with
h
, we may in fact assume
σ(g) = σ(Φ)
n
. So we know σ(Φ) has eigenvalue with absolute value 1.
Let
V
1
V
=
C
n
be a sum of eigenspaces for
σ
(Φ)
n
with all eigenvalues
having absolute value 1. Since
σ
n
) is central, we know
V
1
is invariant, and
hence
V
is irreducible. So
V
1
=
V
. So all eigenvalues of
σ
(Φ) have eigenvalue 1.
Since
V
is irreducible, we know it is
F
-semisimple. So
σ
(Φ) is semisimple. So
hσ(Φ)i is bounded. So im σ is bounded.
5 Modular forms and representation theory
Recall that a modular form is a holomorphic function f : H C such that
f(z) = j(γ, z)
k
f(γ(z))
for all γ in some congruence subgroup of SL
2
(Z), where
γ =
a b
c d
, γ(z) =
az + b
cz + d
, j(γ, z) = cz + d.
Let M
k
be the set of all such f.
Consider the group
GL
2
(Q)
+
= {g GL
2
(Q) : det g > 0}.
This acts on M
k
on the left by
g : f 7→ j
1
(g
1
, z)
k
f(g
1
(z)), j
1
(g, z) = |det g|
1/2
j(g, z).
The factor of
|det g|
1/2
makes the diagonal
diag
(
Q
×
>0
) act trivially. Note that
later we will consider some
g
with negative determinant, so we put the absolute
value sign in.
For any
f M
k
, the stabilizer contains some Γ(
N
), which we can think of
as some continuity condition. To make this precise, we can form the completion
of
GL
2
(
Q
)
+
with respect to
{
Γ(
N
) :
N
1
}
, and the action extends to a
representation π
0
of this completion. In fact, this completion is
G
0
= {g GL
2
(A
Q
) : det(g) Q
×
>0
}.
This is a closed subgroup of G = GL
2
(A
Q
), and in fact
G = G
0
·
ˆ
Z 0
0 1
.
In fact G is a semidirect product of the groups.
The group
G
0
seems quite nasty, since the determinant condition is rather
unnatural. It would be nice to get a representation of
G
itself, and the easy way
to do so is by induction. What is this? By definition, it is
Ind
G
G
0
(M
k
) = {ϕ : G M
k
: h G
0
, ϕ(hg) = π
0
(h)ϕ(g)}.
Equivalently, this consists of functions
F
:
H × G C
such that for all
γ
GL
2
(Q)
+
, we have
j
1
(γ, z)
k
F (γ(z), γg) = F (z, g),
and for every g G, there is some open compact K G such that
F (z, g) = F (z, gh) for all h K,
and that F is holomorphic in z (and at the cusps).
To get rid of the plus, we can just replace
GL
2
(
Q
)
+
, H
with
GL
2
(
Q
)
, C \R
=
H
±
. These objects are called adelic modular forms.
If
F
is an adelic modular form, and we fix
g
, then the function
f
(
z
) =
F
(
z, g
) is a usual modular form. Conversely, if
F
invariant under
ker
(
G
2
(
ˆ
Z
)
GL
2
(
Z/N Z
)), then
F
corresponds to a tuple of Γ(
N
)-modular forms indexed
by (
Z/N Z
)
×
. This has an action of
G
=
Q
0
p
GL
2
(
Q
p
) (which is the restricted
product with respect to GL
2
(Z
p
)).
The adelic modular forms contain the cusp forms, and any
f M
k
generates
a subrepresentation of the space of adelic forms.
Theorem.
(i)
The space
V
f
of adelic cusp forms generated by
f S
k
1
(
N
)) is irreducible
iff f is a T
p
eigenvector for all p - n.
(ii)
This gives a bijection between irreducible
G
-invariant spaces of adelic cusp
forms and Atkin–Lehner newforms.
Note that it is important to stick to cusp forms, where there is an inner
product, because if we look at the space of adelic modular forms, it is not
completely decomposable.
Now suppose (
π, V
) is an irreducible admissible representation of
GL
2
(
A
Q
) =
Q
0
G
p
=
GL
2
(
Q
p
), and take a maximal compact subgroups
K
0
p
=
GL
2
(
Z
p
)
GL
2
(
Q
p
). Then general facts about irreducible representations of products imply
irreducibility (and admissibility) is equivalent to the existence of irreducible
admissible representations (π
p
, V
p
) of G
p
for all p such that
(i)
For almost all
p
,
dim V
K
0
p
p
1 (for
G
p
=
GL
n
(
Q
p
), this implies the
dimension is 1). Fix some non-zero f
0
p
V
K
0
p
p
.
(ii) We have
π =
0
p
π
p
,
the restricted tensor product, which is generated by
N
p
v
p
with
v
p
=
f
0
p
for almost all p. To be precise,
0
p
π
p
= lim
finite S
O
pS
π
p
.
The use of
v
p
is to identify smaller tensor products with larger tensor
products.
Note that (i) is equivalent to the assertion that (
π
p
, V
p
) is an irreducible
principal series representation
Ind
G
p
B
p
(
χ
1
, χ
2
) where
χ
i
are unramified characters.
These unramified characters are determined by χ
p
(p) = α
p,i
.
If
f
=
P
a
n
q
n
S
k
1
(
N
)) is a normalized eigenform with character
ω
:
(
Z/N Z
)
×
C
×
, and if
f
corresponds to
π
=
N
0
π
p
, then for every
p - N
, we
have π
p
= Ind
G
p
B
p
(χ
1
, χ
2
) is an unramified principal series, and
a
p
= p
(k1)/2
(α
p,1
+ α
p,2
)
ω(p)
1
= α
p,1
α
p,2
.
We can now translate difficult theorems about modular forms into representation
theory.
Example. The Ramanujan conjecture (proved by Deligne) says
|a
p
| 2p
(k1)/2
.
If we look at the formulae above, since
ω
(
p
)
1
is a root of unity, this is equivalent
to the statement that |α
p,i
| = 1. This is true iff π
p
is tempered.
We said above that if
dim V
K
0
p
p
1, then in fact it is equal to 1. There is a
generalization of that.
Theorem
(Local Atkin–Lehner theorem)
.
If (
π, V
) is an irreducible representa-
tion of
GL
2
(
F
), where
F/Q
p
and
dim V
=
, then there exists a unique
n
π
>
0
such that
V
K
n
=
(
0 n < n
π
one-dimensional n = n
π
, K
n
=
g
∗ ∗
0 1
mod
n
.
Taking the product of these invariant vectors for
n
=
n
π
over all
p
gives Atkin–
Lehner newform.
What about the other primes? i.e. the primes at infinity?
We have a map
f
:
H
±
×GL
2
(
A
Q
)
C
. Writing
H
=
SL
2
(
R
)
/SO
(2), which
has an action of Γ, we can convert this to an action of
SO
(2) on Γ
\SL
2
(
R
).
Consider the function
Φ
f
: GL
2
(R) × GL
2
(A
Q
) = GL
2
(A
Q
) C
given by
Φ
f
(h
, h
) = j
1
(h
, i)
k
f(h
(i), h
).
Then this is invariant under GL
2
(Q)
+
, i.e.
Φ
f
(γh
, γh
) = Φ(h
, h
).
Now if we take
k
θ
=
cos θ sin θ
sin θ cos θ
SO(2),
then
Φ
f
(h
k
θ
, h
) = e
ikθ
Φ
f
(h
, h
).
So we get invariance under
γ
, but we need to change what
SO
(2) does. In other
words, Φ
f
is now a function
Φ
f
: GL
2
(Q)\G
2
(A
Q
) C
satisfying various properties:
It generates a finite-dimensional representation of SO(2)
It is invariant under an open subset of GL
2
(A
Q
)
It satisfies growth condition and cuspidality, etc.
By the Cauchy–Riemann equations, the holomorphicity condition of
f
says Φ
f
satisfies some differential equation. In particular, that says Φ
f
is an eigenfunction
for the Casimir in the universal enveloping algebra of
sl
2
. These conditions
together define the space of automorphic forms.
Example. Take the non-holomorphic Eisenstein series
E(z, s) =
X
(c,d)6=(0,0)
1
|cz + d|
2s
.
This is a real analytic function on Γ
\H C
. Using the above process, we get
an automorphic form on
GL
2
(
Q
)
\GL
2
(
A
Q
) with
k
= 0. So it actually invariant
under SO(2). It satisfies
E = s(1 s)E.
There exist automorphic cusp forms which are invariant under
SO
(2), known
as Maass forms. They also satisfy a differential equation with a Laplacian
eigenvalue
λ
. A famous conjecture of Selberg says
λ
1
4
. What this is equivalent
to is that the representation of GL
2
(R) they generate is tempered.
In terms of representation theory, this looks quite similar to Ramanujan’s
conjecture!