IV Topics in Number Theory - Class field theory

1Class field theory

IV Topics in Number Theory

1.3 Global class field theory

We now proceed to discuss global class field theory.

Definition

(Global field)

A global field is a number field or

(

) for a smooth

projective absolutely irreducible curve C/F

, i.e. a finite extension of F

(t).

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

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

discussions will mostly focus on those.

Definition

(Place)

Let

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

is a number field, we say a valuation

is a finite place if it is the valuation

at a prime

p C O

. A valuation

is an infinite place if comes from a complex

or real embedding of

. We write Σ

for the set of places of

, and Σ

∞

and

K,∞

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

v - ∞

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

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

closed points of the curve.

v ∈

is a place, then there is a completion

K → K

. If

is infinite,

then

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

is a non-

Archimedean local field.

Example.

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

∞

given by the embedding

Q → R

. If

, then we get the embedding

Q → Q

into the p-adic completion.

Notation.

is a finite place, we write

⊆ K

for the valuation ring of the

completion.

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

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

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

(and

hence K) is by

|x|

= q

−v(x)

where



is the cardinality of the residue field at

. For example, if

and

then q

= p, and |p|

In the Archimedean case, if

is real, then we set

|x|

to be the usual absolute

value; if K

is complex, then we take |x|

= x¯x = |x|

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

product formula:

Proposition (Product formula). If x ∈ K

, then

v∈Σ

|x|

= 1.

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

, and then show

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

all finite extensions of Q.

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

make the following definition:

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

)

v∈K

: x

∈ O

for all but finitely many v ∈ Σ

∞

We can write this as K

∞

K or A

K,∞

× A

∞

, where

∞

= A

K,∞

= K ⊗

R = R

× C

consists of the product over the infinite places, and

K = A

∞

v-∞

[

S⊆Σ

∞

finite

v∈S

v∈Σ

∞

This contains

v-∞

. In the case of a number field,

is the profinite

completion of O

. More precisely, if K is a number field then

= lim

/a = lim O

/NO

= O

⊗

where the last equality follows from the fact that O

is a finite Z-module.

Definition (Idele). The ideles is the restricted product

= A

)

∈

: x

∈ O

for almost all v

These objects come with natural topologies. On

, we take

∞

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

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

, we

take

∞

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

subspace topology under the inclusion

→ A

. Instead, it is induced by the

inclusion

→ A

× A

x 7→ (x, x

−1

It is a basic fact that K

⊆ J

is a discrete subgroup.

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

= J

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

Note that J

comes with a natural absolute value

| · |

: J

→ R

) 7→

v∈Σ

The product formula implies that

{

}

. So this in fact a map

→ R

Moreover, we have

Theorem. The map | · |

: C

→ R

has compact kernel.

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

we will later construct a continuous surjection

→ Cl

(

) to the ideal class

group of

. In particular, this implies

(

) is finite! In fact, the theorem is

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

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

= R

Suppose we have an idele

= (

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

rational number

y ∈ Q

such that

sgn

(

) =

sgn

(

∞

) and

(

) =

(

). So we

have

= Q

Here we think of

as being embedded diagonally into

, and as we have

previously mentioned,

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

description of the idele class group

= R

and

is the kernel | · |

From the decomposition above, we see that

has a maximal connected

subgroup

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

1. For a general

, we write

for the maximal connected subgroup, and then

) = C

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

) =

= lim

(Z/nZ)

= lim

Gal(Q(ζ

)/Q) = Gal(Q(ζ

∞

)/Q).

The field

(

∞

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

this is in fact the maximal abelian extension of

. Global class field theory is a

generalization of this isomorphism to all fields.

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

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

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

Artin map.

Let

L/K

be a finite Galois extension,

a place of

, and

w | v

a place of

extending

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

infinite places, this means the embedding

is an extension of

. We then have

the decomposition group

Gal(L

) ⊆ Gal(L/K).

L/K

is abelian, since any two places lying above

are conjugate, this depends

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

Art

L/K

: J

−→ Gal(L/K)

)

7−→

Art

where we pick one

for each

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

∈ O

and

L/K

is unramified at

v - ∞

, then

Art

(

) = 1. So the product is in

fact finite.

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

all extensions L/K, and get a continuous map

Art

: J

→ Γ

Theorem

(Artin reciprocity law)

. Art

(

) =

{

}

, so induces a map

→

. Moreover,

(i)

char

(

) =

p >

0, then

Art

is injective, and induces an isomorphism

Art

∼

→ W

, where

is defined as follows: since

is a finite

extension of

(

), and wlog assume

∩ K

≡ k

. Then

defined as the pullback

= Gal(

K/K)

∼

Gal(

k/k)

restr.

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

Art

: π

) =

∼

→ Γ

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

L/K

Art

restr.

Art

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

Art

L/K

)

∼

→ Gal(L/K)

Finally, this is functorial, namely if

∼

→ K

is an isomorphism, then we

have a commutative square

Art

Observe that naturality and functoriality are immediate consequences of the

corresponding results for local class field theory.

As a consequence of the isomorphism, we have a correspondence



finite abelian extensions

L/K



←→



finite index open subgroups

of J

containing K



L 7−→ ker(Art

L/K

: J

→ Gal(L/K))

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

Recall that in local class field theory, if we decompose

hπi × O

, then

the local Artin map sends

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

extension

is unramified iff the local Artin map kills of

. Globally,

this tells us

Proposition.

L/K

is an abelian extension of global fields, which corresponds

to the open subgroup

U ⊆ J

under the Artin map, then

L/K

is unramified at

a finite v - ∞ iff O

⊆ U .

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

Since

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

Definition

(Ramification)

v | ∞

is a real place of

, and

L/K

is a finite

abelian extension, then we say

is ramified if for some (hence all) places

above v, w is complex.

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

say v is inert instead.

With this definition,

L/K

is unramified at a real place

iff

⊆ U

Note that since U is open, it automatically contains R

We can similarly read off splitting information.

Proposition.

is finite and unramified, then

splits completely iff

⊆ U

Proof. v

splits completely iff

for all

w | v

, iff

Art

(

) =

{

}

Example.

We will use global class field theory to compute all

extensions

L/Q which are unramified outside 5 and 7.

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

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

cubic diophantine problem to solve.

While

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

extension as a chain

2 = hσi

Since

L/Q

is unramified outside 5 and 7, we know that

must be one of

(

√

(

√

−7

) and

(

√

−35

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

are the possibilities for

. We shall only do the case

(

√

−7

) here. If we

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

of K do not work.

So fix

(

√

−7

). We want

L/K

to be cyclic of degree 3, and

must act

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

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

U ≤ J

index 3 such that

⊆ U

for all

v -

35. We also need

(

) =

, or else the

composite extension would not even be Galois, and

has to acts as

−

1 on

∼

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

We know

(

√

−7

) has has class number 1, and the units are

1. So we

know

{±1}

By assumption, we know

contains

v-35

. So we have to look at the places

that divide 35. In O

√

−7)

, the prime 5 is inert and 7 is ramified.

Since 5 is inert, we know

is an unramified quadratic extension. So

we can write

(5)

= F

× (1 + 5O

(5)

)

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

for the

quotient to have order 3. On

acts as the Frobenius

(

) =

. Since

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

6. This gives an index 3 subgroup

⊆ O

(5)

. Moreover, on here,

acts by

x 7→ x

= x

−1

. Thus, we can take

U =

v6=(5)

× U

and this gives an

extension of

that is unramified outside 5 and 7. It is an

exercise to explicitly identify this extension.

We turn to the prime 7 = −

√

−7

. Since this is ramified, we have

(

√

−7)

= F



1 + (

√

−7)O

√

−7



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

acts trivially on

. So

U must contain O

(

√

−7)

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

We previously explicitly described

. It would be nice to have

a similar description of

for an arbitrary

. The connected component will

come from the infinite places

∞

v|∞

. The connected component is given

×,0

∞

= (R

)

× (C

)

where there are r

real places and r

complex ones. Thus, we find that

Proposition.

{±1}

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

content homomorphism.

Definition

(Content homomorphism)

The content homomorphism is the map

c : J

→ fractional ideals of K

)

7→

v-∞

v(x

)

where

is the prime ideal corresponding to

. We ignore the infinite places

completely.

Observe that

(

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

the kernel of the content map is

∞

, by definition. So we have a short

exact sequence

1 →

{±1}

→ C

→ Cl(K) → 1.

(

√

−D

), then

is finite, and in particular is closed. But

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

Returning to the case

, our favorite abelian extensions are those of

the form L = Q(ζ

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

∼

→ Gal(L/Q)

∼

(Z/NZ)

By local class field theory for

, we see that with our normalizations, this is

just the quotient map, whose kernel is

(1 + N

p-N

p|N

(1 + NZ

)

⊆

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

the quotient map.

These subgroups of

are rather special ones. First of all (1 +

)

form

a neighbourhood of the identity in

. Thus, any open subgroup contains a

subgroup of this form. Equivalently, every abelian extension of

is contained

in Q(ζ

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

For a general number field

, we want to write down an explicit basis for

open subgroups of 1 in π

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

m =

v∈Σ

· (v)

of places of K, where m

≥ 0 are integers.

Given a modulus m, we define the subgroup

v|∞,m

×,0

v|∞,m

v-∞,m

(1+p

)

v-∞,m

⊆ J

Then essentially by definition of the topology of

, any open subgroup of

containing K

×,0

∞

contains some U

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

Definition

(

∞

))

a C O

is an ideal, we write

(

∞

) for the modulus with

= v(a) for all v - ∞, and m

= 1 for all v | ∞.

and

= (

)(

∞

), then we simply get

(1 +

)

, and

= (Z/nZ)

corresponding to the abelian extension Q(ζ

In general, we define

Definition

(Ray class field)

L/K

is abelian with

Gal

(

L/K

)

∼

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

Definition

(Conductor)

corresponds to

U ⊆ J

, then

U ⊇ K

for

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