equivalence of form class group and class group

equivalence of form class group and class group

There are only a finite number of

reduced

primitive

positive

integral binary quadratic forms

of a given negative

http://planetmath.org/node/IntegralBinaryQuadraticForms

discriminant

Δ

. Given

Δ

, call this number

h

Δ

, the

form of

Δ

.

Thus, for example, since there is only one

reduced form

of discriminant

-

163

, we have that

h

-

163

=

1

.

It turns out that the set of reduced forms of a given negative discriminant can be turned into an

abelian group

, called the

,

𝒞

Δ

, by defining a “

multiplication

” on forms that is based on

generalizations

of

identities

such as

(

2

⁢

x

2

+

2

⁢

x

⁢

y

+

3

⁢

y

2

)

⁢

(

2

⁢

z

2

+

2

⁢

z

⁢

w

+

3

⁢

w

2

)

=

(

2

⁢

x

⁢

z

+

x

⁢

w

+

y

⁢

z

+

3

⁢

y

⁢

w

)

2

+

5

⁢

(

x

⁢

w

+

y

⁢

z

)

2

where all of these forms have discriminant

-

20

.

Now, given an

algebraic extension

K

of

ℚ

,

ideal classes

of

𝒪

K

also form an abelian group, called the

http://planetmath.org/node/IdealClass

ideal class group

of

K

,

𝒞

K

. The order of

𝒞

K

is called the

class number

of

K

and is denoted

h

K

. See the ideal class entry for more detail.

For an algebraic extension

K

/

ℚ

, one also defines the

http://planetmath.org/node/DiscriminantOfANumberField

discriminant of the

extension

,

d

K

. For

quadratic extensions

K

=

ℚ

⁢

[

n

]

, where

n

is assumed

squarefree

, the discriminant can be explicitly computed to be

d

K

=

{

4

⁢

n

if

⁢

n

≡

2

,

3

(

mod

4

)

n

if

⁢

n

≡

1

(

mod

4

)

For

imaginary quadratic extensions

, the form

class group

and the class group turn out to be the same!

Theorem 1

.

Let

K

=

Q

⁢

(

n

)

,

n

<

0

squarefree, be a quadratic extension. Then

C

K

, the class group of

K

, is

isomorphic

to the group of reduced forms of discriminant

d

K

,

C

d

K

.

One can in fact exhibit an explicit correspondence

𝒞

d

K

→

𝒞

K

:

a

⁢

x

2

+

b

⁢

x

⁢

y

+

c

⁢

y

2

↦

(

a

,

b

+

d

K

2

)

Note in particular that the simplest, or principal, form of discriminant

d

K

(

x

2

-

d

K

⁢

y

2

or

x

2

+

x

⁢

y

+

1

-

d

K

4

⁢

y

2

) maps to the ideal

(

1

)

=

𝒪

K

; these forms are the identities in

𝒞

d

K

. Showing that the map is 1-1 and onto is not difficult; showing that it is a group isomorphism is more difficult but nevertheless essentially amounts to a computation.

This

theorem

allows us to simply compute at least the size of the class group for quadratic extensions

K

by computing the number of reduced forms of discriminant

d

K

. For example, suppose

K

=

ℚ

⁢

(

-

23

)

. Since

-

23

≡

1

(

4

)

,

𝒪

K

=

ℤ

⁢

[

1

+

-

23

2

]

and

d

K

=

-

23

.

What are the forms of discriminant

-

23

?

|

b

|

≤

a

≤

23

3

<

8

<

3

, and

b

is odd, so

b

=

±

1

.

4

⁢

a

⁢

c

-

b

2

=

23

, so

a

⁢

c

=

6

. We thus get three reduced forms:

(

1

,

1

,

6

)

(

2

,

1

,

3

)

(

2

,

-

1

,

3

)

reduced since

|

b

|

≠

a

,

a

≠

c

Note that

(

1

,

-

1

,

6

)

is not reduced, since

b

<

0

but

|

b

|

=

a

.

So we know that the order of the class group

𝒞

K

is

3

, so

𝒞

K

≅

ℤ

/

3

⁢

ℤ

.

We can use the explicit correspondence above to find representatives of the three elements of the class group using the map from forms to ideals.

(

1

,

1

,

6

)

→

(

1

,

1

+

-

23

2

)

=

(

1

)

(

2

,

1

,

3

)

→

(

2

,

1

+

-

23

2

)

(

2

,

-

1

,

3

)

→

(

2

,

-

1

+

-

23

2

)

In fact, a more general form of Theorem

1

is true. If

K

is an

algebraic number field

,

A

⊂

𝒪

K

, then

A

is not a

Dedekind domain

unless

A

=

𝒪

K

. But even in this case, if one considers only those ideals that are

invertible

in

A

, one can define a group

structure

in a

similar

way; this is once again called the class group of

A

. In the case that

K

is a quadratic extension, these subrings of

𝒪

K

are called

orders

of

K

.

It is the case that each discriminant

Δ

<

0

,

Δ

≡

0

,

1

(

mod

4

)

corresponds to a unique order in a quadratic extension of

ℚ

. Specifically,

Theorem 2

.

Let

Δ

<

0

,

Δ

≡

0

,

1

(

mod

4

)

. Write

Δ

=

m

2

⁢

Δ

′

where

Δ

′

is squarefree. Let

K

=

Q

⁢

(

Δ

′

)

. Then

𝒪

Δ

=

{

ℤ

⁢

[

m

2

⁢

Δ

′

]

,

Δ

′

≡

2

,

3

(

mod

4

)

ℤ

⁢

[

m

⁢

1

+

Δ

′

2

]

,

Δ

′

≡

1

(

mod

4

)

is a subring of

O

K

, and

C

Δ

≅

C

O

Δ

.
(Note that if

Δ

′

≡

2

,

3

(

mod

4

)

, then

m

must be even. For otherwise,

m

2

≡

1

(

mod

4

)

and thus

Δ

≡

2

,

3

(

mod

4

)

, which is impossible. Thus

m

/

2

is an

integer

in this case)

This reduces to the first theorem in the event that

Δ

=

d

K

.

Thus there is a

1

-

1

correspondence between discriminants

Δ

<

0

and orders of

quadratic fields

; in particular, the

ring of algebraic integers

of any quadratic field corresponds to the forms of discriminant equal to the discriminant of the field.

Title

equivalence of form class group and class group

Canonical name

EquivalenceOfFormClassGroupAndClassGroup

Date of creation

2013-03-22 16:56:27

Last modified on

2013-03-22 16:56:27

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11E12

Classification

msc 11E16

Classification

msc 11R29

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