integral binary quadratic forms

integral binary quadratic forms

An

integral binary quadratic form

is a

quadratic form

(q.v.) in two variables over

ℤ

, i.e. a

polynomial

F

⁢

(

x

,

y

)

=

a

⁢

x

2

+

b

⁢

x

⁢

y

+

c

⁢

y

2

,

a

,

b

,

c

∈

ℤ

F

is said to be

primitive

if its coefficients are relatively prime, i.e.

gcd

⁡

(

a

,

b

,

c

)

=

1

, and is said to

represent

an

integer

n

if there are

r

,

s

∈

ℤ

such that

F

⁢

(

r

,

s

)

=

n

. If

gcd

⁡

(

r

,

s

)

=

1

,

F

is said to represent

n

properly

. The theory of integral binary quadratic forms was developed by Gauss, Lagrange, and Legendre.

In what follows, “form” means “integral binary quadratic form”.

Following the article on quadratic forms, two such forms

F

⁢

(

x

,

y

)

and

G

⁢

(

x

,

y

)

are

equivalent

if there is a matrix

M

∈

G

⁢

L

⁢

(

2

,

ℤ

)

such that

G

⁢

(

x

,

y

)

=

F

⁢

(

M

⁢

(

x

,

y

)

T

)

Matrices in

G

⁢

L

⁢

(

2

,

ℤ

)

are matrices with determinant

±

1

. So if

α

,

β

,

γ

,

δ

∈

ℤ

and

d

⁢

e

⁢

t

⁢

(

α

γ

β

δ

)

=

±

1

then if

G

⁢

(

x

,

y

)

=

F

⁢

(

α

⁢

x

+

β

⁢

y

,

γ

⁢

x

+

δ

⁢

y

)

it follows that

G

is equivalent to

F

. If

M

∈

S

⁢

L

⁢

(

2

,

ℤ

)

(i.e.

det

⁡

M

=

1

), we say that

F

and

G

are

properly equivalent

, written

F

∼

G

; otherwise, they are

improperly equivalent

.

Note that while both equivalence and proper equivalence are

equivalence relations

, improper equivalence is not. For if

F

is improperly equivalent to

G

and

G

is improperly equivalent to

H

, then the

product

of the transformation matrices has determinant

1

, so that

F

is properly equivalent to

H

. Since proper equivalence is an equivalence relation, we will say that two forms are in the same

class

if they are properly equivalent.

G

⁢

L

⁢

(

2

,

ℤ

)

is generated as a

multiplicative group

by the two matrices

(

1

0

1

1

)

,

(

0

1

1

0

)

so in particular we see that we can construct all equivalence

transformations

by composing the following three transformations:

Transformation

Matrix

Determinant

(

x

,

y

)

↦

(

y

,

x

)

(

0

1

1

0

)

-

1

(

x

,

y

)

↦

(

y

,

-

x

)

(

0

-

1

1

0

)

1

(

x

,

y

)

↦

(

x

+

d

⁢

y

,

y

)

(

1

0

d

1

)

1

Example

: Let

F

⁢

(

x

,

y

)

=

x

2

+

x

⁢

y

+

6

⁢

y

2

,

G

⁢

(

x

,

y

)

=

82

⁢

x

2

+

51

⁢

x

⁢

y

+

8

⁢

y

2

. Then

G

⁢

(

x

,

y

)

=

F

⁢

(

α

⁢

x

+

β

⁢

y

,

γ

⁢

x

+

δ

⁢

y

)

=

F

⁢

(

4

⁢

x

+

y

,

3

⁢

x

+

y

)

so

(

α

γ

β

δ

)

=

(

4

3

1

1

)

The transformations to map

F

into

G

are

x

2

+

x

⁢

y

+

6

⁢

y

2

(

x

,

y

)

↦

(

x

+

y

,

y

)

x

2

+

3

⁢

x

⁢

y

+

8

⁢

y

2

(

x

,

y

)

↦

(

y

,

x

)

8

⁢

x

2

+

3

⁢

x

⁢

y

+

y

2

(

x

,

y

)

↦

(

x

+

3

⁢

y

,

y

)

8

⁢

x

2

+

51

⁢

x

⁢

y

+

82

⁢

y

2

(

x

,

y

)

↦

(

y

,

x

)

82

⁢

x

2

+

51

⁢

x

⁢

y

+

8

⁢

y

2

and

(

4

3

1

1

)

=

(

0

1

1

0

)

⁢

(

1

0

3

1

)

⁢

(

0

1

1

0

)

⁢

(

1

0

1

1

)

Title

integral binary quadratic forms

Canonical name

IntegralBinaryQuadraticForms

Date of creation

2013-03-22 16:55:44

Last modified on

2013-03-22 16:55:44

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

10

Author

rm50 (10146)

Entry type

Topic

Classification

msc 11E16

Classification

msc 11E12

Related topic

RepresentationOfIntegersByEquivalentIntegralBinaryQuadraticForms

Related topic

ReducedIntegralBinaryQuadraticForms

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