aarjaneiro/planetmath huggingface dataset:
example of Fermat’s last theorem

example of Fermat’s last theorem

Fermat stated that for any

n

>

2

the

Diophantine equation

x

n

+

y

n

=

z

n

has no solution in

positive integers

. For

n

=

4

this follows from the following

Theorem 1

.

x

4

+

y

4

=

z

2

has no solution in positive integers.

Proof.

Suppose we had a

positive

z

such that

x

4

+

y

4

=

z

2

holds. We may assume

gcd

⁡

(

x

,

y

,

z

)

=

1

. Then

z

must be odd, and

x

,

y

have opposite parity. Since

(

x

2

)

2

+

(

y

2

)

2

=

z

2

is a

primitive Pythagorean triple

, we have

x

2

=

2

⁢

p

⁢

q

,

y

2

=

q

2

-

p

2

,

z

=

p

2

+

q

2

(1)

where

p

,

q

∈

ℕ

,

p

<

q

are

coprime

and have opposite parity. Since

y

2

+

p

2

=

q

2

is a primitive Pythagorean triple, we have coprime

s

,

r

∈

ℕ

,

s

<

r

of opposite parity satisfying

q

=

r

2

+

s

2

,

y

=

r

2

-

s

2

,

p

=

2

⁢

r

⁢

s

.

(2)

From

gcd

⁡

(

r

2

,

s

2

)

=

1

it follows that

gcd

⁡

(

r

2

,

r

2

+

s

2

)

=

1

=

gcd

⁡

(

s

2

,

r

2

+

s

2

)

, which implies

gcd

⁡

(

r

⁢

s

,

r

2

+

s

2

)

=

1

. Since

(

x

2

)

2

=

p

⁢

q

2

=

r

⁢

s

⁢

(

r

2

+

s

2

)

is a square, each of

r

,

s

,

r

2

+

s

2

is a square.

Setting

Z

2

=

q

,

X

2

=

r

,

Y

2

=

s

q

=

r

2

+

s

2

leads to

Z

2

=

X

4

+

Y

4

(3)

where

Z

2

=

q

<

q

2

+

p

2

=

z

<

z

2

. Thus, equation

3

gives a solution where

Z

<

z

. Applying the above steps repeatedly would produce an

infinite

sequence

z

>

Z

>

z

2

>

…

of positive integers, each of which was the sum of two

fourth powers

. But there cannot be infinitely many positive integers smaller than a given one; in particular this contradicts to the fact that there must exist a smallest

z

for which (

1

) is solvable. So there are no solutions in positive integers for this equation.
∎

A

consequence

of the above

theorem

is that the area of a

right triangle

with

integer

sides is not a square; equivalently, a right triangle with

rational

sides has an area which is not the square of a rational.

Title

example of Fermat’s last theorem

Canonical name

ExampleOfFermatsLastTheorem

Date of creation

2013-03-22 14:09:51

Last modified on

2013-03-22 14:09:51

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

9

Author

Thomas Heye (1234)

Entry type

Example

Classification

msc 11F80

Classification

msc 14H52

Classification

msc 11D41

Related topic

InfiniteDescent

Related topic

X4Y4z2HasNoSolutionsInPositiveIntegers

Generated on Fri Feb 9 15:34:21 2018 by

LaTeXML
Kummer’s lemma

Kummer’s lemma

The following result is a key ingredient in the proof of Fermat’s last

theorem

for

regular primes

. More concretely, the lemma is needed to show the so-called second case of Fermat, i.e.

x

p

+

y

p

=

z

p

does not have any non-trivial solutions in

ℤ

with

p

>

2

a regular prime and

p

|

x

y

z

. It is due to Ernst Kummer, thus the name.

Theorem

(Kummer’s Lemma).

Let

p

>

2

be a prime, let

ζ

p

be a primitive

p

th

root of unity

and let

K

=

Q

⁢

(

ζ

p

)

be the corresponding

cyclotomic field

. Let

E

be the group of algebraic units of the

ring of integers

O

K

. Suppose that

p

is a regular prime. If a unit

ϵ

∈

E

is congruent modulo

p

to a

rational integer

, then

ϵ

is the

p

th power of another unit also

E

.

For a proof, see

[

Washington

]

, Theorem 5.36. The reader may also be interested in

generalizations

due to

[

Washington 1992

]

and

[

Ozaki 1997

]

.

References

Ozaki 1997

Ozaki, M.,

Kummer’s lemma for

Z

p

-

extensions

over totally real number fields

, Acta Arith. 81 (1997), no. 1, 37–44.

Washington

Washington L. C.,

Introduction

to Cyclotomic
Fields

, Second Edition, Springer-Verlag, New York.

Washington 1992

Washington, L. C.,

Kummer’s lemma for prime power cyclotomic fields

, J.

Number Theory

40 (1992), no. 2, 165–173.

Title

Kummer’s lemma

Canonical

name

KummersLemma

Date of creation

2013-03-22 15:55:21

Last modified on

2013-03-22 15:55:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11F80

Classification

msc 14H52

Classification

msc 11D41

Generated on Fri Feb 9 15:34:24 2018 by

LaTeXML
complex multiplication

complex multiplication

Let

E

be an

elliptic curve

. The

endomorphism ring

of

E

,
denoted

End

⁡

(

E

)

, is the set of all

regular maps

ϕ

:

E

→

E

such that

ϕ

⁢

(

O

)

=

O

, where

O

∈

E

is the

identity element

for the group structure of

E

. Note that this is
indeed a ring under

addition

(

(

ϕ

+

ψ

)

⁢

(

P

)

=

ϕ

⁢

(

P

)

+

ψ

⁢

(

P

)

) and composition of maps.

The following theorem implies that every

endomorphism

is also a

group endomorphism

:

Theorem 1

Let

E

1

,

E

2

be elliptic curves, and let

ϕ

:

E

1

→

E

2

be a regular map such that

ϕ

⁢

(

O

E

1

)

=

O

E

2

. Then

ϕ

is also a

group homomorphism

,
i.e.

∀

P

,

Q

∈

E

1

,

ϕ

⁢

(

P

+

E

1

Q

)

=

ϕ

⁢

(

P

)

+

E

2

ϕ

⁢

(

Q

)

.

[Proof: See

[

2

]

, Theorem 4.8, page 75]

If

End

⁡

(

E

)

is

isomorphic

(as a ring) to an order (

http://planetmath.org/OrderInAnAlgebra

)

R

in a

quadratic imaginary
field

K

then we say that the elliptic curve E has

complex
multiplication

by

K

(or complex multiplication by

R

).

Note

:

End

⁡

(

E

)

always contains a

subring

isomorphic to

ℤ

, formed by the

multiplication

by n

maps:

[

n

]

:

E

→

E

,

[

n

]

⁢

P

=

n

⋅

P

and, in general, these are all the maps in the endomorphism ring of

E

.

Example

: Fix

d

∈

ℤ

. Let

E

be the elliptic
curve defined by

y

2

=

x

3

-

d

⁢

x

then this curve has complex multiplication by

ℚ

⁢

(

i

)

(more concretely by

ℤ

⁢

(

i

)

). Besides the multiplication
by

n

maps,

End

⁡

(

E

)

contains a genuine new element:

[

i

]

:

E

→

E

,

[

i

]

⁢

(

x

,

y

)

=

(

-

x

,

i

⁢

y

)

(the name

complex multiplication

comes from the fact that we
are “multiplying” the points in the curve by a

complex number

,

i

in this case).

References

1

James Milne,

Elliptic Curves

, online course notes.

http://www.jmilne.org/math/CourseNotes/math679.html

http://www.jmilne.org/math/CourseNotes/math679.html

2

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

3

Joseph H. Silverman,

Advanced Topics in
the Arithmetic of Elliptic Curves

. Springer-Verlag, New York,
1994.

4

Goro Shimura,

Introduction to the
Arithmetic Theory of Automorphic Functions

. Princeton University
Press, Princeton, New Jersey, 1971.

Title

complex multiplication

Canonical name

ComplexMultiplication

Date of creation

2013-03-22 13:41:35

Last modified on

2013-03-22 13:41:35

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Definition

Classification

msc 11G05

Related topic

EllipticCurve

Related topic

KroneckerWeberTheorem

Related topic

OrderInAnAlgebra

Related topic

ArithmeticOfEllipticCurves

Defines

complex multiplication

Defines

endomorphism ring

Generated on Fri Feb 9 15:34:30 2018 by

LaTeXML
examples of elliptic curves with complex multiplication

examples of elliptic curves with complex multiplication

Here we show some

elliptic curves

defined over

ℚ

which have

complex multiplication

by a

quadratic imaginary field

K

of

class number

1

(with

End

⁡

(

E

)

exactly

isomorphic

to the full

ring of integers

𝒪

K

).

K

Curve

ℚ

⁢

(

-

1

)

y

2

=

x

3

+

x

ℚ

⁢

(

-

2

)

y

2

=

x

3

+

4

⁢

x

2

+

2

⁢

x

ℚ

⁢

(

-

3

)

y

2

+

y

=

x

3

ℚ

⁢

(

-

7

)

y

2

+

x

⁢

y

=

x

3

-

x

2

-

2

⁢

x

-

1

ℚ

⁢

(

-

11

)

y

2

+

y

=

x

3

-

x

2

-

7

⁢

x

+

10

ℚ

⁢

(

-

19

)

y

2

+

y

=

x

3

-

38

⁢

x

+

90

ℚ

⁢

(

-

43

)

y

2

+

y

=

x

3

-

860

⁢

x

+

9707

ℚ

⁢

(

-

67

)

y

2

+

y

=

x

3

-

7370

⁢

x

+

243528

ℚ

⁢

(

-

163

)

y

2

+

y

=

x

3

-

2174420

⁢

x

+

1234136692

Title

examples of elliptic curves with complex multiplication

Canonical name

ExamplesOfEllipticCurvesWithComplexMultiplication

Date of creation

2013-03-22 14:22:47

Last modified on

2013-03-22 14:22:47

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Example

Classification

msc 11G05

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 15:34:32 2018 by

LaTeXML
Siegel’s theorem

Siegel’s theorem

The following

theorem

is a very deep application of Roth’s theorem. Let

K

be a

number field

and let

S

be a

finite set

of places of

K

. Let

R

S

be the

http://planetmath.org/node/RingOfSIntegers

ring of

S

-

integers

in

K

. Let

C

/

K

be a smooth

projective curve

of genus

g

defined over

K

and let

f

be a non-constant

function

in the

function field

of

C

/

K

, i.e.

f

∈

K

⁢

(

C

)

.

Theorem

(Siegel’s Theorem).

Assume that

C

/

K

has genus

g

≥

1

. Then the set

{

P

∈

C

⁢

(

K

)

:

f

⁢

(

P

)

∈

R

S

}

is finite.

In particular, when

f

is the coordinate functions

x

⁢

(

P

)

and

y

⁢

(

P

)

, Siegel’s theorem implies that a curve of genus

≥

1

has only finitely many integral points. For example, this shows that an

elliptic curve

defined over

ℚ

can only have finitely many points defined over

ℤ

.

Title

Siegel’s theorem

Canonical name

SiegelsTheorem

Date of creation

2013-03-22 15:57:24

Last modified on

2013-03-22 15:57:24

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11G05

Related topic

FaltingsTheorem

Generated on Fri Feb 9 15:34:33 2018 by

LaTeXML
the j-invariant classifies elliptic curves up to isomorphism

the

j

-invariant classifies elliptic curves up to isomorphism

In this entry, an

isomorphism

over

K

should be understood in the sense of the entry

isomorphism of varieties

.

Theorem 1

.

Let

K

be a field, and let

K

¯

be a fixed

algebraic closure

of

K

.

1.

Two

elliptic curves

E

1

and

E

2

are isomorphic (

http://planetmath.org/IsomorphismOfVarieties

) (over

K

¯

) if and only if they have the same

j

-invariant, i.e.

j

⁢

(

E

1

)

=

j

⁢

(

E

2

)

.

2.

Let

j

0

∈

K

¯

be fixed. There exists an elliptic curve

E

defined over the field

K

⁢

(

j

0

)

such that

j

⁢

(

E

)

=

j

0

.

Proof.

For part

2

:

•

For

j

0

=

0

, the curve

E

0

:

y

2

+

y

=

x

3

satisfies

j

⁢

(

E

)

=

0

;

•

For

j

0

=

1728

, the curve

E

1728

:

y

2

=

x

3

+

x

satisfies

j

⁢

(

E

1728

)

=

1728

;

•

If

j

0

≠

0

,

1728

consider the elliptic curve:

E

=

E

j

0

:

y

2

+

x

⁢

y

=

x

3

-

36

j

0

-

1728

⁢

x

-

1

j

0

-

1728

.

It satisfies

j

⁢

(

E

)

=

j

0

and it is defined over

K

⁢

(

j

0

)

.

∎

Title

the

j

-invariant classifies elliptic curves up to isomorphism

Canonical name

TheJinvariantClassifiesEllipticCurvesUpToIsomorphism

Date of creation

2013-03-22 15:06:25

Last modified on

2013-03-22 15:06:25

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11G05

Classification

msc 14H52

Related topic

IsomorphismOfVarieties

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 15:34:38 2018 by

LaTeXML
canonical height on an elliptic curve

canonical height on an elliptic curve

Let

E

/

ℚ

be an

elliptic curve

. It is often useful to have a notion of

height

of a point, in order to talk about the

arithmetic

complexity of a point

P

in

E

⁢

(

ℚ

)

. For this, one defines

height functions

. For example, in

ℚ

one can define a height by

H

⁢

(

p

/

q

)

=

m

⁢

a

⁢

x

⁢

(

|

p

|

,

|

q

|

)

.

Following the example of

ℚ

, one may define a height on

E

/

ℚ

by

h

x

⁢

(

P

)

=

{

log

⁡

H

⁢

(

x

⁢

(

P

)

)

if

⁢

P

≠

O

0

if

⁢

P

=

O

.

In fact, given any

even function

f

:

E

⁢

(

ℚ

)

→

ℝ

on

E

⁢

(

ℚ

)

(i.e.

f

⁢

(

P

)

=

f

⁢

(

-

P

)

for any

P

∈

E

⁢

(

ℚ

)

) one can define a height by:

h

f

⁢

(

P

)

=

log

⁡

H

⁢

(

f

⁢

(

P

)

)

.

However, one can refine this

definition

so that the height function satisfies some very nice

properties

(see below).

Definition

.

Let

Q

be a

number field

and let

E

be an elliptic curve defined over

Q

. The

canonical height

(or Néron-Tate height) on

E

/

Q

, denoted by

h

^

, is the

function

on

E

⁢

(

Q

)

(with real values) defined by:

h

^

⁢

(

P

)

=

1

deg

⁡

f

⁢

lim

N

→

∞

⁡

h

f

⁢

(

[

2

N

]

⁢

P

)

4

N

for any even function

f

:

E

⁢

(

Q

)

→

R

.

The fact that the definition does not depend on the choice of even function

f

is due to J. Tate. In particular, one can simply choose

f

to be the

x

-function, whose degree is

2

. The canonical height satisfies the following properties:

Theorem

.

Let

E

/

Q

and let

h

^

be the canonical height on

E

. Then:

1.

The height

h

^

satisfies the parallelogram law:

h

^

⁢

(

P

+

Q

)

+

h

^

⁢

(

P

-

Q

)

=

2

⁢

h

^

⁢

(

P

)

+

2

⁢

h

^

⁢

(

Q

)

for all

P

,

Q

∈

E

⁢

(

ℚ

¯

)

.

2.

For all

m

∈

ℤ

and all

P

∈

E

⁢

(

ℚ

¯

)

:

h

^

⁢

(

[

m

]

⁢

P

)

=

m

2

⁢

h

^

⁢

(

P

)

.

3.

The height

h

^

is even and the pairing:

⟨

⋅

,

⋅

⟩

:

E

⁢

(

ℚ

¯

)

×

E

⁢

(

ℚ

¯

)

→

ℝ

,

⟨

P

,

Q

⟩

=

h

^

⁢

(

P

+

Q

)

-

h

^

⁢

(

P

)

-

h

^

⁢

(

Q

)

is

bilinear

(usually called the Néron-Tate pairing on

E

/

ℚ

).

4.

For all

P

∈

E

⁢

(

ℚ

¯

)

one has

h

^

⁢

(

P

)

≥

0

and

h

^

⁢

(

P

)

=

0

if and only if

P

is a

torsion

point.

Title

canonical height on an elliptic curve

Canonical

name

CanonicalHeightOnAnEllipticCurve

Date of creation

2013-03-22 16:23:20

Last modified on

2013-03-22 16:23:20

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11G07

Classification

msc 11G05

Classification

msc 14H52

Synonym

Neron-Tate height

Related topic

HeightFunction

Related topic

RegulatorOfAnEllipticCurve

Defines

canonical height

Generated on Fri Feb 9 15:34:43 2018 by

LaTeXML
regulator of an elliptic curve

regulator of an elliptic curve

Let

E

/

ℚ

be an

elliptic curve

, let

E

⁢

(

ℚ

)

be the group of

rational points

on the curve and let

⟨

⋅

,

⋅

⟩

be the Néron-Tate pairing:

⟨

P

,

Q

⟩

=

h

^

⁢

(

P

+

Q

)

-

h

^

⁢

(

P

)

-

h

^

⁢

(

Q

)

where

h

^

is the

canonical height

on the elliptic curve

E

/

ℚ

.

Definition 1

.

Let

E

/

Q

be an elliptic curve and let

{

P

1

,

…

,

P

r

}

be a set of

generators

of the free part of

E

⁢

(

Q

)

, i.e. the points

P

i

generate

E

⁢

(

Q

)

modulo the

torsion subgroup

E

tors

⁢

(

Q

)

. The

height matrix

of

E

/

Q

is the

r

×

r

matrix

H

whose

i

⁢

j

th

component

is

⟨

P

i

,

P

j

⟩

, i.e.

H

=

(

⟨

P

i

,

P

j

⟩

)

.

If

r

=

0

then we define

H

=

1

.

Definition 2

.

The of

E

/

Q

(or the elliptic regulator), denoted by

Reg

⁡

(

E

/

Q

)

or

R

E

/

Q

is defined by

Reg

⁡

(

E

/

ℚ

)

=

det

⁡

(

H

)

where

H

is the height matrix.

Notice the similarities with the

regulator

of a

number field

. The regulator of an elliptic curve is the volume of a

fundamental domain

for

E

⁢

(

ℚ

)

modulo

torsion

, with respect to the

quadratic form

defined by the Néron-Tate pairing.

Title

regulator of an elliptic curve

Canonical name

RegulatorOfAnEllipticCurve

Date of creation

2013-03-22 16:23:24

Last modified on

2013-03-22 16:23:24

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 11G07

Classification

msc 11G05

Classification

msc 14H52

Related topic

CanonicalHeightOnAnEllipticCurve

Related topic

BirchAndSwinnertonDyerConjecture

Related topic

Regulator

Defines

elliptic regulator

Defines

height matrix

Generated on Fri Feb 9 15:34:48 2018 by

LaTeXML
even lattice

even lattice

A

lattice

L

in

ℝ

n

is if the norm of every vector in

L

is an

even integer

. Note that because of the

polarization identity

(

x

,

y

)

=

(

x

+

y

,

x

+

y

)

-

(

x

,

x

)

-

(

y

,

y

)

2

it follows that all

scalar products

in an

even lattice

are .

Title

even lattice

Canonical name

EvenLattice

Date of creation

2013-03-22 18:43:29

Last modified on

2013-03-22 18:43:29

Owner

monster (22721)

Last modified by

monster (22721)

Numerical id

4

Author

monster (22721)

Entry type

Definition

Classification

msc 11H06

Generated on Fri Feb 9 15:34:50 2018 by

LaTeXML
lattice in ℝ^n

lattice in

ℝ

n

Definition

.

A lattice in

R

n

is an

n

-dimensional additive

free group

over

Z

which generates

R

n

over

R

.

Example

: The following is an example of a lattice

ℒ

⊂

ℝ

2

, generated by

w

1

=

(

1

,

2

)

,

w

2

=

(

4

,

1

)

.

ℒ

=

{

α

⁢

w

1

+

β

⁢

w

2

∣

α

,

β

∈

ℤ

}

Title

lattice in

ℝ

n

Canonical name

LatticeInmathbbRn

Date of creation

2013-03-22 13:52:14

Last modified on

2013-03-22 13:52:14

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Definition

Classification

msc 11H06

Synonym

lattice

Synonym

grid

Related topic

MinkowskisTheorem

Related topic

PicksTheorem

Related topic

ProductOfPosets

Defines

lattice in

ℝ

n

Generated on Fri Feb 9 15:34:52 2018 by

LaTeXML
Minkowski’s constant

Minkowski’s constant

The following is a corollary to the famous Minkowski’s theorem on lattices and convex regions. It was also found by Minkowski and sometimes also called Minkowski’s theorem.

Theorem 1

(Minkowski’s Theorem).

Let

K

be a

number field

and let

D

K

be its

discriminant

. Let

n

=

r

1

+

2

⁢

r

2

be the degree of

K

over

Q

, where

r

1

and

r

2

are the number of

real and complex embeddings

, respectively. The

class group

of

K

is denoted by

Cl

⁡

(

K

)

. In any ideal class

C

∈

Cl

⁡

(

K

)

, there exists an ideal

A

∈

C

such that:

|

𝐍

⁢

(

𝔄

)

|

≤

M

K

⁢

|

D

K

|

where

N

⁢

(

A

)

denotes the

absolute norm

of

A

and

M

K

=

n

!

n

n

⁢

(

4

π

)

r

2

.

Definition 1

.

The constant

M

K

, as in the theorem, is usually called the Minkowski’s constant.

In the applications, one uses Stirling’s formula to find approximations of Minkowski’s constant. The following is an immediate corollary of Theorem

1

.

Corollary 1

.

Let

K

be an arbitrary number field. Then the

absolute value

of the discriminant of

K

,

D

K

, is greater than

1

, i.e.

|

D

K

|

>

1

. In particular, there is at least one

rational prime

p

∈

Z

which ramifies in

K

.

See the entry on discriminants (

http://planetmath.org/DiscriminantOfANumberField

) for the relationship between

D

K

and the

ramification

of primes.

Title

Minkowski’s constant

Canonical name

MinkowskisConstant

Date of creation

2013-03-22 15:05:33

Last modified on

2013-03-22 15:05:33

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Corollary

Classification

msc 11H06

Classification

msc 11R29

Related topic

IdealClass

Related topic

StirlingsApproximation

Related topic

DiscriminantOfANumberField

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Related topic

ProofOfMinkowskisBound

Defines

Minkowski’s theorem on ideal classes

Generated on Fri Feb 9 15:34:53 2018 by

LaTeXML
Minkowski’s theorem

Minkowski’s theorem

Let

ℒ

∈

ℝ

2

be a

lattice

in the sense of

number theory

, i.e. a 2-dimensional

free group

over

ℤ

which generates

ℝ

2

over

ℝ

. Let

w

1

,

w

2

be

generators

of the lattice

ℒ

. A set

ℱ

of
the form

ℱ

=

{

(

x

,

y

)

∈

ℝ

2

:

(

x

,

y

)

=

α

⁢

w

1

+

β

⁢

w

2

,

0

≤

α

<

1

,

0

≤

β

<

1

}

is usually called a

fundamental domain

or

fundamental parallelogram

for the lattice

ℒ

.

Theorem 1

(Minkowski’s Theorem).

Let

L

be an arbitrary lattice in

R

2

and
let

Δ

be the area of a fundamental parallelogram. Any
convex region

K

symmetrical about the origin and of
area greater than

4

⁢

Δ

contains points of the lattice

L

other than the origin.

More generally, there is the following

n

-dimensional analogue.

Theorem 2

.

Let

L

be an arbitrary lattice in

R

n

and
let

Δ

be the area of a fundamental parallelopiped. Any
convex region

K

symmetrical about the origin and of
volume greater than

2

n

⁢

Δ

contains points of the lattice

L

other than the origin.

Title

Minkowski’s

theorem

Canonical

name

MinkowskisTheorem

Date of creation

2013-03-22 13:51:42

Last modified on

2013-03-22 13:51:42

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11H06

Synonym

Minkowski’s theorem on convex bodies

Related topic

LatticeInMathbbRn

Related topic

ProofOfMinkowskisBound

Defines

Minkowski’s theorem

Defines

fundamental parallelogram

Generated on Fri Feb 9 15:34:55 2018 by

LaTeXML
proof of Minkowski’s theorem

proof of Minkowski’s theorem

Theorem 1

.

Let

L

be an arbitrary

lattice

in

R

n

and let

Δ

be the area of a fundamental

parallelepiped

. Any

convex region

K

symmetrical about the origin with

μ

⁢

(

K

)

>

2

n

⁢

Δ

contains a point of the lattice

L

other than the origin.

Proof.

Let

D

be any fundamental parallelepiped. Then obviously

ℝ

n

=

∐

x

∈

ℒ

(

D

+

x

)

(where

∐

means

disjoint union

) and thus

1

2

⁢

𝔎

=

∐

x

∈

ℒ

(

1

2

⁢

𝔎

∩

(

D

+

x

)

)

.

Now, note that

1

2

⁢

𝔎

∩

(

D

+

x

)

=

(

(

1

2

⁢

𝔎

-

x

)

∩

D

)

-

x

(draw a picture!) and thus, since measure is preserved by

translation

,

μ

⁢

(

1

2

⁢

𝔎

∩

(

D

+

x

)

)

=

μ

⁢

(

(

1

2

⁢

𝔎

-

x

)

∩

D

)

so that if all the

1

2

⁢

𝔎

-

x

are

disjoint

, we have

2

-

n

⁢

μ

⁢

(

𝔎

)

=

μ

⁢

(

1

2

⁢

𝔎

)

=

μ

⁢

(

∐

x

∈

ℒ

(

1

2

⁢

𝔎

∩

(

D

+

x

)

)

)

=

∑

x

∈

ℒ

μ

⁢

(

(

1

2

⁢

𝔎

-

x

)

∩

D

)

≤

μ

⁢

(

D

)

=

Δ

which is a

contradiction

. Thus there must exist

x

≠

y

∈

ℒ

and

c

1

,

c

2

∈

𝔎

such that

1

2

⁢

c

1

-

x

=

1

2

⁢

c

2

-

y

.

Thus

x

-

y

=

1

2

⁢

(

c

2

-

c

1

)

∈

𝔎

since

𝔎

is convex and centrally

symmetric

, and certainly

x

-

y

∈

ℒ

, so we have found a nonzero element of

𝔎

∩

Λ

.

Corollary 1

.

Let

L

be an arbitrary lattice in

R

n

and let

Δ

be the area of a fundamental parallelepiped. Any compact convex region

K

symmetrical about the origin with

μ

⁢

(

K

)

≥

2

n

⁢

Δ

contains a point of the lattice

L

other than the origin.

Note that this

corollary

requires that

𝔎

be compact in

addition

to being convex and centrally symmetric, but slightly relaxes the volume condition on

𝔎

.

Proof.

Apply the previous case to

C

n

=

(

1

+

1

n

)

⁢

C

, i.e. dilate

C

. This gives a

sequence

of points

x

1

,

x

2

,

…

,

x

n

,

…

with

x

i

∈

Λ

∩

C

i

-

{

0

}

. But

Λ

is discrete, so there must be a

subsequence

constant

at a nonzero

element

x

∈

Λ

⁢

⋂

(

⋂

i

=

1

∞

C

i

-

{

0

}

)

=

Λ

∩

C

¯

-

{

0

}

.

Since

C

is compact and thus closed,

x

∈

C

.

Title

proof of Minkowski’s

theorem

Canonical

name

ProofOfMinkowskisTheorem

Date of creation

2013-03-22 17:53:41

Last modified on

2013-03-22 17:53:41

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Proof

Classification

msc 11H06

Generated on Fri Feb 9 15:34:58 2018 by

LaTeXML