aarjaneiro/planetmath huggingface dataset:
group

group

Group.

A group is a pair

(

G

,

*

)

, where

G

is a non-empty set and “

*

”
is a

binary operation

on

G

, such that the following conditions hold:

•

For any

a

,

b

in

G

,

a

*

b

belongs to

G

. (The

operation

“

*

” is closed).

•

For any

a

,

b

,

c

∈

G

,

(

a

*

b

)

*

c

=

a

*

(

b

*

c

)

.  (

Associativity

of
the operation).

•

There is an

element

e

∈

G

such that

g

*

e

=

e

*

g

=

g

for any

g

∈

G

. (

Existence

of

identity element

).

•

For any

g

∈

G

there exists an element

h

such that

g

*

h

=

h

*

g

=

e

.  (Existence of

inverses

).

If

G

is a group under *, then * is referred to as the

group
operation

of

G

.

Usually, the symbol “

*

” is omitted and we write

a

⁢

b

for

a

*

b

.  Sometimes, the symbol “

+

” is used to represent the
operation, especially when the group is

abelian

.

It can be proved that there is only one identity element, and that for
every element there is only one inverse.  Because of this we usually
denote the inverse of

a

as

a

-

1

or

-

a

when we are using

additive

notation.  The identity element is also called

neutral
element

due to its behavior with respect to the operation, and thus

a

-

1

is sometimes (although uncommonly) called the

neutralizing element

of

a

. An element of a group besides the
identity element is sometimes called a

non-trivial element

.

Groups often arise as the

symmetry groups

of other mathematical objects; the study of such situations uses

group actions

.  In fact, much of the study of groups themselves is conducted using group actions.

Title

group

Canonical name

Group

Date of creation

2013-03-22 11:42:53

Last modified on

2013-03-22 11:42:53

Owner

drini (3)

Last modified by

drini (3)

Numerical id

34

Author

drini (3)

Entry type

Definition

Classification

msc 14F99

Classification

msc 08A99

Classification

msc 20A05

Classification

msc 20-00

Classification

msc 83C99

Classification

msc 32C05

Related topic

Subgroup

Related topic

CyclicGroup

Related topic

Simple

Related topic

SymmetricGroup

Related topic

FreeGroup

Related topic

Ring

Related topic

Field

Related topic

GroupHomomorphism

Related topic

LagrangesTheorem

Related topic

IdentityElement

Related topic

ProperSubgroup

Related topic

Groupoid

Related topic

FundamentalGroup

Related topic

TopologicalGroup

Related topic

LieGroup

Related topic

ProofThatGInGImpliesThatLangleGRangleLeG

Related topic

GeneralizedCyclicGroup

Defines

identity

Defines

inverse

Defines

neutralizing element

Defines

non-trivial element

Defines

nontrivial element

Defines

group operation

Generated on Fri Feb 9 17:58:10 2018 by

LaTeXML
Leray spectral sequence for an affine morphism

Leray spectral sequence for an affine morphism

Let

f

:

X

→

Y

be an

affine morphism

of schemes (that is, every point has an affine

neighborhood

whose preimage is affine). Then

R

i

⁢

f

*

=

0

for every

i

>

0

, and the

Leray spectral sequence

tells us that

H

i

⁢

(

X

,

ℱ

)

=

H

i

⁢

(

Y

,

f

*

⁢

ℱ

)

.

Title

Leray spectral sequence for an affine morphism

Canonical name

LeraySpectralSequenceForAnAffineMorphism

Date of creation

2013-03-22 14:08:46

Last modified on

2013-03-22 14:08:46

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

5

Author

archibal (4430)

Entry type

Example

Classification

msc 14F99

Classification

msc 18G40

Classification

msc 55T05

Generated on Fri Feb 9 17:58:12 2018 by

LaTeXML
Faltings’ theorem

Faltings’ theorem

Let

K

be a

number field

and let

C

/

K

be a

non-singular

curve defined over

K

and genus

g

. When the genus is

0

, the curve is

isomorphic

to

ℙ

1

(over an

algebraic closure

K

¯

) and therefore

C

⁢

(

K

)

is either empty or equal to

ℙ

1

⁢

(

K

)

(in particular

C

⁢

(

K

)

is

infinite

). If the genus of

C

is

1

and

C

⁢

(

K

)

contains at least one point over

K

then

C

/

K

is an

elliptic curve

and the

Mordell-Weil theorem

shows that

C

⁢

(

K

)

is a

finitely generated

abelian group

(in particular,

C

⁢

(

K

)

may be finite or infinite). However, if

g

≥

2

, Mordell conjectured in

1922

that

C

⁢

(

K

)

cannot be infinite. This was first proven by Faltings in

1983

.

Theorem

(Faltings’ Theorem (Mordell’s conjecture)).

Let

K

be a number field and let

C

/

K

be a non-singular curve defined over

K

of genus

g

≥

2

. Then

C

⁢

(

K

)

is finite.

The reader may also be interested in Siegel’s

theorem

.

Title

Faltings’ theorem

Canonical name

FaltingsTheorem

Date of creation

2013-03-22 15:57:21

Last modified on

2013-03-22 15:57:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14G05

Classification

msc 14H99

Synonym

Mordell’s conjecture

Related topic

SiegelsTheorem

Generated on Fri Feb 9 17:58:13 2018 by

LaTeXML
Hasse principle

Hasse principle

Let

V

be an

algebraic variety

defined over a field

K

. By

V

⁢

(

K

)

we denote the set of points on

V

defined over

K

. Let

K

¯

be an

algebraic closure

of

K

. For a

valuation

ν

of

K

, we write

K

ν

for the

completion

of

K

at

ν

. In
this case, we can also consider

V

defined over

K

ν

and
talk about

V

⁢

(

K

ν

)

.

Definition 1

.

1.

If

V

⁢

(

K

)

is not empty we say that

V

is

soluble

in

K

.

2.

If

V

⁢

(

K

ν

)

is not empty then we say that

V

is

locally soluble

at

ν

.

3.

If

V

is locally soluble for all

ν

then we say that

V

satisfies the

Hasse condition

, or we say that

V

/

K

is

everywhere locally soluble

.

The

Hasse Principle

is the idea (or desire) that an
everywhere locally soluble

variety

V

must have a

rational point

,
i.e. a point defined over

K

. Unfortunately this is not true,
there are examples of varieties that satisfy the Hasse condition
but have no rational points.

Example

: A quadric (of any dimension) satisfies the Hasse
condition. This was proved by Minkowski for quadrics over

ℚ

and by Hasse for quadrics over a

number field

.

References

1

Swinnerton-Dyer,

Diophantine Equations

: Progress and Problems

,

http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf

online notes.

Title

Hasse principle

Canonical name

HassePrinciple

Date of creation

2013-03-22 13:50:39

Last modified on

2013-03-22 13:50:39

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Definition

Classification

msc 14G05

Related topic

HasseMinkowskiTheorem

Defines

Hasse principle

Defines

Hasse condition

Defines

locally soluble

Generated on Fri Feb 9 17:58:15 2018 by

LaTeXML
Weil conjectures

Weil conjectures

Conjectures made by Weil on the form of the

zeta function

of a

variety

over a

finite field

. Specifically he thought it should be

rational

, it should split into

polynomial

parts with

integral

coefficients, with roots of a certain magnitude, and of degree = Betti number. Further there should be a functional equation.

(Flesh this out ;))

Title

Weil conjectures

Canonical name

WeilConjectures

Date of creation

2013-03-22 15:43:17

Last modified on

2013-03-22 15:43:17

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

6

Author

nerdy2 (62)

Entry type

Conjecture

Classification

msc 14G15

Classification

msc 14G10

Classification

msc 14G05

Generated on Fri Feb 9 17:58:16 2018 by

LaTeXML
global field

global field

A

global field

is either a

global number field

or
a

global function field

.

A

global number field

is a

finite extension

of the

rationals

.

A

global function field

is a finite extension of the

field of
rational functions

k

⁢

(

t

)

of a

finite field

k

.

Title

global field

Canonical

name

GlobalField

Date of creation

2013-03-22 16:51:54

Last modified on

2013-03-22 16:51:54

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 14G25

Classification

msc 14G20

Defines

global number field

Defines

global function field

Generated on Fri Feb 9 17:58:18 2018 by

LaTeXML
Frobenius morphism

Frobenius morphism

Let

K

be a field of

characteristic

p

>

0

and let

q

=

p

r

. Let

C

be a curve defined over

K

contained in

ℙ

N

, the

projective space

of dimension

N

. Define the

homogeneous ideal

of

C

to be (the

ideal generated by

):

I

⁢

(

C

)

=

{

f

∈

K

⁢

[

X

0

,

…

,

X

N

]

∣

∀

P

∈

C

,

f

⁢

(

P

)

=

0

,

f

⁢

is homogeneous

}

For

f

∈

K

⁢

[

X

0

,

…

,

X

N

]

, of the form

f

=

∑

i

a

i

⁢

X

0

i

0

⁢

…

⁢

X

N

i

N

we define

f

(

q

)

=

∑

i

a

i

q

⁢

X

0

i

0

⁢

…

⁢

X

N

i

N

. We define a new curve

C

(

q

)

as
the

zero set

of the ideal (generated by):

I

⁢

(

C

(

q

)

)

=

{

f

(

q

)

∣

f

∈

I

⁢

(

C

)

}

Definition 1

.

The

q

t

⁢

h

-power

Frobenius morphism

is defined to be:

ϕ

:

C

→

C

(

q

)

ϕ

⁢

(

[

x

0

,

…

,

x

N

]

)

=

[

x

0

q

,

…

⁢

x

N

q

]

In order to check that the Frobenius morphism is well defined we
need to prove that

P

=

[

x

0

,

…

,

x

N

]

∈

C

⇒

ϕ

⁢

(

P

)

=

[

x

0

q

,

…

⁢

x

N

q

]

∈

C

(

q

)

This is equivalent to
proving that for any

g

∈

I

⁢

(

C

(

q

)

)

we have

g

⁢

(

ϕ

⁢

(

P

)

)

=

0

.
Without loss of generality we can assume that

g

is a

generator

of

I

⁢

(

C

(

q

)

)

, i.e.

g

is of the form

g

=

f

(

q

)

for some

f

∈

I

⁢

(

C

)

. Then:

g

⁢

(

ϕ

⁢

(

P

)

)

=

f

(

q

)

⁢

(

ϕ

⁢

(

P

)

)

=

f

(

q

)

⁢

(

[

x

0

q

,

…

,

x

N

q

]

)

=

(

f

(

[

x

0

,

…

,

x

N

]

)

)

q

,

[

a

q

+

b

q

=

(

a

+

b

)

q

in characteristic

p

]

=

(

f

⁢

(

P

)

)

q

=

0

,

[

P

∈

C

,

f

∈

I

(

C

)

]

as desired.

Example

: Suppose

E

is an

elliptic curve

defined over

K

=

𝔽

q

, the field of

p

r

elements. In this case the

Frobenius map

is an

automorphism

of

K

, therefore

E

=

E

(

q

)

Hence the Frobenius morphism is an endomorphism (or

isogeny

) of the elliptic curve.

References

1

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

Title

Frobenius morphism

Canonical name

FrobeniusMorphism

Date of creation

2013-03-22 13:51:45

Last modified on

2013-03-22 13:51:45

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H37

Related topic

FrobeniusAutomorphism

Related topic

FrobeniusMap

Related topic

ArithmeticOfEllipticCurves

Defines

Frobenius morphism

Generated on Fri Feb 9 17:58:21 2018 by

LaTeXML
Archimedean spiral

Archimedean spiral

An

Archimedean spiral

is a spiral with the

polar equation

r

=

a

⁢

θ

1

/

t

,

where

a

is a real,

r

is the radial distance,

θ

is the angle, and

t

is a constant.

The curvature of an Archimedean spiral is given by the formula

t

⁢

θ

1

-

1

/

t

⁢

(

t

2

⁢

θ

2

+

t

+

1

)

a

⁢

(

t

2

⁢

θ

2

+

1

)

3

/

2

.

Title

Archimedean spiral

Canonical name

ArchimedeanSpiral

Date of creation

2013-03-22 14:05:55

Last modified on

2013-03-22 14:05:55

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 14H45

Generated on Fri Feb 9 17:58:22 2018 by

LaTeXML
torsion (space curve)

torsion (space curve)

Let

I

⊂

R

be an

interval

and let

γ

:

I

→

ℝ

3

be a

parameterized space curve

, assumed to be

regular

(

http://planetmath.org/SpaceCurve

) and free of

points of inflection

. We
interpret

γ

⁢

(

t

)

as the

trajectory

of a particle moving through
3-dimensional space. Let

T

⁢

(

t

)

,

N

⁢

(

t

)

,

B

⁢

(

t

)

denote the corresponding

moving trihedron

. The speed of this particle is given by

∥

γ

′

⁢

(

t

)

∥

.

In order for a moving particle to escape the

osculating plane

, it is
necessary for the particle to “roll” along the axis of its

tangent
vector

, thereby lifting the normal

acceleration

vector out of the
osculating plane. The “rate of roll”, that is to say the rate at
which the osculating plane rotates about the tangent vector, is given
by

B

⁢

(

t

)

⋅

N

′

⁢

(

t

)

; it is a number that depends on the
speed of the particle. The rate of roll relative to the particle’s
speed is the quantity

τ

⁢

(

t

)

=

B

⁢

(

t

)

⋅

N

′

⁢

(

t

)

∥

γ

′

⁢

(

t

)

∥

=

(

γ

′

⁢

(

t

)

×

γ

′′

⁢

(

t

)

)

⋅

γ

′′′

⁢

(

t

)

∥

γ

′

⁢

(

t

)

×

γ

′′

⁢

(

t

)

∥

2

,

called the

torsion

of the curve, a quantity
that is invariant with respect to reparameterization. The torsion

τ

⁢

(

t

)

is, therefore, a measure of an intrinsic

property

of the
oriented space curve, another

real number

that can be covariantly
assigned to the point

γ

⁢

(

t

)

.

Title

torsion (space curve)

Canonical

name

TorsionspaceCurve

Date of creation

2013-03-22 12:15:05

Last modified on

2013-03-22 12:15:05

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 14H50

Synonym

torsion

Related topic

SpaceCurve

Related topic

CurvatureOfACurve

Generated on Fri Feb 9 17:58:27 2018 by

LaTeXML
bound for the rank of an elliptic curve

bound for the rank of an elliptic curve

Theorem

.

Let

E

/

Q

be an

elliptic curve

given by the equation:

E

:

y

2

=

x

⁢

(

x

-

t

)

⁢

(

x

-

s

)

,

with

⁢

t

,

s

∈

ℤ

and suppose that

E

has

s

=

m

+

a

primes of

bad reduction

, with

m

and

a

being the number of primes with

multiplicative

and additive reduction respectively. Then the rank of

E

, denoted by

R

E

, satisfies:

R

E

≤

m

+

2

⁢

a

-

1

Example

.

As an application of the theorem above, we can prove that

E

1

:

y

2

=

x

3

-

x

has only finitely many

rational

solutions. Indeed, the

discriminant

of

E

1

,

Δ

=

64

, is only

divisible

by

p

=

2

, which is a prime of (bad) multiplicative reduction. Therefore

R

E

1

=

0

. Moreover, the

Nagell-Lutz theorem

implies that the only

torsion

points on

E

1

are those of order

2

. Hence, the only

rational points

on

E

1

are:

{

𝒪

,

(

0

,

0

)

,

(

1

,

0

)

,

(

-

1

,

0

)

}

.

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

Title

bound for the rank of an elliptic curve

Canonical name

BoundForTheRankOfAnEllipticCurve

Date of creation

2013-03-22 14:24:25

Last modified on

2013-03-22 14:24:25

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:58:34 2018 by

LaTeXML
conductor of an elliptic curve

conductor of an elliptic curve

Let

E

be an

elliptic curve

over

ℚ

. For each prime

p

∈

ℤ

define the quantity

f

p

as follows:

f

p

=

{

0

⁢

, if

E

has good reduction at

p

,

1

⁢

, if

E

has multiplicative reduction at

p

,

2

⁢

, if

E

has additive reduction at

p

, and

p

≠

2

,

3

,

2

+

δ

p

⁢

, if

E

has additive reduction at

p

=

2

⁢

o

⁢

r

⁢

3

.

where

δ

p

depends on

wild ramification

in the action of the

inertia group

at

p

of

Gal

⁡

(

ℚ

¯

/

ℚ

)

on the

Tate
module

T

p

⁢

(

E

)

.

Definition

.

The

conductor

N

E

/

Q

of

E

/

Q

is defined to
be:

N

E

/

ℚ

=

∏

p

p

f

p

where the product is over all primes and the exponent

f

p

is
defined as above.

Example

.

Let

E

/

ℚ

:

y

2

+

y

=

x

3

-

x

2

+

2

⁢

x

-

2

. The primes of

bad
reduction

for

E

are

p

=

5

and

7

. The

reduction

at

p

=

5

is

additive

, while the reduction at

p

=

7

is

multiplicative

. Hence

N

E

/

ℚ

=

25

⋅

7

=

175

.

References

1

James Milne,

Elliptic Curves

,

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

online course
notes.

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.

Title

conductor of an elliptic curve

Canonical

name

ConductorOfAnEllipticCurve

Date of creation

2013-03-22 13:49:51

Last modified on

2013-03-22 13:49:51

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Synonym

conductor

Related topic

EllipticCurve

Related topic

LSeriesOfAnEllipticCurve

Related topic

ArithmeticOfEllipticCurves

Defines

conductor of an elliptic curve

Generated on Fri Feb 9 17:58:36 2018 by

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examples of torsion subgroups of elliptic curves

examples of torsion subgroups of elliptic curves

Mazur’s theorem shows that given an

elliptic curve

defined over the

rationals

, the only possible

torsion subgroups

are the following:

ℤ

/

N

⁢

ℤ

with

⁢

1

<

N

<

11

⁢

or

⁢

N

=

12

ℤ

/

2

⁢

ℤ

⊕

ℤ

/

2

⁢

N

⁢

ℤ

⁢

with

⁢

0

<

N

<

5

Here we show examples of curves with the torsion subgroups mentioned above:

CURVE

TORSION SUBGROUP

GENERATORS

y

2

=

x

3

-

2

trivial

𝒪

y

2

=

x

3

+

8

ℤ

/

2

⁢

ℤ

[

[

-

2

,

0

]

]

y

2

=

x

3

+

4

ℤ

/

3

⁢

ℤ

[

[

0

,

2

]

]

y

2

=

x

3

+

4

⁢

x

ℤ

/

4

⁢

ℤ

[

[

2

,

4

]

]

y

2

-

y

=

x

3

-

x

2

ℤ

/

5

⁢

ℤ

[

[

0

,

1

]

]

y

2

=

x

3

+

1

ℤ

/

6

⁢

ℤ

[

[

2

,

3

]

]

y

2

=

x

3

-

43

⁢

x

+

166

ℤ

/

7

⁢

ℤ

[

[

3

,

8

]

]

y

2

+

7

⁢

x

⁢

y

=

x

3

+

16

⁢

x

ℤ

/

8

⁢

ℤ

[

[

-

2

,

10

]

]

y

2

+

x

⁢

y

+

y

=

x

3

-

x

2

-

14

⁢

x

+

29

ℤ

/

9

⁢

ℤ

[

[

3

,

1

]

]

y

2

+

x

⁢

y

=

x

3

-

45

⁢

x

+

81

ℤ

/

10

⁢

ℤ

[

[

0

,

9

]

]

y

2

+

43

⁢

x

⁢

y

-

210

⁢

y

=

x

3

-

210

⁢

x

2

ℤ

/

12

⁢

ℤ

[

[

0

,

210

]

]

y

2

=

x

3

-

4

⁢

x

ℤ

/

2

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

[

[

2

,

0

]

,

[

0

,

0

]

]

y

2

=

x

3

+

2

⁢

x

2

-

3

⁢

x

ℤ

/

4

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

[

[

3

,

6

]

,

[

0

,

0

]

]

y

2

+

5

⁢

x

⁢

y

-

6

⁢

y

=

x

3

-

3

⁢

x

2

ℤ

/

6

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

[

[

-

3

,

18

]

,

[

2

,

-

2

]

]

y

2

+

17

⁢

x

⁢

y

-

120

⁢

y

=

x

3

-

60

⁢

x

2

ℤ

/

8

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

[

[

30

,

-

90

]

,

[

-

40

,

400

]

]

Title

examples of torsion subgroups of elliptic curves

Canonical name

ExamplesOfTorsionSubgroupsOfEllipticCurves

Date of creation

2013-03-22 14:22:44

Last modified on

2013-03-22 14:22:44

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Example

Classification

msc 14H52

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:58:41 2018 by

LaTeXML
false counterexamples to Fermat’s last theorem

false counterexamples to Fermat’s last theorem

Like Martin Gardner’s famous 1975 joke that

e

π

⁢

163

is an

integer

, hoax

counterexamples

to Fermat’s last

theorem

typically depend on the loss of machine precision. The following false counterexamples should check out on most

scientific calculators

.

1782

12

+

1841

12

=

1922

12

6107

6

+

8919

6

=

9066

6

3987

12

+

4365

12

=

4472

12

Executing the left side of the first equation on a typical scientific calculator and then taking the 12th root of that result will yield 1922. But on software calculators, such as the

Mac OS Calculator

, the 12th root is given as 1921.99999995495. Rising points out that it is not necessary to carry out any calculations in order to see that the first equation is false: “The left side adds an

even number

and an

odd number

; thus that sum must be odd. The right side is even.” In fact,

modular arithmetic

can be used to show all these equations are false: the left side of the second one is

congruent

to

5

mod

9

while the right side is congruent to

3

mod

9

;

casting out nines

also disproves the third equation.

Taxicab numbers

involving

1

x

in one of the

expressions

, like

Ramanujan

’s friend 1729, can be used to benchmark machine precision. Any scientific calculator will readily show that

10

3

+

9

3

≠

12

3

, since

1729

3

≈

12.002314

, and for some larger taxicab number a false counterexample will appear. A

computer algebra system

, on the other hand, can determine the falsehood of a counterexample even if it lacks the machine precision to resolve the numerical

difference

. For example, on

Mathematica

, assuming

x, y, z, n

have been defined:

TrueQ[x^n + y^n == z^n]

should return

False

.

Some of these false counterexamples have appeared on episodes of

The Simpsons

, such as the first one, which appeared in the

Homer

3

segment of “Treehouse of Horror VI,” first aired October 29, 1995, more than a year after

Andrew Wiles

and his colleagues announced the corrected proof of Fermat’s last theorem.

References

1

Gerald R. Rising,

Inside Your

Calculator

: From Simple Programs to Significant Insights

. Hoboken, New Jersey: John Wiley & Sons (2007): Appendix D

2

Ray Richmond & Antonia Coffman,

The Simpsons: A

Complete

Guide to Our Favorite Family

. New York: HarperCollings (1997): 187

Title

false counterexamples to Fermat’s last theorem

Canonical

name

FalseCounterexamplesToFermatsLastTheorem

Date of creation

2013-03-22 18:05:46

Last modified on

2013-03-22 18:05:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 14H52

Classification

msc 11D41

Classification

msc 11F80

Generated on Fri Feb 9 17:58:43 2018 by

LaTeXML
Hasse’s bound for elliptic curves over finite fields

Hasse’s bound for elliptic curves over finite fields

Let

E

be an

elliptic curve

defined over a

finite field

𝔽

q

with

q

=

p

r

elements (

p

∈

ℤ

is a prime). The
following

theorem

gives a bound of the size of

E

⁢

(

𝔽

q

)

,

N

q

, i.e. the number points of

E

defined over

𝔽

q

.
This was first conjectured by Emil Artin (in his thesis!) and
proved by Helmut Hasse in the 1930’s.

Theorem 1

(Hasse).

∣

N

q

-

q

-

1

∣

≤

2

⁢

q

Remark

: Let

a

p

=

p

+

1

-

N

p

as in the

definition

of the
L-series of an ellitpic curve. Then Hasse’s bound reads:

∣

a

p

∣

≤

2

⁢

p

This fact is key for the convergence of the L-series of

E

.

Title

Hasse’s bound for elliptic curves over finite fields

Canonical

name

HassesBoundForEllipticCurvesOverFiniteFields

Date of creation

2013-03-22 13:55:41

Last modified on

2013-03-22 13:55:41

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Synonym

Hasse’s bound

Related topic

LSeriesOfAnEllipticCurve

Related topic

EllipticCurve

Related topic

BadReduction

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:58:45 2018 by

LaTeXML