elliptic curve

elliptic curve

1

Basics

An

elliptic curve

over a field

K

is a projective

nonsingular

algebraic curve

E

over

K

of genus 1 together with a point

O

of

E

defined over

K

. The word “genus” is taken here in the

algebraic geometry

sense, and has no

relation

with the topological notion of genus (defined as

1

-

χ

/

2

, where

χ

is the Euler characteristic) except when the field of definition

K

is the

complex numbers

ℂ

.

Using the

Riemann-Roch theorem for curves

, one can show that every elliptic curve

E

is the

zero set

of a

Weierstrass equation

of the form

E

:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

,

for some

a

i

∈

K

, where the

polynomial

on the

right hand side

has no double roots. When

K

has

characteristic

other than 2 or 3, one can further simpify this Weierstrass equation into the form

E

:

y

2

=

x

3

-

27

⁢

c

4

⁢

x

-

54

⁢

c

6

.

The extremely strange numbering of the

coefficients

is an artifact of the process by which the above equations are derived. Also, note that these equation are for affine curves; to

translate

them to

projective curves

, one has to homogenize the equations (replace

x

with

X

/

Z

, and

y

with

Y

/

Z

).

2

Examples

We present here some pictures of elliptic curves over the field

ℝ

of

real numbers

. These pictures are in some sense

not

representative of most of the elliptic curves that people work with, since many of the interesting cases tend to be of elliptic curves over algebraically closed fields. However, curves over the complex numbers (or, even worse, over algebraically closed fields in characteristic

p

) are very difficult to graph in three

dimensions

, let alone two.

Figure

1

is a graph of the elliptic curve

y

2

=

x

3

-

x

.

Figure 1:

Graph of

y

2

=

x

⁢

(

x

-

1

)

⁢

(

x

+

1

)

Figure

2

shows the graph of

y

2

=

x

3

-

x

+

1

:

Figure 2:

Graph of

y

2

=

x

3

-

x

+

1

Finally, Figures

3

and

4

are examples of algebraic curves that are

not

elliptic curves. Both of these curves have singularities at the origin.

Figure 3:

Graph of

y

2

=

x

2

⁢

(

x

+

1

)

. Has two

tangents

at the origin.

Figure 4:

Graph of

y

2

=

x

3

. Has a cusp at the origin.

3

The Group Law

The points on an elliptic curve have a natural group

structure

, which makes the elliptic curve into an

abelian variety

. There are many

equivalent

ways to define this group structure; two of the most common are:

•

Every Weyl

divisor

on

E

is linearly equivalent to a unique divisor of the form

[

P

]

-

[

O

]

for some

P

∈

E

, where

O

∈

E

is the

base point

. The

divisor class group

of

E

then yields a group structure on the points of

E

, by way of this correspondence.

•

Let

O

∈

E

denote the base point. Then one can show that every line joining two points on

E

intersects

a unique third point of

E

(after properly accounting for tangent lines as a

multiple

intersection

). For any two points

P

,

Q

∈

E

, define their sum as:

(a)

Form the line between

P

and

Q

; let

R

be the third point on

E

that intersects this line;

(b)

Form the line between

O

and

R

; define

P

+

Q

to be the third point on

E

that intersects this line.

This

addition

operation

yields a

group operation

on the points of

E

having the base point

O

for

identity

.

4

Elliptic Curves over

ℂ

Over the complex numbers, the general correspondence between

algebraic

and

analytic

theory specializes in the elliptic curves case to yield some very useful insights into the structure of elliptic curves over

ℂ

. The starting point for this investigation is the Weierstrass

𝔭

–

function

, which we define here.

Definition 1

.

A

lattice

in

ℂ

is a

subgroup

L

of the

additive group

ℂ

which is generated by two

elements

ω

1

,

ω

2

∈

ℂ

that are

linearly independent

over

ℝ

.

Definition 2

.

For any lattice

L

in

ℂ

, the

Weierstrass

p

L

–function

of

L

is the function

𝔭

L

:

ℂ

⟶

ℂ

given by

𝔭

L

⁢

(

z

)

:=

1

z

2

+

∑

ω

∈

L

∖

{

0

}

(

1

(

z

-

ω

)

2

-

1

ω

2

)

.

When the lattice

L

is clear from

context

, it is customary to suppress it from the notation and simply write

𝔭

for the Weierstrass

𝔭

–function.

Properties of the Weierstrass

p

–function:

•

𝔭

⁢

(

z

)

is a meromorphic function with double poles at points in

L

.

•

𝔭

⁢

(

z

)

is

constant

on each coset of

ℂ

/

L

.

•

𝔭

⁢

(

z

)

satisfies

the differential equation

𝔭

′

⁢

(

z

)

2

=

4

⁢

𝔭

⁢

(

z

)

3

-

g

2

⁢

𝔭

⁢

(

z

)

-

g

3

where the constants

g

2

and

g

3

are given by

g

2

:=

60

⁢

∑

ω

∈

L

∖

{

0

}

1

ω

4

g

3

:=

140

⁢

∑

ω

∈

L

∖

{

0

}

1

ω

6

The last property above implies that, for any

z

∈

ℂ

/

L

, the point

(

𝔭

⁢

(

z

)

,

𝔭

′

⁢

(

z

)

)

lies on the elliptic curve

E

:

y

2

=

4

⁢

x

3

-

g

2

⁢

x

-

g

3

. Let

ϕ

:

ℂ

/

L

⟶

E

be the map given by

ϕ

⁢

(

z

)

:=

{

(

𝔭

⁢

(

z

)

,

𝔭

′

⁢

(

z

)

)

z

∉

L

∞

z

∈

L

(where

∞

denotes the point at infinity on

E

). Then

ϕ

is actually a

bijection

(!), and moreover the map

ϕ

:

ℂ

/

L

⟶

E

is an

isomorphism

of Riemann surfaces as well as a group isomorphism (with the addition operation on

ℂ

/

L

inherited from

ℂ

, and the elliptic curve group operation on

E

).

We can go even further: it turns out that

every

elliptic curve

E

over

ℂ

can be obtained in this way from some lattice

L

. More precisely, the following is true:

Theorem 3

.

1.

For every elliptic curve

E

:

y

2

=

4

⁢

x

3

-

b

⁢

x

-

c

over

ℂ

, there is a unique lattice

L

⊂

ℂ

whose constants

g

2

and

g

3

satisfy

b

=

g

2

and

c

=

g

3

.

2.

Two elliptic curves

E

and

E

′

over

ℂ

are isomorphic if and only if their corresponding lattices

L

and

L

′

satisfy the equation

L

′

=

α

⁢

L

for some scalar

α

∈

ℂ

.

References

1

Dale Husemoller,

Elliptic Curves

. Springer–Verlag, New York, 1997.

2

James Milne,

Elliptic Curves

, online course notes.

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

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

3

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer–Verlag, New York, 1986.

Title

elliptic curve

Canonical name

EllipticCurve

Date of creation

2013-03-22 12:03:02

Last modified on

2013-03-22 12:03:02

Owner

djao (24)

Last modified by

djao (24)

Numerical id

32

Author

djao (24)

Entry type

Definition

Classification

msc 11G07

Classification

msc 11G05

Classification

msc 14H52

Related topic

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JInvariant

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MordellWeilTheorem

Related topic

ConductorOfAnEllipticCurve

Defines

𝔭

-function

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