rank of an elliptic curve

rank of an elliptic curve

Let

K

be a

number field

and let

E

be an

elliptic curve

over

K

. By

E

⁢

(

K

)

we denote the set of points in

E

with coordinates
in

K

.

Theorem 1

(Mordell-Weil).

E

⁢

(

K

)

is a

finitely generated

abelian
group

.

Proof.

The proof of this theorem is fairly involved. The
main two ingredients are the so called “

weak Mordell-Weil theorem

”
(see below), the concept of

height function

for abelian groups and
the “descent” theorem.

See

[

2

]

, Chapter VIII, page
189.
∎

Theorem 2

(Weak Mordell-Weil).

E

⁢

(

K

)

/

m

⁢

E

⁢

(

K

)

is
finite for all

m

≥

2

.

The

Mordell-Weil theorem

implies that for any elliptic curve

E

/

K

the group of points has the following structure:

E

⁢

(

K

)

≃

E

torsion

⁢

(

K

)

⁢

⊕

ℤ

R

where

E

torsion

⁢

(

K

)

denotes the set of points of

finite order

(or

torsion group

),
and

R

is a non-negative

integer

which is called the

r

⁢

a

⁢

n

⁢

k

of the
elliptic curve. It is not known how big this number

R

can get
for elliptic curves over

ℚ

. The largest rank known for
an elliptic curve over

ℚ

is 28

http://www.math.hr/ duje/tors/tors.html

Elkies (2006).

Note: see Mazur’s theorem for an account of the possible torsion subgroups over

ℚ

.

Examples

:

1.

The elliptic curve

E

1

/

ℚ

:

y

2

=

x

3

+

6

has rank 0
and

E

1

⁢

(

ℚ

)

≃

0

.

2.

Let

E

2

/

ℚ

:

y

2

=

x

3

+

1

, then

E

2

⁢

(

ℚ

)

≃

ℤ

/

6

⁢

ℤ

. The torsion group
is generated by the point

(

2

,

3

)

.

3.

Let

E

3

/

ℚ

:

y

2

=

x

3

+

109858299531561

, then

E

3

⁢

(

ℚ

)

≃

ℤ

/

3

⁢

ℤ

⁢

⊕

ℤ

5

. See

http://math.bu.edu/people/alozano/

Torsion

.html

generators

here.

4.

Let

E

4

/

ℚ

:

y

2

+

1951

/

164

⁢

x

⁢

y

-

3222367

/

40344

⁢

y

=

x

3

+

3537

/

164

⁢

x

2

-

40302641

/

121032

⁢

x

, then

E

4

⁢

(

ℚ

)

≃

ℤ

10

. See

http://math.bu.edu/people/alozano/Examples.html

generators
here.

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

rank of an elliptic curve

Canonical name

RankOfAnEllipticCurve

Date of creation

2013-03-22 13:49:12

Last modified on

2013-03-22 13:49:12

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

14

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Synonym

rank

Related topic

EllipticCurve

Related topic

HeightFunction

Related topic

MordellWeilTheorem

Related topic

SelmerGroup

Related topic

MazursTheoremOnTorsionOfEllipticCurves

Related topic

NagellLutzTheorem

Related topic

ArithmeticOfEllipticCurves

Defines

weak Mordell-Weil theorem

Defines

rank of an elliptic curve

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