the arithmetic of elliptic curves

the arithmetic of elliptic curves

The Arithmetic of Elliptic Curves

An

elliptic curve

over a field

K

is a projective

nonsingular

curve

E

defined over

K

of genus

1

together with a point

O

∈

E

defined over

K

. In the simple case

K

=

ℚ

every elliptic curve is

isomorphic

(over

ℚ

) to a curve defined by an

equation

of the form:

y

2

=

x

3

+

A

⁢

x

+

B

where

A

,

B

are

integers

. The most remarkable feature of an elliptic curve is the fact that the group of points can be given the

structure

of a group.

The theory of elliptic curves is a very rich mix of

algebraic geometry

and

number theory

(

arithmetic geometry

). As in many other areas of number theory, the

concepts

are simple to state but the theory is extremely deep and beautiful. The intrinsic

arithmetic

of the points on an elliptic curve is absolutely compelling. The most prominent mathematicians of our time have contributed in the development of the theory. The ultimate goal of the theory is to completely understand the structure of the points on the elliptic curve over any field

F

and being able to find them.

1.1

Basic Definitions

1.

For a basic exposition of the subject the reader should start with the entry elliptic curve (

http://planetmath.org/EllipticCurve

) (defines elliptic curve, the group law and gives some examples with graphs, also treats elliptic curves over the

complex numbers

).

2.

Some basic objects attached to an elliptic curve:

j

-

invariant

(

http://planetmath.org/JInvariant

),

discriminant

and invariant differential. The

j

-invariant classifies elliptic curves up to

isomorphism

(

http://planetmath.org/JInvariantClassifiesEllipticCurvesUpToIsomorphism

).

3.

Isogeny

, the

dual isogeny

and the

Frobenius morphism

.

4.

Elliptic curves over

finite fields

:

good reduction

, bad reduction, multiplicative reduction, additive reduction, cusp, node.

5.

One of the most important objects that one can

associate

to an elliptic curve is the

L

-series (

http://planetmath.org/LSeriesOfAnEllipticCurve

) (the entry defines the

L

-series of an elliptic curve (

http://planetmath.org/LSeriesOfAnEllipticCurve

) and also talks about analytic continuation).

6.

The

conductor

of an elliptic curve is an integer quantity that measures the arithmetic

complexity

of the curve (the entry contains examples).

7.

The

Tate module

of an elliptic curve (it is also defined in the entry

inverse limit

).

8.

The

canonical height

on an elliptic curve (over

ℚ

).

9.

The

height matrix

and the elliptic regulator of an elliptic curve.

1.2

Elliptic Curves over Finite Fields

1.

See bad reduction (

http://planetmath.org/BadReduction2

).

2.

The criterion of Néron-Ogg-Shafarevich.

3.

Supersingular

reduction

.

4.

Hasse’s bound for elliptic curves over finite fields.

1.3

The Mordell-Weil Group

E

⁢

(

K

)

1.

The structure of

E

⁢

(

K

)

is given by the

Mordell-Weil theorem

(see also this entry (

http://planetmath.org/RankOfAnEllipticCurve

)). The main two ingredients of the proof of the

theorem

are the concept of height function and the so-called descent theorem.

2.

The free rank of the

abelian group

E

⁢

(

K

)

is called the rank of an elliptic curve (

http://planetmath.org/RankOfAnEllipticCurve

) (the entry contains examples).

3.

Together with the Mordell-Weil group, one defines two other rather important groups: the

Selmer groups

and the Tate-Shafarevich group. The Tate-Shafarevich group (or “Sha”) measures the failure of the

Hasse principle

on the elliptic curve.

4.

Some examples:

Mordell curves

.

1.4

The Torsion Subgroup of

E

⁢

(

K

)

1.

The

Nagell-Lutz Theorem

.

2.

Mazur’s theorem on

torsion

of elliptic curves (a classification of all possible torsion subgroups).

3.

Examples of torsion subgroups of elliptic curves

(includes examples of all possible

subgroups

).

4.

A way to determine the torsion group: the torsion subgroup of an elliptic curve injects in the reduction of the curve.

1.5

Computing the Rank

1.

Read about the rank (

http://planetmath.org/RankOfAnEllipticCurve

).

2.

A bound for the rank of an elliptic curve.

1.6

Complex Multiplication

1.

Definition of the

endomorphism ring

(

http://planetmath.org/EndomorphismRing

) and

complex multiplication

.

2.

Examples of elliptic curves with complex multiplication

.

3.

A

connection

between complex multiplication and

class field theory

:

abelian extensions of quadratic imaginary number fields

.

4.

Definition of Grössencharacters, in general.

1.7

Famous Problems and Conjectures

1.

Fermat’s Last Theorem was finally solved using the theory of elliptic curves and

modular forms

.

2.

The

Birch and Swinnerton-Dyer conjecture

(relating the

L

-series of an elliptic curve with the

algebraic

rank).

3.

The

Taniyama-Shimura-Weil Conjecture

(now a theorem!).

1.8

Cryptography

1.

Cryptography and Number Theory

.

2.

The

elliptic curve discrete logarithm problem

.

3.

The

Diffie-Hellman key exchange

.

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.

Note: If you want to contribute to this entry, please send an email to the author (alozano).

Title

the arithmetic of elliptic curves

Canonical name

TheArithmeticOfEllipticCurves

Date of creation

2013-03-22 15:06:19

Last modified on

2013-03-22 15:06:19

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Topic

Classification

msc 14H52

Classification

msc 11G05

Synonym

concepts in the theory of elliptic curves

Related topic

EllipticCurve

Related topic

CriterionOfNeronOggShafarevich

Related topic

HassesBoundForEllipticCurvesOverFiniteFields

Related topic

BirchAndSwinnertonDyerConjecture2

Related topic

RankOfAnEllipticCurve

Related topic

MazursTheoremOnTorsionOfEllipticCurves

Related topic

TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve

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