Birch and Swinnerton-Dyer conjecture

Birch and Swinnerton-Dyer conjecture

Let

E

be an

elliptic curve

over

ℚ

, and let

L

⁢

(

E

,

s

)

be the L-series attached to

E

.

Conjecture 1

(Birch and Swinnerton-Dyer).

1.

L

⁢

(

E

,

s

)

has a zero at

s

=

1

of order equal to the rank of

E

⁢

(

ℚ

)

.

2.

Let

R

=

rank

⁡

(

E

⁢

(

ℚ

)

)

. Then the

residue

of

L

⁢

(

E

,

s

)

at

s

=

1

, i.e.

lim

s

→

1

⁡

(

s

-

1

)

-

R

⁢

L

⁢

(

E

,

s

)

has a concrete
expression involving the following

invariants

of

E

: the real
period, the

Tate-Shafarevich group

, the

elliptic regulator

and the

Neron model of

E

.

J. Tate said about this conjecture: “

This remarkable conjecture relates the behavior of a

function

L

at a point where it is not at present known to be defined to the order of a group (Sha) which is not known to be finite!

” The precise statement of the conjecture asserts that:

lim

s

→

1

⁡

L

⁢

(

E

,

s

)

(

s

-

1

)

R

=

|

Sha

|

⋅

Ω

⋅

Reg

⁡

(

E

/

ℚ

)

⋅

∏

p

c

p

|

E

tors

⁢

(

ℚ

)

|

2

where

•

R

is the rank of

E

/

ℚ

.

•

Ω

is either the real period or twice the real period of a

minimal model

for

E

, depending on whether

E

⁢

(

ℝ

)

is

connected

or not.

•

|

Sha

|

is the order of the Tate-Shafarevich group of

E

/

ℚ

.

•

Reg

⁡

(

E

/

ℚ

)

is the

http://planetmath.org/node/RegulatorOfAnEllipticCurve

elliptic regulator of

E

⁢

(

ℚ

)

.

•

|

E

tors

⁢

(

ℚ

)

|

is the number of

torsion

points on

E

/

ℚ

(including the

point at infinity

O

).

•

c

p

is an

elementary

local factor, equal to the

cardinality

of

E

⁢

(

ℚ

p

)

/

E

0

⁢

(

ℚ

p

)

, where

E

0

⁢

(

ℚ

p

)

is the set of points in

E

⁢

(

ℚ

p

)

whose

reduction

modulo

p

is

non-singular

in

E

⁢

(

𝔽

p

)

. Notice that if

p

is a prime of

good reduction

for

E

/

ℚ

then

c

p

=

1

, so only

c

p

≠

1

only for finitely many primes

p

. The number

c

p

is usually called the Tamagawa number of

E

at

p

.

The following is an easy

consequence

of the B-SD conjecture:

Conjecture 2

(Parity Conjecture).

The

root number

of

E

, denoted by

w

, indicates the parity of
the rank of the elliptic curve, this is,

w

=

1

if and only if the
rank is even.

There has been a great amount of research towards the B-SD conjecture.
For example, there are some particular cases which are already
known:

Theorem 1

(Coates, Wiles).

Suppose

E

is an elliptic curve defined over an

imaginary quadratic
field

K

, with

complex multiplication

by

K

, and

L

⁢

(

E

,

s

)

is the
L-series of

E

. If

L

⁢

(

E

,

1

)

≠

0

then

E

⁢

(

K

)

is finite.

References

1

Claymath Institute,

Description

,

http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/

online.

2

J. Coates, A. Wiles,

On the Conjecture of
Birch and Swinnerton-Dyer

, Inv. Math. 39, 223-251 (1977).

3

Keith Devlin,

The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time

, 189 - 212, Perseus Books Group, New York (2002).

4

James Milne,

Elliptic Curves

,

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

online course
notes.

5

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

6

Joseph H. Silverman,

Advanced Topics in
the Arithmetic of Elliptic Curves

. Springer-Verlag, New York,
1994.

Title

Birch and Swinnerton-Dyer conjecture

Canonical name

BirchAndSwinnertonDyerConjecture

Date of creation

2013-03-22 13:49:46

Last modified on

2013-03-22 13:49:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

16

Author

alozano (2414)

Entry type

Conjecture

Classification

msc 14H52

Synonym

BS-D conjecture

Related topic

EllipticCurve

Related topic

RegulatorOfAnEllipticCurve

Related topic

MordellCurve

Related topic

ArithmeticOfEllipticCurves

Defines

Birch and Swinnerton-Dyer conjecture

Defines

parity conjecture

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