L-series of an elliptic curve

L-series of an elliptic curve

Let

E

be an

elliptic curve

over

ℚ

with

Weierstrass
equation

:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

with

coefficients

a

i

∈

ℤ

. For

p

a prime in

ℤ

, define

N

p

as the number of points in the

reduction

of the curve modulo

p

, this is, the number of points in:

{

O

}

∪

{

(

x

,

y

)

∈

𝔽

p

2

:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

-

x

3

-

a

2

⁢

x

2

-

a

4

⁢

x

-

a

6

≡

0

⁢

m

⁢

o

⁢

d

⁢

p

}

where

O

is the point at infinity. Also, let

a

p

=

p

+

1

-

N

p

. We define the

local part at

p

of
the L-series

to be:

L

p

⁢

(

T

)

=

{

1

-

a

p

⁢

T

+

p

⁢

T

2

⁢

, if

E

has good reduction at

p

,

1

-

T

⁢

, if

E

has split multiplicative reduction at

p

,

1

+

T

⁢

, if

E

has non-split multiplicative reduction at

p

,

1

⁢

, if

E

has additive reduction at

p

.

Definition

.

The L-series of the elliptic curve

E

is defined to
be:

L

⁢

(

E

,

s

)

=

∏

p

1

L

p

⁢

(

p

-

s

)

where the product is over all primes.

Note: The product converges and gives an

analytic function

for all

R

⁢

e

⁢

(

s

)

>

3

/

2

. This follows from the fact that

∣

a

p

∣

≤

2

⁢

p

. However, far more is true:

Theorem

(Taylor, Wiles).

The L-series

L

⁢

(

E

,

s

)

has an

analytic continuation

to the entire

complex plane

, and it satisfies the following functional equation.
Define

Λ

⁢

(

E

,

s

)

=

(

N

E

/

ℚ

)

s

/

2

⁢

(

2

⁢

π

)

-

s

⁢

Γ

⁢

(

s

)

⁢

L

⁢

(

E

,

s

)

where

N

E

/

Q

is the

conductor

of

E

and

Γ

is
the

Gamma function

. Then:

Λ

⁢

(

E

,

s

)

=

w

⁢

Λ

⁢

(

E

,

2

-

s

)

w

⁢

i

⁢

t

⁢

h

⁢

w

=

±

1

The number

w

above is usually called the

root number

of

E

, and it has an important conjectural meaning (see

Birch and
Swinnerton-Dyer conjecture

).

This result was known for elliptic curves having

complex
multiplication

(Deuring, Weil) until the general result was
finally proven.

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.

4

Goro Shimura,

Introduction to the
Arithmetic Theory of

Automorphic Functions

. Princeton University
Press, Princeton, New Jersey, 1971.

Title

L-series of an elliptic curve

Canonical name

LseriesOfAnEllipticCurve

Date of creation

2013-03-22 13:49:43

Last modified on

2013-03-22 13:49:43

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Synonym

L-function of an elliptic curve

Related topic

EllipticCurve

Related topic

DirichletLSeries

Related topic

ConductorOfAnEllipticCurve

Related topic

HassesBoundForEllipticCurvesOverFiniteFields

Related topic

ArithmeticOfEllipticCurves

Defines

L-series of an elliptic curve

Defines

local part of the L-series

Defines

root number

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