the Grössencharacter associated to a CM elliptic curve

the Grössencharacter associated to a CM elliptic curve

Let

K

be a

quadratic imaginary field

and let

A

/

F

be an

elliptic curve

defined over a

number field

F

(such that

K

⊂

F

), with

complex multiplication

by

K

. The so-called
‘Main Theorem of Complex Multiplication’ (

[

2

]

, Thm. 8.2)
implies the existence of a Grössencharacter of

F

,

ψ

A

/

F

:

𝒜

F

∗

→

ℂ

∗

associated to the
curve

A

/

F

satisfying several interesting properties which we
collect in the following statement.

Theorem

(

[

2

]

, Thm. 9.1, Prop. 10.4, Cor. 10.4.1).

Let

℘

be a prime of

F

of

good reduction

for

A

/

F

, i.e. the

reduction

A

~

/

F

of

A

modulo

℘

is
smooth. There exists a Grössencharacter of

F

,

ψ

A

/

F

:

A

F

∗

→

C

∗

, such that:

1.

ψ

A

/

F

is

unramified

at a prime

𝔔

of

F

if and only if

A

/

F

has good reduction at

𝔔

;

2.

ψ

A

/

F

⁢

(

℘

)

belongs to

𝒪

K

, thus

multiplication

by

[

ψ

A

/

F

⁢

(

℘

)

]

is a well defined

endomorphism

of

A

/

F

.
Moreover

N

ℚ

F

⁢

(

℘

)

=

N

ℚ

K

⁢

(

ψ

A

/

F

⁢

(

℘

)

)

;

3.

the following

diagram

is

commutative

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

>

[

d

]

⁢

\ar

⁢

@

-

>

[

r

]

[

ψ

A

/

F

⁢

(

℘

)

]

⁢

&

⁢

A

⁢

\ar

⁢

@

-

>

[

d

]

⁢

&

⁢

A

~

⁢

\ar

⁢

@

-

>

[

r

]

ϕ

℘

⁢

&

⁢

A

~

⁢

&

where

ϕ

℘

:

A

~

→

A

~

be the

N

ℚ

F

⁢

(

℘

)

-power

Frobenius map

and the vertical maps are
reduction mod

℘

;

4.

let

|

A

~

⁢

(

𝒪

F

/

℘

)

|

be the number of
points in

A

~

over the

finite field

𝒪

F

/

℘

and put

a

℘

=

N

ℚ

F

⁢

(

℘

)

+

1

-

|

A

~

⁢

(

𝒪

F

/

℘

)

|

. Then

a

℘

=

ψ

A

/

F

⁢

(

℘

)

+

ψ

A

/

F

⁢

(

℘

)

¯

=

2

⋅

ℜ

⁡

(

ψ

A

/

F

⁢

(

℘

)

)

.

5.

(due to Deuring) let

L

⁢

(

A

/

F

,

s

)

be the

L

-function
associated to the elliptic curve

A

/

F

. If

K

⊂

F

then

L

⁢

(

A

/

F

,

s

)

=

L

⁢

(

ψ

A

/

F

,

s

)

⁢

L

⁢

(

ψ

A

/

F

¯

,

s

)

. If

K

⊈

F

, and

F

′

=

F

⁢

K

, then

L

⁢

(

E

/

F

,

s

)

=

L

⁢

(

ψ

A

/

F

′

,

s

)

.

In particular, if

h

K

=

1

then

A

is defined over

K

(actually,
it may be defined over

ℚ

),

ψ

A

/

K

⁢

(

℘

)

is a

generator

of

℘

(by part (2), and the explicit generator can be pinned
down using part (4)). Thus, if

e

is the number of

roots of unity

in

K

, then

ψ

A

/

K

k

⁢

(

℘

)

=

α

k

where

α

is

any

generator of

℘

. Also, by part (5),

L

⁢

(

A

/

ℚ

,

s

)

=

L

⁢

(

ψ

A

/

K

,

s

)

.

References

1

J. H. Silverman,

The

Arithmetic of
Elliptic Curves

, Springer-Verlag, New York.

2

J. H. Silverman,

Advanced Topics in
the Arithmetic of Elliptic Curves

. Springer-Verlag, New York,
1994.

Title

the Grössencharacter associated to a CM elliptic curve

Canonical name

TheGrossencharacterAssociatedToACMEllipticCurve

Date of creation

2013-03-22 15:45:29

Last modified on

2013-03-22 15:45:29

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 11G05

Related topic

Grossencharacter

Related topic

EllipticCurve

Defines

grossencharacter associated to an elliptic curve

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