minimal model for an elliptic curve

minimal model for an elliptic curve

Let

K

be a

local field

,

complete

with respect to a

discrete valuation

ν

(for example,

K

could be

ℚ

p

, the field of

http://planetmath.org/node/PAdicIntegers

p

-adic numbers, which is complete with respect to the

http://planetmath.org/node/PAdicValuation

p

-adic

valuation

).

Let

E

/

K

be an

elliptic curve

defined over

K

given by a

Weierstrass equation

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

where

a

1

,

a

2

,

a

3

,

a

4

,

a

6

are

constants

in

K

. By a suitable change of variables, we may assume that

ν

⁢

(

a

i

)

≥

0

. As it is pointed out in

http://planetmath.org/node/WeierstrassEquationOfAnEllipticCurve

this entry, any other Weierstrass equation for

E

is obtained by a change of variables of the form

x

=

u

2

⁢

x

′

+

r

,

y

=

u

3

⁢

y

′

+

s

⁢

u

2

⁢

x

′

+

t

with

u

,

r

,

s

,

t

∈

K

and

u

≠

0

. Moreover, by

Proposition

2 in the same entry, the

discriminants

of both equations

satisfy

Δ

=

u

12

⁢

Δ

′

, so they only differ by a

12

th power of a non-zero number in

K

. Let us define a set:

S

=

{

ν

⁢

(

Δ

)

:

Δ

⁢

is the discriminant of a Weierstrass eq. for

E

and

⁢

ν

⁢

(

Δ

)

≥

0

}

Since

ν

is a discrete valuation, the set

S

is a set of non-negative

integers

, therefore it has a minimum value

m

∈

S

. Moreover, by the remark above,

m

satisfies

0

≤

m

<

12

and

m

is the unique number

t

∈

S

with

0

≤

t

<

12

.

Definition

.

Let

E

/

K

be an elliptic curve over a local field

K

, complete with respect to a discrete valuation

ν

. A Weierstrass equation for

E

with discriminant

Δ

is said to be a

minimal model

for

E

(at

ν

) if

ν

⁢

(

Δ

)

=

m

, the minimum of the set

S

above.

It follows from the discussion above that every elliptic curve over a local field

K

has a minimal model over

K

.

Definition

.

Let

F

be a

number field

and let

ν

be an

infinite

or

finite place

(

archimedean

or non-archimedean prime) of

F

. Let

E

/

F

be an elliptic curve over

F

. A given Weierstrass model for

E

/

F

is said to be

minimal

at

ν

if the same model is minimal over

F

ν

, the

completion

of

F

at

ν

. A Weierstrass equation for

E

/

F

is said to be minimal if it is minimal at

ν

for all places

ν

of

F

.

It can be shown that all elliptic curves over

ℚ

have a global minimal model. However, this is not true over general number fields. There exist elliptic curves over a number field

F

which do not have a global minimal model (i.e. any given model is not minimal at

ν

for every

ν

).

Title

minimal model for an elliptic curve

Canonical name

MinimalModelForAnEllipticCurve

Date of creation

2013-03-22 15:48:03

Last modified on

2013-03-22 15:48:03

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Classification

msc 11G05

Classification

msc 11G07

Synonym

minimal equation

Defines

minimal model

Generated on Fri Feb 9 17:58:57 2018 by

LaTeXML