criterion of Néron-Ogg-Shafarevich

criterion of Néron-Ogg-Shafarevich

In this entry, we use the following notation.

K

is a

local field

,

complete

with respect to a

discrete valuation

ν

,

R

is the

ring of integers

of

K

,

ℳ

is the

maximal ideal

of

R

and

𝔽

is the

residue field

of

R

.

Definition

.

Let

Ξ

be a set on which

Gal

⁡

(

K

¯

/

K

)

acts. We say that

Ξ

is

unramified

at

ν

if the action of the

inertia group

I

ν

on

Ξ

is

trivial

, i.e.

ζ

σ

=

ζ

for all

σ

∈

I

ν

and for all

ζ

∈

Ξ

.

Theorem

(Criterion of N

𝐞

´

ron-Ogg-Shafarevich).

Let

E

/

K

be an

elliptic curve

defined over

K

. The following are

equivalent

:

1.

E

has

good reduction

over

K

;

2.

E

⁢

[

m

]

is unramified at

ν

for all

m

≥

1

,

gcd

⁡

(

m

,

char

⁡

(

𝔽

)

)

=

1

;

3.

The

Tate module

T

l

⁢

(

E

)

is unramified at

ν

for
some (all) l,

l

≠

char

⁡

(

𝔽

)

;

4.

E

⁢

[

m

]

is unramified at

ν

for infinitely many

integers

m

≥

1

,

gcd

⁡

(

m

,

char

⁡

(

𝔽

)

)

=

1

.

Corollary

.

Let

E

/

K

be an elliptic curve. Then

E

has
potential good reduction if and only if the inertia group

I

ν

acts on

T

l

⁢

(

E

)

through a finite

quotient

for some
prime

l

≠

char

⁡

(

F

)

.

Proof of Corollary.

(

⇒

) Assume that

E

has potential good
reduction. By definition, there exists a

finite extension

of

K

, call it

K

′

, such that

E

/

K

′

has good reduction. We
can extend

K

′

(if necessary) so

K

′

/

K

is a Galois finite extension.

Let

ν

′

and

I

ν

′

be the corresponding

valuation

and inertia group for

K

′

. Then the

theorem

above (
(1)

⇒

(3) ) implies that

T

l

⁢

(

E

)

is unramified at

ν

′

for all

l

,

l

≠

char

⁡

(

𝔽

)

=

char

⁡

(

𝔽

′

)

(since

𝔽

′

is a finite extension of

𝔽

). So

I

ν

′

acts
trivially on

T

l

⁢

(

E

)

for all

l

≠

char

⁡

(

𝔽

′

)

. Thus

I

ν

↪

T

l

⁢

(

E

)

factors through the finite
quotient

I

ν

/

I

ν

′

.

(

⇐

) Let

l

≠

char

⁡

(

𝔽

)

, and assume

I

ν

↪

T

l

⁢

(

E

)

factors through a finite quotient,
say

I

ν

/

J

. Let

K

¯

J

be the

fixed field

of

J

, then

K

¯

J

/

K

¯

I

ν

is a finite extension, so we can find a finite
extension

K

′

/

K

so that

K

¯

J

=

K

′

⁢

K

¯

I

ν

. So
the inertia group of

K

′

is equal to

J

, and

J

acts trivially on

T

l

⁢

(

E

)

. Hence the criterion (
(3)

⇒

(1) ) implies that

E

has good reduction over

K

′

, and since

K

′

/

K

is finite,

E

has
potential good reduction.
∎

Proposition

.

Let

E

/

K

be an elliptic curve. Then

E

has
potential good reduction if and only if its

j

-

invariant

is

integral

( i.e.

j

⁢

(

E

)

∈

R

).

Proof.

(

⇐

) Assume

char

⁡

(

𝔽

)

≠

2

, it is easy to prove
that we can extend

K

to a finite extension

K

′

so that

E

has a

Weierstrass equation

:

E

:

y

2

=

x

⁢

(

x

-

1

)

⁢

(

x

-

λ

)

λ

≠

0

,

1

(1)

Since we are assuming

j

⁢

(

E

)

∈

R

, and:

(

1

-

λ

⁢

(

1

-

λ

)

)

3

-

j

⁢

λ

2

⁢

(

1

-

λ

)

2

=

0

(2)

then

λ

∈

R

and

λ

≠

0

,

1

mod

ℳ

′

(

⇒

Δ

′

∈

(

R

′

)

*

). Hence

E

/

K

′

has
good reduction, i.e.

E

has potential good
reduction.

(

⇒

) Assume that

E

has potential good reduction,
so there exists

K

′

so that

E

/

K

′

has good
reduction. Let

Δ

′

,

c

4

′

the usual quantities associated
to the Weierstrass equation over

K

′

. Since

E

/

K

′

has good reduction,

Δ

′

∈

(

R

′

)

*

, and so

j

⁢

(

E

)

=

(

c

4

′

)

3

Δ

′

∈

R

′

. But since

E

is defined over

K

,

j

⁢

(

E

)

∈

K

, so

j

⁢

(

E

)

∈

K

⁢

⋂

R

′

=

R

.
∎

Title

criterion of Néron-Ogg-Shafarevich

Canonical name

CriterionOfNeronOggShafarevich

Date of creation

2013-03-22 17:14:58

Last modified on

2013-03-22 17:14:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 14H52

Synonym

criterion of Neron-Ogg-Shafarevich

Related topic

EllipticCurve

Related topic

ArithmeticOfEllipticCurves

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