bad reduction

bad reduction

1

Singular Cubic Curves

Let

E

be a

cubic curve

over a field

K

with

Weierstrass
equation

f

⁢

(

x

,

y

)

=

0

, where:

f

⁢

(

x

,

y

)

=

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

-

x

3

-

a

2

⁢

x

2

-

a

4

⁢

x

-

a

6

which has a

singular point

P

=

(

x

0

,

y

0

)

. This is

equivalent

to:

∂

⁡

f

/

∂

⁡

x

⁢

(

P

)

=

∂

⁡

f

/

∂

⁡

y

⁢

(

P

)

=

0

and so we can write the

Taylor expansion

of

f

⁢

(

x

,

y

)

at

(

x

0

,

y

0

)

as follows:

f

⁢

(

x

,

y

)

-

f

⁢

(

x

0

,

y

0

)

=

λ

1

⁢

(

x

-

x

0

)

2

+

λ

2

⁢

(

x

-

x

0

)

⁢

(

y

-

y

0

)

+

λ

3

⁢

(

y

-

y

0

)

2

-

(

x

-

x

0

)

3

=

[

(

y

-

y

0

)

-

α

⁢

(

x

-

x

0

)

]

⁢

[

(

y

-

y

0

)

-

β

⁢

(

x

-

x

0

)

]

-

(

x

-

x

0

)

3

for some

λ

i

∈

K

and

α

,

β

∈

K

¯

(an

algebraic closure

of

K

).

Definition 1

.

The singular point

P

is a

node

if

α

≠

β

. In this
case there are two different tangent lines to

E

at

P

, namely:

y

-

y

0

=

α

⁢

(

x

-

x

0

)

,

y

-

y

0

=

β

⁢

(

x

-

x

0

)

If

α

=

β

then we say that

P

is a

cusp

, and there is a
unique tangent line at

P

.

Note: See the entry for

elliptic curve

for examples of cusps and
nodes.

There is a very simple criterion to know whether a cubic curve in

Weierstrass form

is

singular

and to differentiate nodes from
cusps:

Proposition 1

.

Let

E

/

K

be given by a Weierstrass equation, and let

Δ

be
the

discriminant

and

c

4

as in the definition of

Δ

. Then:

1.

E

is singular if and only if

Δ

=

0

,

2.

E

has a node if and only if

Δ

=

0

and

c

4

≠

0

,

3.

E

has a cusp if and only if

Δ

=

0

=

c

4

.

Proof.

See

[

2

]

, chapter III,

Proposition

1.4, page 50.
∎

2

Reduction of Elliptic Curves

Let

E

/

ℚ

be an elliptic curve (we could work over any

number field

K

, but we choose

ℚ

for simplicity in the
exposition). Assume that

E

has a

minimal model

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

in

ℤ

. Let

p

be a prime in

ℤ

. By reducing
each of the coefficients

a

i

modulo

p

we obtain the equation
of a cubic curve

E

~

over the

finite field

𝔽

p

(the field with

p

elements

).

Definition 2

.

1.

If

E

~

is a

non-singular

curve then

E

~

is an elliptic curve over

𝔽

p

and we say that

E

has

good reduction

at

p

. Otherwise, we say that

E

has

bad
reduction

at

p

.

2.

If

E

~

has a cusp then we say that

E

has

additive reduction

at

p

.

3.

If

E

~

has a node then we say that

E

has

multiplicative reduction

at

p

. If the slopes of the tangent
lines (

α

and

β

as above) are in

𝔽

p

then
the

reduction

is said to be

split

multiplicative

(and

non-split

otherwise).

From

Proposition 1

we deduce the following:

Corollary 1

.

Let

E

/

Q

be an elliptic curve with coefficients in

Z

. Let

p

∈

Z

be a prime. If

E

has bad
reduction at

p

then

p

∣

Δ

.

Examples

:

1.

E

1

:

y

2

=

x

3

+

35

⁢

x

+

5

has good reduction at

p

=

7

.

2.

However

E

1

has bad reduction at

p

=

5

, and the reduction
is

additive

(since modulo

5

we can write the equation as

[

(

y

-

0

)

-

0

⁢

(

x

-

0

)

]

2

-

x

3

and the slope is

0

).

3.

The elliptic curve

E

2

:

y

2

=

x

3

-

x

2

+

35

has bad
multiplicative reduction at

5

and

7

. The reduction at

5

is split,
while the reduction at

7

is non-split. Indeed, modulo

5

we could
write the equation as

[

(

y

-

0

)

-

2

⁢

(

x

-

0

)

]

⁢

[

(

y

-

0

)

+

2

⁢

(

x

-

0

)

]

-

x

3

, being
the slopes

2

and

-

2

. However, for

p

=

7

the slopes are not in

𝔽

7

(

-

1

is not in

𝔽

7

).

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

bad reduction

Canonical name

BadReduction

Date of creation

2013-03-22 13:49:21

Last modified on

2013-03-22 13:49:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

12

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Related topic

EllipticCurve

Related topic

JInvariant

Related topic

HassesBoundForEllipticCurvesOverFiniteFields

Related topic

TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve

Related topic

ArithmeticOfEllipticCurves

Related topic

SingularPointsOfPlaneCurve

Defines

bad reduction

Defines

good reduction

Defines

cusp

Defines

node

Defines

multiplicative reduction

Defines

additive reduction

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