Selmer group

Selmer group

Given an

elliptic curve

E

we can define two very interesting and
important groups, the

Selmer group

and the

Tate-Shafarevich group

, which together provide a measure of
the failure of the

Hasse principle

for elliptic curves, by
measuring whether the curve is everywhere locally soluble. Here we
present the construction of these groups.

Let

E

,

E

′

be elliptic curves defined over

ℚ

and let

ℚ

¯

be an

algebraic closure

of

ℚ

. Let

ϕ

:

E

→

E

′

be an non-constant

isogeny

(for example, we
can let

E

=

E

′

and think of

ϕ

as being the “

multiplication

by

n

” map,

[

n

]

:

E

→

E

). The following standard result
asserts that

ϕ

is surjective over

ℚ

¯

:

Theorem 1

.

Let

C

1

,

C

2

be curves defined over an algebraically closed field

K

and let

ψ

:

C

1

→

C

2

be a

morphism

(or

algebraic map

) of curves. Then

ψ

is either constant or
surjective.

Proof.

See

[

4

]

, Chapter II.6.8.
∎

Since

ϕ

:

E

⁢

(

ℚ

¯

)

→

E

′

⁢

(

ℚ

¯

)

is
non-constant, it must be surjective and we obtain the following

exact sequence

:

0

→

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

→

E

⁢

(

ℚ

¯

)

→

E

′

⁢

(

ℚ

¯

)

→

0

(

1

)

where

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

=

Ker

⁡

ϕ

. Let

G

=

Gal

⁡

(

ℚ

¯

/

ℚ

)

, the

absolute Galois group

of

ℚ

, and consider the

i

t

⁢

h

-

cohomology group

H

i

⁢

(

G

,

E

⁢

(

ℚ

¯

)

)

(we
abbreviate by

H

i

⁢

(

G

,

E

)

). Using

equation

(

1

)

we obtain the
following

long exact sequence

(see

Proposition

1 in

group
cohomology

):

0

→

H

0

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

→

H

0

⁢

(

G

,

E

)

→

H

0

⁢

(

G

,

E

′

)

→

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

→

H

1

⁢

(

G

,

E

)

→

H

1

⁢

(

G

,

E

′

)

(

2

)

Note that

H

0

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

=

(

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

G

=

E

⁢

(

ℚ

)

⁢

[

ϕ

]

and similarly

H

0

⁢

(

G

,

E

)

=

E

⁢

(

ℚ

)

,

H

0

⁢

(

G

,

E

′

)

=

E

′

⁢

(

ℚ

)

From

(

2

)

we can obtain an exact sequence:

0

→

E

′

⁢

(

ℚ

)

/

ϕ

⁢

(

E

⁢

(

ℚ

)

)

→

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

→

H

1

⁢

(

G

,

E

)

⁢

[

ϕ

]

→

0

We could repeat the same procedure but this time for

E

,

E

′

defined over

ℚ

p

,for some

prime number

p

, and obtain
a similar exact sequence but with

coefficients

in

ℚ

p

which relates to the original in the following

commutative diagram

(here

G

p

=

Gal

⁡

(

ℚ

p

¯

/

ℚ

p

)

):

0

→

E

′

⁢

(

ℚ

)

/

ϕ

⁢

(

E

⁢

(

ℚ

)

)

→

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

→

H

1

⁢

(

G

,

E

)

⁢

[

ϕ

]

→

0

↓

↓

↓

0

→

E

′

⁢

(

ℚ

p

)

/

ϕ

⁢

(

E

⁢

(

ℚ

p

)

)

→

H

1

⁢

(

G

p

,

E

⁢

(

ℚ

p

¯

)

⁢

[

ϕ

]

)

→

H

1

⁢

(

G

p

,

E

)

⁢

[

ϕ

]

→

0

The goal here is to find a

finite

group containing

E

′

⁢

(

ℚ

)

/

ϕ

⁢

(

E

⁢

(

ℚ

)

)

. Unfortunately

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

is not necessarily finite. With
this purpose in mind, we define the

ϕ

-

Selmer group

:

S

ϕ

⁢

(

E

/

ℚ

)

=

Ker

⁡

(

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

→

∏

p

H

1

⁢

(

G

p

,

E

)

)

Equivalently, the

ϕ

-

Selmer group

is the set of
elements

γ

of

H

1

⁢

(

G

,

E

⁢

(

ℚ

¯

)

⁢

[

ϕ

]

)

whose
image

γ

p

in

H

1

⁢

(

G

p

,

E

⁢

(

ℚ

𝕡

¯

)

⁢

[

ϕ

]

)

comes
from some element in

E

⁢

(

ℚ

p

)

.

Finally, by imitation of the

definition

of the Selmer group, we
define the

Tate-Shafarevich group

:

T

⁢

S

⁢

(

E

/

ℚ

)

=

Ker

⁡

(

H

1

⁢

(

G

,

E

)

→

∏

p

H

1

⁢

(

G

p

,

E

)

)

The Tate-Shafarevich group is precisely the group that measures
the Hasse principle in the elliptic curve

E

. It is unknown if
this group is finite.

References

1

J.P. Serre,

Galois Cohomology

,
Springer-Verlag, New York.

2

James Milne,

Elliptic Curves

,

http://www.jmilne.org/math/CourseNotes/math679.html

online course
notes.

3

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

4

R. Hartshorne,

Algebraic Geometry

,
Springer-Verlag, 1977.

Title

Selmer group

Canonical

name

SelmerGroup

Date of creation

2013-03-22 13:50:55

Last modified on

2013-03-22 13:50:55

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 14H52

Related topic

GroupCohomology

Related topic

RankOfAnEllipticCurve

Related topic

ArithmeticOfEllipticCurves

Defines

Selmer group

Defines

Tate-Shafarevich group

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