Selmer group
Given an elliptic curve 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 be elliptic curves defined over and let
be an algebraic closure of . Let
be an non-constant isogeny
(for example, we
can let and think of as being the “multiplication
by ” map, ). The following standard result
asserts that is surjective over :
Theorem 1.
Let be curves defined over an algebraically closed field and let
be a morphism (or
algebraic map) of curves. Then is either constant or
surjective.
Proof.
See [4], Chapter II.6.8. ∎
Since is
non-constant, it must be surjective and we obtain the following
exact sequence:
where . Let
, the
absolute Galois group of , and consider the
-cohomology group (we
abbreviate by ). Using equation we obtain the
following long exact sequence (see Proposition 1 in group
cohomology
):
Note that
and similarly
From we can obtain an exact sequence:
We could repeat the same procedure but this time for
defined over ,for some prime number , and obtain
a similar exact sequence but with coefficients in
which relates to the original in the following commutative diagram
(here ):
The goal here is to find a finite group containing . Unfortunately is not necessarily finite. With this purpose in mind, we define the -Selmer group:
Equivalently, the -Selmer group is the set of elements of whose image in comes from some element in .
Finally, by imitation of the definition of the Selmer group, we define the Tate-Shafarevich group:
The Tate-Shafarevich group is precisely the group that measures the Hasse principle in the elliptic curve . 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.htmlonline 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 |