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 :
See , 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):
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.
- 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.
|Date of creation||2013-03-22 13:50:55|
|Last modified on||2013-03-22 13:50:55|
|Last modified by||alozano (2414)|