the torsion subgroup of an elliptic curve injects in the reduction of the curve
Let be an elliptic curve defined over and let be a prime. Let
be a minimal Weierstrass equation for , with coefficients . Let be the reduction of modulo (see bad reduction) which is a curve defined over . The curve can also be considered as a curve over the -adics, , and, in fact, the group of rational points injects into . Also, the groups and are related via the reduction map:
There is an exact sequence of abelian groups
where the right-hand side map is restricted to .
Notation: Given an abelian group , we denote by the -torsion of , i.e. the points of order .
Remark: Part of the proposition is quite useful when trying to compute the torsion subgroup of . As we mentioned above, injects into . The proposition can be reworded as follows: for all primes which do not divide , must be injective and therefore the number of -torsion points divides the number of points defined over .
Let be given by
The discriminant of this curve is . Recall that if is a prime of bad reduction, then . Thus the only primes of bad reduction are , so is non-singular for all .
Let and consider the reduction of modulo , . Then we have
where all the coordinates are to be considered modulo (remember the point at infinity!). Hence . Similarly, we can prove that .
so must be 1.
For the case be know that divides . But it is easy to see that if is non-trivial, then divides its order. Since does not divide , we conclude that must be trivial. Similarly is trivial as well. Therefore has trivial torsion subgroup.
and the group is generated by .
|Title||the torsion subgroup of an elliptic curve injects in the reduction of the curve|
|Date of creation||2013-03-22 13:55:47|
|Last modified on||2013-03-22 13:55:47|
|Last modified by||alozano (2414)|