Recall that an elliptic curve is an abelian variety; in particular, its set of points admits a group structure. A point of is a torsion point if it is a torsion element of this group structure.
If is defined over a number field , the Galois theory of interacts very well with the torsion points of , as we describe below.
Let be an elliptic curve over a field , and let be an integer. An -torsion point of is a point of such that , that is, in the (abelian) group structure on (the set of points of) , where denotes the identity element of the group structure (often taken to be the point at infinity).
A torsion point of is a point of which is an -torsion point of for some integer .
For a fixed integer , the set of -torsion points of assembles into an abelian group with respect to the group structure of . The same is true of the set of all torsion points of .
For a fixed integer , the set of -torsion points of is often denoted .
The following observation is used frequently when working with torsion points of an elliptic curve over a number field.
Let be an elliptic curve defined over , the rationals. Then for any integer , there is an isomorphism of abelian groups .
Thus if is a prime, so that is a field , we can think of the set of automorphisms of the abelian group as the matrix group, specifically the general linear group, .
Given an elliptic curve defined over a number field , the following observation ties the Galois theory of to the torsion points of . We shall denote the algebraic closure of by .
Let be an elliptic curve defined over a number field . Let be an extension of . The Galois group acts on all of (the set of points of) , the (abelian group of) torsion points of , and the (abelian group of) -torsion points of for any fixed integer , in the obvious way: given an automorphism of which fixes , we send a point of to .
The principal point is that given and as in Proposition , one can check that is still a point of , and is still an -torsion point for some if is.
Given Remark , we can, as with any group action, reformulate Proposition in the case of -torsion points for a fixed prime as follows.
Let be a prime. Let be an elliptic curve defined over a number field . Let be an extension of . Then the application of automorphisms of to points of determines a group homomorphism .
Given the homomorphism of Corollary , we can take its kernel. This subgroup of determines, by Galois theory, a field extension of . Let us denote it by . In fact, is simply the tensor product of with ; from that point of view, we have demonstrated that this tensor product is a finite Galois extension of .
Created on March 31, 2021 at 01:23:28. See the history of this page for a list of all contributions to it.