is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.In mathematics, an isogeny is a morphism of varieties between two abelian varieties (e.g. elliptic curves) that is surjective and has a finite kernel. Every isogeny is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.
For elliptic curves, this notion can also be formulated as follows:
Let E1 and E2 be elliptic curves over a field k. An isogeny between E1 and E2 is a surjective morphism 
of varieties that preserves basepoints (i.e. f maps the infinite point on E1 to that on E2).
Two elliptic curves E1 and E2 are called isogenous if there is an isogeny 
. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the elliptic curves.