Isogeny

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In algebraic geometry, an isogeny between abelian varieties is a rational map which is also a group homomorphism, with finite kernel.

Elliptic curves

As 1-dimensional abelian varieties, elliptic curves provide a convenient introduction to the theory. If is a non-trivial rational map which maps the zero of E1 to the zero of E1, then it is necessarily a group homomorphism. The kernel of φ is a proper subvariety of E1 and hence a finite set of order d, the degree of φ. Conversely, every finite subgroup of E1 is the kernel of some isogeny.

There is a dual isogeny defined by

the sum being taken on E1 of the d points on the fibre over Q. This is indeed an isogeny, and the composite is just multiplication by d.

The curves E1 and E2 are said to be isogenous: this is an equivalence relation on isomorphism classes of elliptic curves.

Examples

Let E1 be an elliptic curve over a field K of characteristic not 2 or 3 in Weierstrass form.

Degree 2

A subgroup of order 2 on E1 must be of the form where P = (e,0) with e being a root of the cubic in X. Translating so that e=0 and the curve has equation , the map

is an isogeny from E1 to the isogenous curve E2 with equation .

Degree 3

A subgroup of order 3 must be of the form where x is in K but y need not be. We shall assume that (by taking a quadratic twist if necessary). Translating, we can put E in the form . The map

is an isogeny from E1 to the isogenous curve E2 with equation .

Elliptic curves over the complex numbers

An elliptic curve over the complex numbers is isomorphic to a quotient of the complex numbers by some lattice. If E1 = C/L1, and L1 is a sublattice of L2 of index d, then E2 = C/L2 is an isogenous curve. Representing the homothety class of a lattice by a point τ in the upper half-plane, the isogenous curves correspond to the lattices with moduli

with a.c = d and b=0,1,...,c-1.

Elliptic curves over finite fields

Isogenous elliptic curves over a finite field have the same number of points (although not necessarily the same group structure). The converse is also true: this is the Honda-Tate theorem.

References