Isogeny

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 $$\phi: E_1 \rightarrow E_2$$ 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 $$\hat\phi: E_2 \rightarrow E_1$$ defined by


 * $$\hat\phi : Q \mapsto \sum_{P: \phi(P)=Q} P ,\,$$

the sum being taken on E1 of the d points on the fibre over Q. This is indeed an isogeny, and the composite $$\phi \cdot \hat\phi$$ 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 $$\{\mathcal{O}, P \}$$ where P = (e,0) with e being a root of the cubic in X. Translating so that e=0 and the curve has equation $$Y^2 = X^3 + AX^2 + BX$$, the map


 * $$ (X,Y) \mapsto (X+B/X+A,Y-BY/X^2) \,$$

is an isogeny from E1 to the isogenous curve E2 with equation $$Y^2 = X^3 - 2A X^2 + (A^2-4B)X$$.

Degree 3
A subgroup of order 3 must be of the form $$\{\mathcal{O}, (x,\pm y)\}$$ where x is in K but y need not be. We shall assume that $$y \in K$$ (by taking a quadratic twist if necessary). Translating, we can put E in the form $$Y^2 + XY + mY = X^3$$. The map



(X,Y) \mapsto \left(X - {m Y \over X^2} + {m X \over Y},                       Y - {m^2 Y \over X^3} - {m X^3 \over Y^2} \right) $$

is an isogeny from E1 to the isogenous curve E2 with equation $$Y^2 + XY + 3mY = X^3 - 6mX - (m+9m^2)$$.

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


 * $$ \frac{a\tau + b}{c} \,$$

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.