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 $Y^{2}=X^{3}-2AX^{2}+(A^{2}-4B)X$.

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

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

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.