# 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 is a non-trivial rational map which maps the zero of *E*_{1} to the zero of *E*_{1}, then it is necessarily a group homomorphism. The kernel of φ is a proper subvariety of *E*_{1} and hence a finite set of order *d*, the *degree* of φ. Conversely, every finite subgroup of *E*_{1} is the kernel of some isogeny.

There is a *dual isogeny* defined by

the sum being taken on *E*_{1} of the *d* points on the fibre over *Q*. This is indeed an isogeny, and the composite is just multiplication by *d*.

The curves *E*_{1} and *E*_{2} are said to be *isogenous*: this is an equivalence relation on isomorphism classes of elliptic curves.

### Examples

Let *E*_{1} 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 *E*_{1} 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 *E*_{1} to the isogenous curve *E*_{2} 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 *E*_{1} to the isogenous curve *E*_{2} 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 *E*_{1} = **C**/*L*_{1}, and *L*_{1} is a sublattice of *L*_{2} of index *d*, then *E*_{2} = **C**/*L*_{2} 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

- J.W.S. Cassels (1991).
*Lectures on Elliptic Curves*. Cambridge University Press, 58-65. ISBN 0-521-42530-1. - Dale Husemöller (1987).
*Elliptic curves*. Springer-Verlag, 91-96,163. ISBN 0-387-96371-5. - Joseph H. Silverman (1986).
*The Arithmetic of Elliptic Curves*. Springer-Verlag, 70-79,84-90. ISBN 0-387-96203-4.