# 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 $\phi: E_1 \rightarrow E_2$ 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 Y2 = X3 + AX2 + 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 Y2 = X3 − 2AX2 + (A2 − 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 Y2 + XY + mY = X3. 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 Y2 + XY + 3mY = X3 − 6mX − (m + 9m2).

### 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.