# 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 ${\displaystyle \phi :E_{1}\rightarrow E_{2}}$ 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 ${\displaystyle {\hat {\phi }}:E_{2}\rightarrow E_{1}}$ defined by

${\displaystyle {\hat {\phi }}:Q\mapsto \sum _{P:\phi (P)=Q}P,\,}$

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

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

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

#### Degree 3

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

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

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

### 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

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

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.