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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.


[edit] 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.

[edit] Examples

Let E1 be an elliptic curve over a field K of characteristic not 2 or 3 in Weierstrass form.

[edit] 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.

[edit] 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).

[edit] 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.

[edit] 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.

[edit] References

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