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 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 defined by
The curves E1 and E2 are said to be isogenous: this is an equivalence relation on isomorphism classes of elliptic curves.
Let E1 be an elliptic curve over a field K of characteristic not 2 or 3 in Weierstrass form.
A subgroup of order 2 on E1 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 Y2 = X3 + AX2 + BX, the map
is an isogeny from E1 to the isogenous curve E2 with equation Y2 = X3 − 2AX2 + (A2 − 4B)X.
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 Y2 + XY + mY = X3. The map
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
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.
- 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.
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