2.2. Definition 21

but with the understanding that in this new model we may have left

out some points of E at infinity (i.e., those points [X, Y, 0] satisfying

Eq. 2.2).

In general, one can find a change of coordinates that simplifies

Eq. 2.3 enormously:

Proposition 2.2.2. Let E be an elliptic curve, given by Eq. 2.2,

defined over a field K of characteristic different from 2 or 3. Then

there exists a curve E given by

zy2

=

x3

+

Axz2

+

Bz3,

A, B ∈ K with

4A3

+

27B2

= 0

and an invertible change of variables ψ : E → E of the form

ψ([X, Y, Z]) =

f1(X, Y, Z)

g1(X, Y, Z)

,

f2(X, Y, Z)

g2(X, Y, Z)

,

f3(X, Y, Z)

g3(X, Y, Z)

where fi and gi are polynomials with coeﬃcients in K for i = 1,2,3,

and the origin O is sent to the point [0,1,0] of E, i.e., ψ(O) = [0,1,0].

The existence of such a change of variables is a consequence of

the Riemann-Roch theorem of algebraic geometry (for a proof of the

proposition see [Sil86], Chapter III.3). The reference [SiT92], Ch. I.

3, gives an explicit method to find the change of variables ψ : E → E.

See also pages 46-49 of [Mil06].

A projective equation of the form

zy2

=

x3

+

Axz2

+

Bz3,

or

y2

=

x3+Ax+B

in aﬃne coordinates, is called a Weierstrass equation.

From now on, we will often work with an elliptic curve in this form.

Notice that a curve E given by a Weierstrass equation y2 = x3 +Ax+

B is non-singular if and only if 4A3 + 27B2 = 0, and it has a unique

point at infinity, namely [0,1,0], which we shall call the origin O or

the point at infinity of E.

Sometimes we shall use a more general Weierstrass equation

y2

+ a1xy + a3y =

x3

+

a2x2

+ a4x + a6

with ai ∈ Q (we will explain the funky choice of notation for the

coeﬃcients later), but most of the time we will work with equations

of the form

y2

=

x3

+ Ax + B. It is easy to come up with a change

of variables from one form to the other (see Exercise 2.12.3).