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).

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.