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).
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