Prime Numbers

320 Chapter 7 ELLIPTIC CURVE ARITHMETIC
[x, y, z] being a solution, the notation indicating that we consider as identical
any two solutions (x, y, z), (x ′ ,y ′ ,z ′ ) of (7.2) if and only if there is a nonzero
t ∈ F with x ′ = tx, y ′ = ty, z ′ = tz.
The projective solutions of (7.2) are almost exactly the same as the affine
solutions of (7.1). In particular, a solution (x, y) of (7.1) may be identified with
the solution [x, y, 1] of (7.2), and any solution [x, y, z] of (7.2) with z = 0may
be identified with the solution (x/z, y/z) of (7.1). The solutions [x, y, z] with
z = 0 do not correspond to any affine solutions, and are called the “points at
infinity” for the equation.
Equations (7.1) and (7.2) are cumbersome. It is profitable to consider
a change in variables that sends solutions with coordinates in F to like
solutions, and vice versa for the inverse transformation. For example, consider
the Fermat equation for exponent 3, namely,
x 3 + y 3 = z 3 .
Assume we are considering solutions in a field F with characteristic not equal
to 2 or 3. Letting X =12z, Y = 36(x − y), Z = x + y, wehavetheequivalent
equation
Y 2 Z = X 3 − 432Z 3 .
The inverse change of variables is x = 1 1
1 1
1
72Y + 2Z, y = − 72Y + 2Z, z = 12X. The projective curve (7.2) is considered to be “nonsingular” (or “smooth”)
over the field F if even over the algebraic closure of F thereisnopoint
[x, y, z] on the curve where all three partial derivatives vanish. In fact, if the
characteristic of F is not equal to 2 or 3, any nonsingular projective equation
(7.2) with at least one solution in F × F × F (with not all of the coordinates
zero) may be transformed by a change of variables to the standard form
y 2 z = x 3 + axz 2 + bz 3 , a,b ∈ F, (7.3)
where the one given solution of the original equation is sent to [0, 1, 0]. Further,
it is clear that a curve given by (7.3) has just this one point at infinity, [0, 1, 0].
Theaffineformis
y 2 = x 3 + ax + b. (7.4)
Such a form for a cubic curve is called a Weierstrass form. It is sometimes
convenient to replace x with (x + constant), and so get another Weierstrass
form:
y 2 = x 3 + Cx 2 + Ax + B, A,B,C ∈ F. (7.5)
If we have a curve in the form (7.4) and the characteristic of F is not 2 or 3,
then the curve is nonsingular if and only if 4a 3 +27b 2 is not 0; see Exercise 7.3.
If the curve is in the form (7.5), the condition that the curve be nonsingular
is more complicated: It is that 4A 3 +27B 2 − 18ABC − A 2 C 2 +4BC 3 =0.
Whether we are dealing with the affine form (7.4) or (7.5), we use the
notation O to denote the one point at infinity [0, 1, 0] that occurs for the
projective form of the curve.
We now make the fundamental definition for this chapter.