ABSTRACT

Definition 36.1. Let R be a field in which 2 = 1 + 1 6= 0, 3 = 1 + 1 + 1 6= 0. Equivalently R does not contain F2 or F3. A subset Z = Z(R) ⊂ R × R is called an affine elliptic curve if there exist a, b ∈ R with 4a3 + 27b2 6= 0 such that

Z(R) = {(x, y) ∈ R×R; y2 = x3 + ax+ b}. We call Z(R) the elliptic curve over R defined by the equation y2 = x3 + ax + b. Next we define the projective elliptic curve defined by the equation y2 = x3 +ax+b as the set

E(R) = Z(R) ∪ {∞} where ∞ is an element not belonging to Z(R). (We usually drop the word “projective” and we call ∞ the point at infinity on E(R).) If (x, y) ∈ E(R) define (x, y)′ = (x,−y). Also define ∞′ = ∞. Next we define a binary operation ? on E(R) called the chord-tangent operation; we will see that E(R) becomes a group with respect to this operation. First define (x, y) ? (x,−y) = ∞, ∞?(x, y) = (x, y)?∞ = (x, y), and∞?∞ =∞. Also define (x, 0)?(x, 0) =∞. Next assume (x1, y1), (x2, y2) ∈ E(R) with (x2, y2) 6= (x1,−y1) . If (x1, y1) 6= (x2, y2) we let L12 be the unique line passing through (x1, y1) and (x2, y2). Recall that explicitly

L12 = {(x, y) ∈ R×R; y − y1 = m(x− x1)} where

m = (y2 − y1)(x2 − x1)−1. If (x1, y1) = (x2, y2) we let L12 be the “line tangent to Z(R) at (x1, y1)” which is by definition given by the same equation as before except now m is defined to be

m = (3x21 + a)(2y1) −1.