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Facultatea de Matematică Anul II Master, Geometrie Algebrică Curbe eliptice These notes use as main source the text J. S. Milne: Elliptic curves and algebraic geometry, Math679 U. Michigan notes (1996) 1 Definition. First properties Definition 1.1. Let k be a field. An elliptic curve over k can be defined as: • (a) A complete nonsingular curve E of genus 1 over k together with a point O ∈ E(k). • (b) A nonsingular plane projective E of degree 3 together with a point O ∈ E(k). • (c) A nonsingular plane projective curve E of the form Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 . The definition makes sense once we prove Proposition 1.1. Let E be an algebraic curve. The three properties listed in the definition hereinbefore are equivalent. Demonstraţie. Let E be as in (c). Then E(k) contains a canonical element O = (0 : 1 : 0), and the pair (E; O) satisfies the other two properties. This is obvious for (b), and (a) follows readily. Let (E; O) be as in (a). Then there is an isomorphism from E onto a curve as in (c) sending O to (0 : 1 : 0). Let (E; O) be as in (b). Then there is a change of variables transforming E into a curve as in (c) and O into (0 : 1 : 0) (if O is a point of inflection, the change of variables can be taken to be linear). 1 Let (E; O) be a nonsingular cubic curve in P2( k). We assume that O is a point of inflection. Proposition 1.2. (a) After a linear change of variables (with coefficients in k), the point O will be (0 : 1 : 0) and its tangent line will be H∞ : Z = 0; (b) If (0 : 1 : 0) ∈ E(k) and the tangent line to E at (0 : 1 : 0) is H∞ : Z = 0, then the equation of E has the form Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 . Thus, an elliptic curve is isomorphic to a curve of the form E : Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 . and, conversely, every nonsingular such E is an elliptic curve. This is usually referred to as the canonical or Weierstrass equation of the curve. One can show that it is canonical up to a change of variables of the form: X = u2 X 0 + r Y = u3 Y 0 + su2 X 0 + t with u; r; s; t ∈ k and u 6= 0. Theorem 1.3. Assume char k 6= 2, 3. (a) The curve E(a, b) : Y 2 Z = X 3 + aXZ 2 + bZ 3 , a, b ∈ k is nonsingular, and hence (together with O = (0 : 1 : 0)) defines an elliptic curve over k, if and only if 4a3 + 27b2 6= 0. (b) Every elliptic curve over k is isomorphic to one of the form E(a, b). (c) Two elliptic curves E(a, b) and E(a0 , b0 ) are isomorphic if and only if there exists a c ∈ k ∗ such a0 = c4 a, b0 = c6 b; the isomorphism is then (x : y : z) 7→ (c2 x : c3 y : z). For an elliptic curve E, define 4a3 4a3 + 27b2 if E ' E(a, b). Since the expression on the right is unchanged when (a, b) is replaced by (c4 a, c6 b), this is well-defined, and E ' E 0 iff j(E) = j(E 0 ) when k is algebraically closed. j(E) = 1728 2 2 The group law The point at infinity, O, is the zero for the group law. The group law is determined by: P + Q + R = O ⇔ P ; Q; Rlieonastraightline. If P = (x : y : z), then −P = (x : −y : z). In particular, −P = P , i.e., P has order 2 if and only if y = 0. The points of order two are the points (x : 0 : 1) where x is a root of X 3 + aX + b. Proposition 2.1. • P + O = O + P = P for all P ∈ E(a, b) ( O is the zero element of the group). • Let P = (x, y) 6= O. Then −P := (x, −y) ; P + (−P ) = O • Let P (x1 , y1 ) şi Q(x2 , y2 ) be points different to O, P 6= −Q. Then P + Q = (x3 , y3 ), where x3 = λ2 − x1 − x2 with ( λ := , y2 −y1 x2 −x1 3x21 +a 2y1 y3 = λ(x1 − x3 ) − y1 , dacă P 6= Q dacă P = Q Geometrically, the group law can be described as follows: 3 For example: 4