Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of lines through the origin in Q . For example, the projective plane f Q 2 is a quotient of the unit sphere in Q 3 modulo the relation (X,Y, Z) ∼ (−X,−Y,−Z). We define f R n and f C n similarly. A projective variety is a simultaneous irreducible system of homogeneous polynomial equations in n + 1 variables. The Q -points, R -points and C -points of the projective variety are all solutions of the polynomial system in g Q n, f R n and f C n, respectively; these are (n+ 1)-tuples, not n-tuples as before. A curve is a projective variety corresponding to one homogeneous polynomial equation p(X,Y, Z) = 0. In particular, n + 1 = 3; that is, n = 2. Such a curve is smooth or non-singular if there is no C -point at which the partial derivatives pX , pY , pZ all vanish. For example, the conic