20 2. Elliptic curves then C(C) may be considered as a surface over R, and the genus of C counts the number of holes in the surface. For example, the projective line P1(C) has no holes and g = 0 (the projective plane is homeomorphic to a sphere see Ap- pendix C for a quick introduction to projective geometry), and an elliptic curve has genus 1 (homeomorphic to a torus see Theorem 3.2.5). Surprisingly, the genus of a curve is intimately related with the arithmetic of its points. More precisely, Louis Mordell conjectured that a curve C of genus 2 can only have a finite number of rational solutions. The conjecture was proved by Faltings in 1983. 2.2. Definition Definition 2.2.1. An elliptic curve over Q is a smooth cubic projec- tive curve E defined over Q with at least one rational point O E(Q) that we call the origin. In other words, an elliptic curve is a curve E in the projective plane (see Appendix C) given by a cubic polynomial F (X, Y, Z) = 0 with rational coefficients, i.e., F (X, Y, Z) = aX3 + bX2Y + cXY 2 + dY 3 (2.2) +eX2Z + fXY Z + gY 2 Z +hXZ2 + jY Z2 + kZ3 = 0, with coefficients a, b, c, . . . Q, and such that E is smooth i.e., the tangent vector ( ∂F ∂X (P), ∂F ∂Y (P), ∂F ∂Z (P) ) does not vanish at any P E (see Appendix C.5 for a brief introduction to singularities and non- singular or smooth curves). If the coefficients a, b, c, . . . are in a field K, then we say that E is defined over K (and write E/K). Even though the fact that E is a projective curve is crucial, we usually consider just affine charts of E, e.g. those points of the form {[X, Y, 1]}, and study instead the affine curve given by aX3 + bX2Y + cXY 2 + dY 3 (2.3) +eX2 + fXY + gY 2 + hX + jY + k = 0
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