4 - Elliptic curves

Abstract

This chapter defines the notion of an elliptic curve over an arbitrary field, for example over the field Q of rational numbers or over a finite field. Adopting the more naive points of view of the “analytic geometry” due to Descartes and Fermat, and the projective geometry of Chasles and Poncelet, the chapter takes advantage of the fact that, in the projective plane, an elliptic curve can be considered as a hypersurface, that is, as an object given by a non-zero homogeneous polynomial F(X, Y, Z) ∊ k[X, Y, Z] of a certain type, and the readers will study these hypersurfaces using the notion of an intersection product.