Elliptic Curves

Abstract

Next to the quadrics these are the most studied curves. They are the object of an extensive and deep theory with many connections to analysis and arithmetic (Husemöller [Hus], Lang [L], Silverman [S₁], [S₂]). On the role of elliptic curves in cryptography, see Koblitz [K] and Washington [W]. After choosing a point O,an elliptic curve may be given a group structure using a geometric construction. We first concern ourselves with this construction. Finally, we classify elliptic curves up to a coordinate transformation. This chapter contains only the rudiments of the algebraic theory of elliptic curves.