Complete Systems of Two Addition Laws for Elliptic Curves Original Research Article

Let E be an elliptic curve over a field k, given in Weierstrass form. The addition law E × E → E can be described by a finite number of triples of bihomogeneous polynomials, each triple being valid on an open subset of E × E. In this paper we prove that the minimal number of triples that is needed equals two, and that this can only be achieved with polynomials of bidegree (2,2). Our approach leads to a specific set of two triples with the desired property.