%0 Journal Article
%T Complete addition laws on abelian varieties
%A Christophe Arene
%A David Kohel
%A Christophe Ritzenthaler
%J Mathematics
%D 2011
%I arXiv
%X We prove that under any projective embedding of an abelian variety A of dimension g, a complete system of addition laws has cardinality at least g+1, generalizing of a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in P^2. In contrast with this geometric constraint, we moreover prove that if k is any field with infinite absolute Galois group, then there exists, for every abelian variety A/k, a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or embedding in P^15, respectively, up to a finite number of counterexamples for |k| less or equal to 5.
%U http://arxiv.org/abs/1102.2349v2