Complete addition laws on abelian varieties

Source abstract

Abstract We prove that under any projective embedding of an abelian variety A of dimension g , a complete set of addition laws has cardinality at least g +1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ 2 . In contrast, we prove, moreover, 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 the embedding in ℙ 15 , respectively, up to a finite number of counterexamples for ∣ k ∣≤5 .