Complete addition laws on abelian varieties

TL;DR

This paper establishes a lower bound on the number of addition laws needed for projective embeddings of abelian varieties and shows the existence of universal addition laws over fields with infinite Galois groups.

Contribution

It generalizes known results for elliptic curves to higher-dimensional abelian varieties and proves the existence of universal addition laws over certain fields.

Findings

Minimum of g+1 addition laws for embeddings of dimension g

Existence of universal addition laws over fields with infinite Galois groups

Specific embeddings for dimensions 1 and 2 with finite exceptions

Abstract

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.