Complete addition laws on abelian varieties

2011/02/11 by Christophe Arene, Arene, Christophe, David Kohel +3
Mathematics · #11G20 #11T71 #14K15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G20 #msc:11T71 #msc:14K15

paper · pdf · doi:10.48550/arxiv.1102.2349

9 pages. Finale version, accepted for publication in LMS Journal of Computation and Mathematics

arxiv created 2012/04/20 · arxiv updated 2012/04/23

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 P². 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 P¹5, respectively, up to a finite number of counterexamples for |k| less or equal to 5.

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