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Complete addition laws on abelian varieties

Part of: Arithmetic algebraic geometry Finite fields and commutative rings (number-theoretic aspects) Abelian varieties and schemes

Published online by Cambridge University Press:  01 September 2012

Christophe Arene ,
David Kohel  and
Christophe Ritzenthaler
Christophe Arene
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: arene.christophe@gmail.com)
David Kohel
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: kohel@iml.univ-mrs.fr)
Christophe Ritzenthaler
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: ritzenth@iml.univ-mrs.fr)

Abstract

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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 ℙ2. 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 ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

MSC classification

Secondary: 11G20: Curves over finite and local fields 14K15: Arithmetic ground fields 11T71: Algebraic coding theory; cryptography
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 308 - 316
Copyright
Copyright © London Mathematical Society 2012

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