A counterexample to the local-global principle of linear dependence for abelian varieties

2009/05/04 by Peter Jossen, Antonella Perucca, Jossen, Peter +1
Mathematics · #11G05 #11G10 #14K15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G10 #msc:14K15

paper · pdf · doi:10.48550/arxiv.0905.0409

3 pages

arxiv created 2009/05/04 · openalex publication_date 2009/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Gajda in 2002 asked whether it is true that the point P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all but finitely many primes p of k. We answer negatively to Gajda's question.

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