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

TL;DR

This paper provides a counterexample to Gajda's question, demonstrating that local conditions modulo primes do not always determine global membership in subgroups of abelian varieties over number fields.

Contribution

The authors construct a specific counterexample showing the failure of the local-global principle for linear dependence in abelian varieties, answering Gajda's question negatively.

Findings

Counterexample disproves the local-global principle

Local conditions modulo primes are insufficient for global dependence

Highlights limitations of local-global approaches in abelian varieties

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.