Detecting linear dependence on a simple abelian variety

TL;DR

This paper proves a criterion for linear dependence in simple abelian varieties over number fields, showing that local dependence modulo almost all primes implies global dependence.

Contribution

It establishes a new criterion linking local and global properties of points on simple abelian varieties over number fields.

Findings

If P is in X modulo almost all primes, then P is in X over the number field.

The result applies to geometrically simple abelian varieties.

It advances understanding of local-global principles in arithmetic geometry.

Abstract

Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P be an element of A(k). We prove that if P belongs to X modulo almost all primes of k then P already belongs to X.