The constant of the support problem for abelian varieties

TL;DR

This paper investigates the support problem for abelian varieties over number fields, focusing on the minimal scalar c relating points P and Q under reduction modulo primes, and provides refined counterexamples.

Contribution

It analyzes the minimal constant c in the support problem for abelian varieties and constructs refined counterexamples to previous conjectures.

Findings

Determined bounds for the minimal value of c.

Constructed explicit counterexamples to support conjectures.

Enhanced understanding of the support problem in abelian varieties.

Abstract

Let A be an abelian variety defined over a number field K and let P and Q be points in A(K) satisfying the following condition: for all but finitely many primes p of K, the order of (Q mod p) divides the order of (P mod p). Larsen proved that there exists a positive integer c such that cQ is in the End_K(A)-module generated by P. We study the minimal value of c and construct some refined counterexamples.