Two variants of the support problem for products of abelian varieties and tori

TL;DR

This paper investigates support problems for points on products of abelian varieties and tori over number fields, establishing conditions under which points are related via endomorphisms and scalar multiplication, with weaker assumptions explored.

Contribution

It introduces variants of the support problem, including radical and l-adic support problems, and proves related results under these weaker conditions.

Findings

Existence of an endomorphism and scalar linking P and Q under divisibility conditions

Extension of results to radical and l-adic valuation support problems

Broader applicability of support problem solutions with weaker assumptions

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem).