The multilinear support problem for products of abelian varieties and tori

TL;DR

This paper extends the support problem to multiple points on products of abelian varieties and tori, showing that reduction properties imply algebraic dependencies over endomorphism rings.

Contribution

It generalizes Larsen's support problem to several points on products of abelian varieties and tori, linking reduction behavior to algebraic dependencies.

Findings

Dependence relations are detected via reduction orders.

Conditions on reductions imply algebraic dependencies.

Generalization of Larsen's support problem.

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.