On reduction maps and arithmetic dynamics of Mordell–Weil type groups
Volume 210 / 2023
Abstract
We investigate reductions of points in Mordell–Weil type groups over number fields. Central examples of such groups are the groups of $S$-units in a number field, Mordell–Weil groups of abelian varieties, and odd algebraic $K$-theory groups.
We establish two local-global principles in arithmetic dynamics of Mordell–Weil type groups.
The first principle concerns intersection of orbits. We show that if two orbits intersect locally then they intersect globally. This is similar to a local-global formulation of the dynamical Mordell–Lang conjecture.
The second principle is the affirmative answer to a generalization, for Mordell–Weil type groups, of a question whether inclusion of supports of corresponding terms of two Lehmer–Pierce sequences implies a relation between their generators.
One of the technical tools used in the proofs is a result on the orders of images of linearly independent points via reduction maps. As a by-product of its proof, we obtain a result on potential independence of Kummer towers.
