Unit Equations on Quaternions

Abstract

Abstract A classical result about unit equations says that if Γ1 and Γ2 are finitely generated subgroups of ${\mathbb C}^\times$, then the equation x + y = 1 has only finitely many solutions with x ∈ Γ1 and y ∈ Γ2. We study a non-commutative analogue of the result, where $\Gamma_1,\Gamma_2$ are finitely generated subsemigroups of the multiplicative group of a quaternion algebra. We prove an analogous conclusion when both semigroups are generated by algebraic quaternions with norms greater than 1 and one of the semigroups is commutative. As an application in dynamics, we prove that if f and g are endomorphisms of a curve C of genus 1 over an algebraically closed field k, and deg( f), deg(g)≥ 2, then f and g have a common iterate if and only if some forward orbit of f on C(k) has infinite intersection with an orbit of g.