Unit equations on quaternions

Abstract

A classical result about unit equations says that if ${\Gamma }_1$ and ${\Gamma }_2$ are finitely generated subgroups of ${\mathbb{C}}^{\times }$, then the equation $x+y=1$ has only finitely many solutions with $x\in {\Gamma }_1$ and $y\in {\Gamma }_2$. We study a noncommutative 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 $\mathrm{deg}(f),\mathrm{deg}(g)\ge 2$, 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 $g$.

14 pages