Unit equations on quaternions

TL;DR

This paper extends classical unit equation results to noncommutative quaternion semigroups, showing finiteness of solutions under specific conditions and applying it to dynamics on elliptic curves.

Contribution

It introduces a noncommutative analogue of unit equations in quaternion algebras and proves finiteness of solutions when semigroups are generated by algebraic quaternions with norms greater than one, with one semigroup being commutative.

Findings

Finiteness of solutions for quaternion semigroup equations under specified conditions

Application to dynamics: common iterates of endomorphisms on elliptic curves

Extension of classical results to noncommutative algebraic structures

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 $\mathrm{deg}(f), \mathrm{deg}(g)\geq 2$, 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 $g$.