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- W2982577275 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." @default.
- W2982577275 created "2019-11-08" @default.
- W2982577275 creator A5019916748 @default.
- W2982577275 date "2020-11-17" @default.
- W2982577275 modified "2023-09-23" @default.
- W2982577275 title "Unit Equations on Quaternions" @default.
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- W2982577275 doi "https://doi.org/10.1093/qmath/haaa043" @default.
- W2982577275 hasPublicationYear "2020" @default.
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