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- Title
Unit equations on quaternions.
- Authors
Huang, Yifeng
- 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.
- Subjects
QUATERNIONS; GROUP algebras; EQUATIONS; ENDOMORPHISMS
- Publication
Quarterly Journal of Mathematics, 2020, Vol 71, Issue 4, p1521
- ISSN
0033-5606
- Publication type
Article
- DOI
10.1093/qmath/haaa043