Unit equations on quaternions
November 30, 2022
A classical theorem in number theory states that for any finitely generated subgroup $\Gamma$ of $C*$, the "unit equation” $x+y=1$ has only finitely many solutions with $x,y\in \Gamma$. One can view it as a statement that relates addition and multiplication of complex numbers in a fundamental way. Our main result (arXiv: 1910.13250) is an analog of this theorem on quaternions, where the multiplication is no longer commutative. We then explain its connection to iterations of self-maps on abelian varieties, and give a result about an orbit intersection problem as an application. The approach to our main result is based on the analysis of the Euclidean norm on quaternions, and Baker’s estimate of linear combinations of logarithms. Time permitting, I will sketch a proof focusing on how the difficulties caused by noncommutativity are miraculously addressed in our case.