The non-unit conjecture for Misiurewicz parameters
Volume 224 / 2026
Acta Arithmetica 224 (2026), 117-139
MSC: Primary 37P15; Secondary 11R09, 37P20
DOI: 10.4064/aa250908-21-5
Published online: 25 June 2026
Abstract
A Misiurewicz parameter is a complex number $c$ for which the critical point $z=0$ under $z^2+c$ is strictly preperiodic. Such parameters play the same role as special points in dynamical moduli spaces that is played by singular moduli (corresponding to CM elliptic curves) as special points on modular curves. Building on our earlier work, we investigate whether the difference of two Misiurewicz parameters can be an algebraic unit. (The corresponding question for singular moduli was recently answered in the negative by Li.) We answer this dynamical question in many new cases under a widely believed irreducibility assumption.
