The Dynamical Mordell--Lang Conjecture for Relatively \'{E}tale Systems over Products of Curves

Abstract

We establish the dynamical Mordell--Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively \'{e}tale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively \'{e}tale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative \'{e}taleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.