The p-adic distance of special points to subvarieties
José Felipe Voloch
TL;DR
This work addresses the problem of a uniform $p$-adic bound for the distance $d(P,X)$ from torsion points $P$ on a semiabelian variety $A$ to a subvariety $X$, highlighting the roots-of-unity paradigm as a unifying method. It surveys progress across settings, including the torus case, general semiabelian varieties, Siegel moduli spaces, and algebraic dynamics, with key techniques ranging from lower bounds for linear forms in roots of unity to perfectoid methods for CM points. The contributions show substantial progress toward a general uniform bound, including complete proofs in the $p$-adic setting for several cases and novel dynamics-based approaches, while outlining remaining challenges for $p$-power torsion and broader Shimura-type contexts. The results underscore connections between unlikely intersection phenomena, $p$-adic approximation, and modern tools from arithmetic geometry and dynamics, with implications for CM points, moduli spaces, and dynamical systems in non-archimedean settings.
Abstract
In a paper of Tate and the author, we conjectured a uniform bound for the p-adic distance of torsion points on a semiabelian variety, not lying in a subvariety, to that subvariety. We survey the progress made on that conjecture and on similar statements in analogous situations.