An Infinitesimal $p$-adic Multiplicative Manin-Mumford Conjecture

Abstract

Our results concern analytic functions on the open unit $p$-adic poly-disc in ${\mathbb{C}}_p^n$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $({\zeta }_1-1,\dots ,{\zeta }_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just the formal multiplicative group and we extend the results to this setting.

arXiv comment14 pages, minor corrections, slightly strengthened the statement of the main theorem. Accepted for publication in Journal de Théorie des Nombres de Bordeaux