An infinitesimal p-adic multiplicative Manin–Mumford Conjecture

Abstract

Our results concern certain analytic functions on the open unit poly-disc in ℂ p n centered at the multiplicative unit and we show such functions only vanish at finitely many n-tuples of roots of unity (ζ 1 -1,...,ζ 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 formal 𝔾 m and we extend the results to this setting.