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

TL;DR

This paper proves a finiteness property for zeros of certain $p$-adic analytic functions near roots of unity, extending classical conjectures and applying to Lubin-Tate formal groups, revealing rigidity phenomena in $p$-adic geometry.

Contribution

It extends the multiplicative Manin-Mumford conjecture to a broader class of $p$-adic analytic functions and formal groups, establishing new rigidity results and generalizing previous polynomial cases.

Findings

Finitely many roots of unity where functions vanish unless along a formal torus

Rigidity results for formal tori in $p$-adic settings

Extension of results to Lubin-Tate formal groups

Abstract

Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(\zeta_1-1,\ldots,\zeta_n-1)$ 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.