An infinitesimal p-adic multiplicative Manin–Mumford Conjecture
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 393-408.

Our results concern certain analytic functions on the open unit poly-disc in 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.

Nos résultats concernent certaines fonctions analytiques sur la boule ouverte unité dans p n centrée en 1. Nous montrons que celles-ci soit ne s’annulent en (ζ 1 -1,...,ζ n -1) que pour un nombre fini de racines de l’unité, soit s’annulent sur tout un translaté du groupe formel multiplicatif. Pour les fonctions polynômiales, cela suit de la conjecture de Manin–Mumford multiplicative. Or nous considérons un ensemble de fonctions bien plus vaste et en particulier déduisons un résultat de rigidité pour les tores formels. De plus, nous étendons ces résultats au-delà du groupe multiplicatif aux groupes formels de type Lubin–Tate.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1030
Classification: 11S31, 13H05, 13F25, 14L05
Keywords: Manin–Mumford, $p$-adic rigidity
Serban, Vlad 1

1 Fields Institute, 222 College Street, Toronto, Ontario M5T3J1, Canada
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Serban, Vlad. An infinitesimal $p$-adic multiplicative Manin–Mumford Conjecture. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 393-408. doi : 10.5802/jtnb.1030. http://www.numdam.org/articles/10.5802/jtnb.1030/

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