Galois extensions of height-one commuting dynamical systems

Ghassan Sarkis; Joel Specter

doi:10.5802/jtnb.831

Ghassan Sarkis ¹; Joel Specter ²

¹ Pomona College 610 North College Avenue Claremont, CA 91711, USA
² Northwestern University 2033 Sheridan Road Evanston, IL 60208, USA

Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 163-178.

Abstract
Résumé

We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$ -adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From an earlier result, this shows that the original two series must in fact be endomorphisms of some height-one formal group.

Nous considérons un système dynamique constitué d’une paire de séries formelles commutant pour la composition, l’une non inversible et l’autre inversible d’ordre infini, de hauteur $1$ à coefficients dans les entiers $p$ -adiques. En supposant que chaque point du système dynamique engendre une extension galoisienne du corps $ℚ_{p}$ , nous montrons que ces extensions sont en fait abéliennes, et, à partir des résultats de la théorie du corps des normes, nous montrons que le système dynamique doit contenir une série d’ordre fini. À partir d’un résultat précédent, cela montre que les deux séries formelles doivent être des endomorphismes d’un groupe formel de hauteur $1$ .

MR Zbl

DOI: 10.5802/jtnb.831

Classification: 11S31, 37P20, 14L05, 11S15

Author's affiliations:

Ghassan Sarkis ¹; Joel Specter ²

¹ Pomona College 610 North College Avenue Claremont, CA 91711, USA
² Northwestern University 2033 Sheridan Road Evanston, IL 60208, USA

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     title = {Galois extensions of height-one commuting dynamical systems},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {163--178},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
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     year = {2013},
     doi = {10.5802/jtnb.831},
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Ghassan Sarkis; Joel Specter. Galois extensions of height-one commuting dynamical systems. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 163-178. doi : 10.5802/jtnb.831. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.831/

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