Galois extensions of height-one commuting dynamical systems

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 (2013)

  • Volume: 25, Issue: 1, page 163-178
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Sarkis, Ghassan, and Specter, Joel. "Galois extensions of height-one commuting dynamical systems." Journal de Théorie des Nombres de Bordeaux 25.1 (2013): 163-178. <http://eudml.org/doc/275770>.

@article{Sarkis2013,
abstract = {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.},
affiliation = {Pomona College 610 North College Avenue Claremont, CA 91711, USA; Northwestern University 2033 Sheridan Road Evanston, IL 60208, USA},
author = {Sarkis, Ghassan, Specter, Joel},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {4},
number = {1},
pages = {163-178},
publisher = {Société Arithmétique de Bordeaux},
title = {Galois extensions of height-one commuting dynamical systems},
url = {http://eudml.org/doc/275770},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Sarkis, Ghassan
AU - Specter, Joel
TI - Galois extensions of height-one commuting dynamical systems
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 163
EP - 178
AB - 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.
LA - eng
UR - http://eudml.org/doc/275770
ER -

References

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  11. J. Lubin, J. Tate, Formal complex multiplication in local fields. Ann. Math. 81 (1965), no. 2, 380–387. Zbl0128.26501MR172878
  12. G. Sarkis, Height one commuting dynamical systems over ℤ p . Bull. Lond. Math. Soc. 42 (2010), no. 3, 381–387. Zbl1200.11087MR2651931
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