Galois extensions of height-one commuting dynamical systems
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 163-178.

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 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.

DOI: 10.5802/jtnb.831
Classification: 11S31,  37P20,  14L05,  11S15
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
@article{JTNB_2013__25_1_163_0,
     author = {Ghassan Sarkis and Joel Specter},
     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},
     number = {1},
     year = {2013},
     doi = {10.5802/jtnb.831},
     mrnumber = {3063836},
     zbl = {1292.37008},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.831/}
}
TY  - JOUR
AU  - Ghassan Sarkis
AU  - Joel Specter
TI  - Galois extensions of height-one commuting dynamical systems
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2013
DA  - 2013///
SP  - 163
EP  - 178
VL  - 25
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.831/
UR  - https://www.ams.org/mathscinet-getitem?mr=3063836
UR  - https://zbmath.org/?q=an%3A1292.37008
UR  - https://doi.org/10.5802/jtnb.831
DO  - 10.5802/jtnb.831
LA  - en
ID  - JTNB_2013__25_1_163_0
ER  - 
%0 Journal Article
%A Ghassan Sarkis
%A Joel Specter
%T Galois extensions of height-one commuting dynamical systems
%J Journal de théorie des nombres de Bordeaux
%D 2013
%P 163-178
%V 25
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.831
%R 10.5802/jtnb.831
%G en
%F JTNB_2013__25_1_163_0
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/

[1] R. Camina, Subgroups of the Nottingham group. J. Algebra 196 (1997), no. 1, 101–113. | MR | Zbl

[2] J.-M. Fontaine, J.-P. Wintenberger, Le “corps des normes” de certaines extensions algébriques de corps locaux. C. R. Acad. Sci. Paris 288 (1979), 367–370. | MR | Zbl

[3] J.-M. Fontaine, J.-P. Wintenberger, Extensions algébriques et corps des normes des extensions APF des corps locaux. C. R. Acad. Sci. Paris 288 (1979), 441–444. | MR | Zbl

[4] G. Klaas, C. R. Leedham-Green, W. Plesken, Linear Pro-p-groups of Finite Width. Lecture Notes in Mathematics 1674 (Springer-Verlag), 1997. | MR | Zbl

[5] B. Klopsch, Automorphisms of the Nottingham group. J. Algebra 223 (2000), no. 1, 37–56. | MR | Zbl

[6] N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions. Springer, New York, 1977. | MR | Zbl

[7] F. Laubie, A. Movahhedi, A. Salinier, Systèmes dynamiques non archimédiens et corps des normes. Compos. Math. 132 (2002), 57–98. | MR | Zbl

[8] J. Lubin, One-parameter formal Lie groups over 𝔓-adic integer rings. Ann. Math. 80 (1964), 464–484. | MR | Zbl

[9] J. Lubin, Nonarchimedean dynamical systems. Compos. Math. 94 (1994), 321–346. | Numdam | MR | Zbl

[10] J. Lubin Torsion in the Nottingham group. Bull. Lond. Math. Soc. 43 (2011), 547–560. | MR

[11] J. Lubin, J. Tate, Formal complex multiplication in local fields. Ann. Math. 81 (1965), no. 2, 380–387. | MR | Zbl

[12] G. Sarkis, Height one commuting dynamical systems over p . Bull. Lond. Math. Soc. 42 (2010), no. 3, 381–387. | MR | Zbl

[13] S. Sen, On automorphisms of local fields. Ann. of Math. (2) 90 (1969), 33–46. | MR | Zbl

[14] J.-P. Serre, Local Fields. Springer, New York, 1979. | MR | Zbl

[15] J.-P. Wintenberger, Extensions abéliennes et groupes dÕautomorphismes de corps locaux. C.R. Acad. Sci. Paris 290 (1980), 201–203. | MR | Zbl

Cited by Sources: