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

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.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.831
Classification : 11S31,  37P20,  14L05,  11S15
@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},
     zbl = {1292.37008},
     mrnumber = {3063836},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.831/}
}
Ghassan Sarkis; Joel Specter. Galois extensions of height-one commuting dynamical systems. Journal de Théorie des Nombres de Bordeaux, Tome 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 1474165 | Zbl 0883.20015

[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 526137 | Zbl 0475.12020

[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 527692 | Zbl 0403.12018

[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 1483894 | Zbl 0901.20013

[5] B. Klopsch, Automorphisms of the Nottingham group. J. Algebra 223 (2000), no. 1, 37–56. | MR 1738250 | Zbl 0965.20021

[6] N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions. Springer, New York, 1977. | MR 466081 | Zbl 0364.12015

[7] F. Laubie, A. Movahhedi, A. Salinier, Systèmes dynamiques non archimédiens et corps des normes. Compos. Math. 132 (2002), 57–98. | MR 1914256 | Zbl 1101.14057

[8] J. Lubin, One-parameter formal Lie groups over 𝔓-adic integer rings. Ann. Math. 80 (1964), 464–484. | MR 168567 | Zbl 0135.07003

[9] J. Lubin, Nonarchimedean dynamical systems. Compos. Math. 94 (1994), 321–346. | Numdam | MR 1310863 | Zbl 0843.58111

[10] J. Lubin Torsion in the Nottingham group. Bull. Lond. Math. Soc. 43 (2011), 547–560. | MR 2820144 | Zbl pre05907311

[11] J. Lubin, J. Tate, Formal complex multiplication in local fields. Ann. Math. 81 (1965), no. 2, 380–387. | MR 172878 | Zbl 0128.26501

[12] G. Sarkis, Height one commuting dynamical systems over p . Bull. Lond. Math. Soc. 42 (2010), no. 3, 381–387. | MR 2651931 | Zbl 1200.11087

[13] S. Sen, On automorphisms of local fields. Ann. of Math. (2) 90 (1969), 33–46. | MR 244214 | Zbl 0199.36301

[14] J.-P. Serre, Local Fields. Springer, New York, 1979. | MR 554237 | Zbl 0423.12016

[15] J.-P. Wintenberger, Extensions abéliennes et groupes dÕautomorphismes de corps locaux. C.R. Acad. Sci. Paris 290 (1980), 201–203. | MR 564309 | Zbl 0428.12012