We give a short proof of a conjecture of Lubin concerning certain families of -adic power series that commute under composition. We prove that if the family is full (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of this group. The proof uses ramification theory and some -adic Hodge theory.
Nous donnons une démonstration courte d’une conjecture de Lubin concernant certaines familles de séries formelles -adiques qui commutent pour la composition. Nous montrons que si la famille est pleine (assez grosse), il existe un groupe formel de Lubin-Tate tel que toutes les séries de la famille sont des endomorphismes de ce groupe. La démonstration utilise la théorie de la ramification et un peu de théorie de Hodge -adique.
Published online:
DOI: 10.5802/pmb.o-2
Keywords: $p$-adic dynamical system, Lubin-Tate formal group, $p$-adic Hodge theory
@article{PMB_2016____19_0, author = {Laurent Berger}, title = {Lubin{\textquoteright}s conjecture for full $p$-adic dynamical systems}, journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres}, pages = {19--24}, publisher = {Presses universitaires de Franche-Comt\'e}, year = {2016}, doi = {10.5802/pmb.o-2}, zbl = {1429.11208}, language = {en}, url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.o-2/} }
TY - JOUR AU - Laurent Berger TI - Lubin’s conjecture for full $p$-adic dynamical systems JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2016 SP - 19 EP - 24 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.o-2/ DO - 10.5802/pmb.o-2 LA - en ID - PMB_2016____19_0 ER -
%0 Journal Article %A Laurent Berger %T Lubin’s conjecture for full $p$-adic dynamical systems %J Publications mathématiques de Besançon. Algèbre et théorie des nombres %D 2016 %P 19-24 %I Presses universitaires de Franche-Comté %U https://pmb.centre-mersenne.org/articles/10.5802/pmb.o-2/ %R 10.5802/pmb.o-2 %G en %F PMB_2016____19_0
Laurent Berger. Lubin’s conjecture for full $p$-adic dynamical systems. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2016), pp. 19-24. doi : 10.5802/pmb.o-2. https://pmb.centre-mersenne.org/articles/10.5802/pmb.o-2/
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