On the analyticity of the maximal extension of a number field with prescribed ramification and splitting

Lim, Donghyeok; Maire, Christian

doi:10.1090/proc/16922

On the analyticity of the maximal extension of a number field with prescribed ramification and splitting
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by Donghyeok Lim and Christian Maire;

Proc. Amer. Math. Soc. 152 (2024), 5013-5024

DOI: https://doi.org/10.1090/proc/16922

Published electronically: September 26, 2024

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Abstract:

We determine all the $p$-adic analytic groups that are realizable as Galois groups of the maximal pro-$p$ extensions of number fields with prescribed ramification and splitting under an assumption which allows us to move away from the Tame Fontaine-Mazur conjecture.

References

Y. Benmerieme and A. Movahhedi, Multi-quadratic $p$-rational number fields, J. Pure Appl. Algebra 225 (2021), no. 9, Paper No. 106657, 17. MR 4192837, DOI 10.1016/j.jpaa.2020.106657
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495
Georges Gras, Les $\theta$-régulateurs locaux d’un nombre algébrique: conjectures $p$-adiques, Canad. J. Math. 68 (2016), no. 3, 571–624 (French, with English and French summaries). MR 3492629, DOI 10.4153/CJM-2015-026-3
G. Gras, Class field theory, from theory to practice, corr. 2nd ed., Springer Monographs in Mathematics, Springer, 2005, xiii+507 pages.
Farshid Hajir and Christian Maire, Prime decomposition and the Iwasawa MU-invariant, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 3, 599–617. MR 3933912, DOI 10.1017/S0305004118000191
Wolfgang N. Herfort and Luis Ribes, On automorphisms of free pro-$p$-groups. I, Proc. Amer. Math. Soc. 108 (1990), no. 2, 287–295. MR 984794, DOI 10.1090/S0002-9939-1990-0984794-6
Helmut Koch, Galois theory of $p$-extensions, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. With a foreword by I. R. Shafarevich; Translated from the 1970 German original by Franz Lemmermeyer; With a postscript by the author and Lemmermeyer. MR 1930372, DOI 10.1007/978-3-662-04967-9
Julien Koperecz, Triquadratic $p$-rational fields, J. Number Theory 242 (2023), 402–408. MR 4490454, DOI 10.1016/j.jnt.2022.04.011
Jon González-Sánchez and Benjamin Klopsch, Analytic pro-$p$ groups of small dimensions, J. Group Theory 12 (2009), no. 5, 711–734. MR 2554763, DOI 10.1515/JGT.2009.006
Christian Maire, Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres, J. Théor. Nombres Bordeaux 17 (2005), no. 2, 575–606 (French, with English and French summaries). MR 2211309, DOI 10.5802/jtnb.509
A. Movahhedi, Sur les $p$-extensions des corps $p$-rationnels, PhD Université Paris VII, 1988.
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
The PARI Group, PARI/GP-2.15.3, Univ. Bordeaux, 2023, http://pari.math.u-bordeaux.fr/.
Alexander Schmidt, Bounded defect in partial Euler characteristics, Bull. London Math. Soc. 28 (1996), no. 5, 463–464. MR 1396144, DOI 10.1112/blms/28.5.463
Kay Wingberg, Galois groups of local and global type, J. Reine Angew. Math. 517 (1999), 223–239. MR 1728539, DOI 10.1515/crll.1999.096