On pro-𝑝 link groups of number fields

Mizusawa, Yasushi

doi:10.1090/tran/7787

On pro-$p$ link groups of number fields
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Trans. Amer. Math. Soc. 372 (2019), 7225-7254 Request permission

Abstract:

As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e., tamely ramified over the intermediate cyclotomic $\mathbb Z_p$-extension of the number field. In some basic cases, such a pro-$p$ Galois group also has a Koch type presentation described by linking numbers and mod $2$ Milnor numbers (Rédei symbols) of primes. Then the pro-$2$ Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.

References

Fumiya Amano, On a certain nilpotent extension over $\textbf {Q}$ of degree 64 and the 4-th multiple residue symbol, Tohoku Math. J. (2) 66 (2014), no. 4, 501–522. MR 3350281, DOI 10.2748/tmj/1432229194
Julien Blondeau, Philippe Lebacque, and Christian Maire, On the cohomological dimension of some pro-$p$-extensions above the cyclotomic $\Bbb Z_p$-extension of a number field, Mosc. Math. J. 13 (2013), no. 4, 601–619, 736–737 (English, with English and Russian summaries). MR 3184074, DOI 10.17323/1609-4514-2013-13-4-601-619
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
Bruce Ferrero, The cyclotomic $\textbf {Z}_{2}$-extension of imaginary quadratic fields, Amer. J. Math. 102 (1980), no. 3, 447–459. MR 573095, DOI 10.2307/2374108
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Patrick Forré, Strongly free sequences and pro-$p$-groups of cohomological dimension 2, J. Reine Angew. Math. 658 (2011), 173–192. MR 2831517, DOI 10.1515/CRELLE.2011.067
Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. MR 53938, DOI 10.2307/1969736
Jochen Gärtner, Rédei symbols and arithmetical mild pro-2-groups, Ann. Math. Qué. 38 (2014), no. 1, 13–36 (English, with English and French summaries). MR 3249409, DOI 10.1007/s40316-014-0021-3
Jochen Gärtner, Higher Massey products in the cohomology of mild pro-$p$-groups, J. Algebra 422 (2015), 788–820. MR 3272101, DOI 10.1016/j.jalgebra.2014.07.023
Robert Gold, The nontriviality of certain $Z_{1}$-extensions, J. Number Theory 6 (1974), 369–373. MR 369316, DOI 10.1016/0022-314X(74)90034-1
Georges Gras, Class field theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. From theory to practice; Translated from the French manuscript by Henri Cohen. MR 1941965, DOI 10.1007/978-3-662-11323-3
Ralph Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, 263–284. MR 401702, DOI 10.2307/2373625
Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638, DOI 10.5802/aif.1299
Jonathan Hillman, Algebraic invariants of links, Series on Knots and Everything, vol. 32, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1932169, DOI 10.1142/9789812776648
Jonathan Hillman, Daniel Matei, and Masanori Morishita, Pro-$p$ link groups and $p$-homology groups, Primes and knots, Contemp. Math., vol. 416, Amer. Math. Soc., Providence, RI, 2006, pp. 121–136. MR 2276139, DOI 10.1090/conm/416/07890
Tsuyoshi Itoh and Yasushi Mizusawa, On tamely ramified pro-$p$-extensions over ${\Bbb Z}_p$-extensions of ${\Bbb Q}$, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 2, 281–294. MR 3177870, DOI 10.1017/S0305004113000637
Jean-François Jaulent, Théorie $l$-adique globale du corps de classes, J. Théor. Nombres Bordeaux 10 (1998), no. 2, 355–397 (French, with English and French summaries). MR 1828250, DOI 10.5802/jtnb.233
Teruhisa Kadokami and Yasushi Mizusawa, Iwasawa type formula for covers of a link in a rational homology sphere, J. Knot Theory Ramifications 17 (2008), no. 10, 1199–1221. MR 2460171, DOI 10.1142/S0218216508006580
Teruhisa Kadokami and Yasushi Mizusawa, On the Iwasawa invariants of a link in the 3-sphere, Kyushu J. Math. 67 (2013), no. 1, 215–226. MR 3089003, DOI 10.2206/kyushujm.67.215
T. Kadokami and Y. Mizusawa, Iwasawa invariants of links, Intelligence of low-dimensional topology, RIMS Kōkyūroku 1911 (2014), 10–17.
Yûji Kida, On cyclotomic $\textbf {Z}_{2}$-extensions of imaginary quadratic fields, Tohoku Math. J. (2) 31 (1979), no. 1, 91–96. MR 526512, DOI 10.2748/tmj/1178229880
Klaus Haberland, Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. With two appendices by Helmut Koch and Thomas Zink. MR 519872
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
John Labute, Mild pro-$p$-groups and Galois groups of $p$-extensions of $\Bbb Q$, J. Reine Angew. Math. 596 (2006), 155–182. MR 2254811, DOI 10.1515/CRELLE.2006.058
John Labute and Ján Mináč, Mild pro-2-groups and 2-extensions of $\Bbb Q$ with restricted ramification, J. Algebra 332 (2011), 136–158. MR 2774682, DOI 10.1016/j.jalgebra.2011.01.019
B. Mazur, Remarks on the Alexander polynomial, unpublished, 1963-64, http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf.
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
Yasushi Mizusawa, On the maximal unramified pro-2-extension of $\Bbb Z_2$-extensions of certain real quadratic fields. II, Acta Arith. 119 (2005), no. 1, 93–107. MR 2163520, DOI 10.4064/aa119-1-7
Yasushi Mizusawa, On the maximal unramified pro-2-extension over the cyclotomic $\Bbb Z_2$-extension of an imaginary quadratic field, J. Théor. Nombres Bordeaux 22 (2010), no. 1, 115–138 (English, with English and French summaries). MR 2675876, DOI 10.5802/jtnb.707
Yasushi Mizusawa, On unramified Galois 2-groups over $\Bbb Z_2$-extensions of real quadratic fields, Proc. Amer. Math. Soc. 138 (2010), no. 9, 3095–3103. MR 2653934, DOI 10.1090/S0002-9939-10-10458-4
Yasushi Mizusawa, Tame pro-2 Galois groups and the basic $\Bbb {Z}_2$-extension, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2423–2461. MR 3748573, DOI 10.1090/tran/7023
Yasushi Mizusawa and Manabu Ozaki, Abelian 2-class field towers over the cyclotomic $\Bbb Z_2$-extensions of imaginary quadratic fields, Math. Ann. 347 (2010), no. 2, 437–453. MR 2606944, DOI 10.1007/s00208-009-0431-8
Yasushi Mizusawa and Manabu Ozaki, On tame pro-$p$ Galois groups over basic $\Bbb {Z}_p$-extensions, Math. Z. 273 (2013), no. 3-4, 1161–1173. MR 3030694, DOI 10.1007/s00209-012-1048-2
Masanori Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141–167. MR 1925911, DOI 10.1515/crll.2002.070
Masanori Morishita, Milnor invariants and Massey products for prime numbers, Compos. Math. 140 (2004), no. 1, 69–83. MR 2004124, DOI 10.1112/S0010437X03000137
Masanori Morishita, Knots and primes, Universitext, Springer, London, 2012. An introduction to arithmetic topology. MR 2905431, DOI 10.1007/978-1-4471-2158-9
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
T. Nguyễn-Quang-Đỗ, Formations de classes et modules d’Iwasawa, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 167–185 (French). MR 756093, DOI 10.1007/BFb0099451
Keiji Okano, Abelian $p$-class field towers over the cyclotomic $\Bbb Z_p$-extensions of imaginary quadratic fields, Acta Arith. 125 (2006), no. 4, 363–381. MR 2271283, DOI 10.4064/aa125-4-5
Manabu Ozaki, Non-abelian Iwasawa theory of $\Bbb Z_p$-extensions, J. Reine Angew. Math. 602 (2007), 59–94. MR 2300452, DOI 10.1515/CRELLE.2007.003
Manabu Ozaki and Hisao Taya, On the Iwasawa $\lambda _2$-invariants of certain families of real quadratic fields, Manuscripta Math. 94 (1997), no. 4, 437–444. MR 1484637, DOI 10.1007/BF02677865
The PARI Group, PARI/GP, version 2.7.4, University of Bordeaux, 2015, http://pari.math.u-bordeaux.fr/.
Ladislaus Rédei, Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper. I, J. Reine Angew. Math. 180 (1939), 1–43 (German). MR 1581597, DOI 10.1515/crll.1939.180.1
Landry Salle, Sur les pro-$p$-extensions à ramification restreinte au-dessus de la $\Bbb Z_p$-extension cyclotomique d’un corps de nombres, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 485–523 (French, with English and French summaries). MR 2477515, DOI 10.5802/jtnb.638
Landry Salle, On maximal tamely ramified pro-2-extensions over the cyclotomic $\Bbb Z_2$-extension of an imaginary quadratic field, Osaka J. Math. 47 (2010), no. 4, 921–942. MR 2791570
Jonathan W. Sands, On the nontriviality of the basic Iwasawa $\lambda$-invariant for an infinitude of imaginary quadratic fields, Acta Arith. 65 (1993), no. 3, 243–248. MR 1254959, DOI 10.4064/aa-65-3-243-248
Alexander Schmidt, Rings of integers of type $K(\pi ,1)$, Doc. Math. 12 (2007), 441–471. MR 2365909
Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431
Romyar T. Sharifi, On Galois groups of unramified pro-$p$ extensions, Math. Ann. 342 (2008), no. 2, 297–308. MR 2425144, DOI 10.1007/s00208-008-0236-1
Jun Ueki, On the Iwasawa $\mu$-invariants of branched $\mathbf {Z}_p$-covers, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 6, 67–72. MR 3508576, DOI 10.3792/pjaa.92.67
Jun Ueki, On the Iwasawa invariants for links and Kida’s formula, Internat. J. Math. 28 (2017), no. 6, 1750035, 30. MR 3663789, DOI 10.1142/S0129167X17500355
Denis Vogel, On the Galois group of 2-extensions with restricted ramification, J. Reine Angew. Math. 581 (2005), 117–150. MR 2132673, DOI 10.1515/crll.2005.2005.581.117
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
K. Wingberg, Arithmetical Koch groups, preprint, 2007, www.mathi.uni-heidelberg.de/groups/arithmetik/paperwingberg/wingberg40-en.html.
Gen Yamamoto, On the vanishing of Iwasawa invariants of absolutely abelian $p$-extensions, Acta Arith. 94 (2000), no. 4, 365–371. MR 1779948, DOI 10.4064/aa-94-4-365-371
Yoshihiko Yamamoto, Divisibility by $16$ of class number of quadratic fields whose $2$-class groups are cyclic, Osaka J. Math. 21 (1984), no. 1, 1–22. MR 736966