MIZUSAWA, YASUSHI

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TitleOn unramified pro-$p$ Galois groups over cyclotomic$\mathbb{Z}_p$-extensions - A survey (Proceedings of theSymposium on Algebraic Number theory and Related Topics)Author(s)MIZUSAWA, YASUSHICitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2007), B4: 223-233Issue Date2007-12URL http://hdl.handle.net/2433/174161RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Kolcvurokv. Bessats1J, 4 223~233 On unramified pro=p Cialois group§ over cyclotornic Zp~extensions - A s11:rvey By "{ASUSHI * Abstract For a fixed number p, we denote the cyclotomic of a number field k. We expect that the Galois group extension over k"'o would information of the maximal unramified about the Galois groups of towers of number fields. In this paper, we will an overview of some topics on v<rith an announcement. of some results in p = 2 case. § 1. Introduction Let p be a fixed m1D ..:1ber and of integers. For a finite extension k of the field Q of rational uu.!HI.Jc< extelJSl•OD. of the number field k. The Galois group r = the cyclotomic is isomorphic to the additive group of and has a ry. The main this paper is the group By choosing a suitable section r '--+ r H ::Y of the natural exact sequence 1 -:. splits since r is a free pro-p group) such that ::Y is an element subgroup of a lying above p, we define an action of r on "(, i.e., we define a continuous homomorphism ..+. • f1 ' A ). r'(k ) ~r . .L --,- ... .ll,UtJ u \ oo 2000 Mathematics Subject Classification(s): 11R23. The author was supported by JSPS Resea-rch Fellowships for Young Scientists. the inertia via the left '''Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan. @ 2007 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved. 224 YASUSHI MIZUSAWA such that = 'I g = a operator group with ¢ ( cf. [15] Then, the Galois group G(koo) is [23] U ). To know the unramified pro-p Galois group G(koo) as a pro-p-F operator group is almost equivalent to c::: G(koo) ><1 r as a pro-p group. n ?. 0, Wi3 by the n-th of the cyclic subextension of degree over k. VIe are also interested in the Galois group G(kn) = Gal(Z(kn)/kn) of the maximal pro-p-extension L(kn) of To borrow the words of [23], the unramified pro-p Galois group is "one of the most on,' 01"n.' in algebraic IlUll'lber . jTJ:-18 seq11er1C€ of :a-extensions associated to the commutator series called p-cl.ass Held tower of the Hilbert p-class field related to the on the p-Sylmv subgroup of the ideal class group of If n is sufficiently there is a induced frorn the restriction ---·~.c-r-···-o· quotient of , and the structure of the relations of pro-p group and the action ofT. By the induced projective G(ko-J) c::: In this paper, we that its structure aS a pro-p-r nn,Pr:'<UW information about the Galois groups c or expect<:htl:ons, we shall see some groups see some examples of in the next section. In the third section, we will or met abelian) Acknowledgements. This paper is written as a of the talk entitled "On the m.axima.l unramified pro-2-extension over the cyclotomic Z2-extension of an imaginary quadratic field" by the author. In the talk, the author announced some results written in subsection 3.2 with mentioning several topics in section 2. The author expresses his gratitude to Professor Ki-ichiro Hashimoto for him an opportunity of talking at the conference and this article. The author also thanks to Doctor Satoshi ON UNRAM!FIED PRO-p GALOIS GROUPS OVE:R CYCLOTOMIC -A SURVEY 225 Related topics 2.L Frmu abelian Iwasawa theo:ry. the action of r induced fron1 ¢, the abelianization X(koo) G(keyco) is considered as an Iwasawa module, a module over the group ring Zp[[T]]. The module with the Galois group of the maximal abelian , and it is proven by lwasawa that [[F]]-module. Then, we can define the Iwasa:wa invariants .\ = Qp), f1 = and the characteristic of the Iwasawa module , where denotes the field of numbers of Zp-extension QcxJ of Q !). Based on the with Alexander polynomial of it is out in [14] that the Iwasawa P(T) is also obtained in the words of pro-p Fox calculus if we have a Gal(L(koo)/1~) For Zp-extensions of any finite extensions of Q, the of Iwasawa. Since = 0)' is to the finiteness of this claim in the words about G(k=) as follows: "p, = 0" conjecture. The Galois group, i.e., the generator rank < 00. Ferrero and [3] is true if /{5 is an abelian extension over Q. This is an advantage of treating cyclotomic if k is a certain CM-field, the Iwasawa re-lated to the p-adic £-functions the theorems of Mazur and 'it~Viles [10] [22], ""U""··v•.J "Iwasawa's main conjecture", Especially, if k is an imaginary quadratic field vvith the associated Dirichlet character X ( f:. w the Teichmiiller we have a power series f(T) E Zp[[T]] constructed from Stickelberger elements such that = (2P(T)) as a principal ideal of Zp[[T]] and the Kubota-Leopoldt's p-adic £-function Lp(s, vJX) f(~(;) 8 - 1), where I'>,: T--+ z; is the restricted 2.2. Nonabelian Iwasawa type formulae. We define the lower central series of G(k®) by putting C( 1l(ke) = G(k@) and C(i+l) = [C(i) (!;@), G(l~:@)] i ?:: 1 inductively. The bracket means a topologically closed commutator subgroup. We also the quotients (!~~) = (k") for i ?:: L 226 YASUSHI M!ZUSAWA In [18], Ozaki defined the i-th Iwasawa module as the action of T induced from and sho-wed some basic properties. Especially, for each i > 1 X(i) X(i) - , with to the restriction mappings. Note that X(1l = X(k00 ). J1 = 0, the i-th Iwasawa module X(il(koo) is a finitely torsion Zp[[F]]-module with = 0 for each i ~ L the i-th lwasawa ;\-invariant is defined as ).,(i) = = dim!Qlv (X(i) ®zp Qp)· By considering the structure of X(i) and putting the following nonabelianization of Iwasavra's formula. Theorem 2.1 (Ozaki such thai: all sufficiently large n. \ }• for each i, we denote above formula holds for all n ~ that {L = 0, and (kn)) = vi - L.Jcf":::=l , Ozaki gave any i ~ l. there integer such that the _,,o.,.,;:-.-r, quotient of w-ell known as the Iwasawa's class number fornmla that i = the action of , Toward a nonabelianization of Iwasawa's main that "INhat kind of functions relate to _;{Ul interested in how p-adic £-functions relate to them .. . In the ease ->- oo)" has of p-class field towers also controls the behavior Ozaki [17] asked Vve are also 2.3. Freeness a:nd infinite p-class field towers. Based on the ·n·"'""''""'"" the Galois groups of Golod and field towers. When the field tower is infinite, eve are interested in the cohomological dimension of the Galois group. On the other hand, Ozaki the following problem: Problem 2.2. Is the grottp ? Especially, -the r'elaiion rank group G(koo) as a pro-p , Zjp7l) < oo '? Though the showed that #G(kn) answer of this problem is not clear yet, and Okano [6] n if oo > d(G(koo)) 2 :» 4T(G(l~00 )), based on the idea of Especially, they investigated the consequences under the assumption that G(koo) is a free pro-p group, i.e., T(G(koo)) = 0, ON UNRAMIFIED PRO-p GALOIS GROUPS OVER CYCLOTOMIC Zp-EXTENS!ONS - A SURVEY 227 Theorem 2.3 (Fujii-Okano Let p be odd, and k a CM-field with the maxi-totally real , and 8 the set Assume that #S = 1, #A.(k+) = 1 and dimwP 2:: 2. If G(koo) -is a free pro-p g-roup, then = c0 all n ;_:::: l. that X(k!,) ::::: Zjp'll, ,\;:::: 1 '1,----,+---,o"~,..,.,_-#,.,~,-;S'"" where o = 1 or 0 to whethe1< k G ( k~)) is a free pro-p group for an element For each odd p, using the result of [25], we can find infinitely many uuc"6 'u'u extensions k of degree 2p the of (2) for the freeness . In to be very delicate. Though the freeness were treated in [17] Remark [19]) and the results · [''"].l "b Slc · ~ 111 :t:u , y , ,.1antL it seems that we have no concrete example of nonabelian free If ) is a nonabelian free pro-p group, we can see that as -t oo. It is a considerable problem to find unbounded as i -t oo. such that 2.4. p-adic analyticity and finite p~class field towe:r-5. For any finite dimen-over Qp and any linear continuous p : -t a of the Fontaine and l\1azur [4]) that the image of p is this claim asserts that: Fontai::ne-IVI:azur conjecbu·e. The Galois group of infinite p-adic analytic quotient. field tower has no Since any group, we can this lem that whether the Galois powerful) or not. For this the group G(k00 ). Theore:rn 2.4 (Wingberg [24]). pro-p group has an open powerful sub-in the statement of ·we are also interested. in the prob-infinite p-adic (resp. the considering Assume that p is odd and k is a CM-field con-taining (p; and that p, = 0 #G(kn) < 00. . If n is sufficiently then On the other hand, under the assumption that both "f.l· = 0" conjecture and Fontaine-Mazur conjecture hold, we can easily show the following by the properties 228 YASUSHI M!ZUSAWA Proposition 2.5, Fonta·ine-Mazur Assume that f-L = 0 the sense is lf then #G(kn) < oo. Put H = Gal(L(kn)/kco n Then H is an open subgroup of G(kn) and to a quotient Since G(koo) has finite rank in the sense [2] Definition 3.12 ( cf. [2] Theorem 3.13, Corollary 8.33), H is also a pro-p group of finite rank ( cf. Exercise 3.1). is p-adic has no infinite nmst be finite. 0 The border between finite cases and infinite cases is one of the main theme in the provides criteria for finiteness of area between nearly free cases and to characterize number fields k v,rith p-adie cases that becomes 2,5. GreenbeJ:·g;s For any tured that #X(koo) < oo by Greenberg [8]. Since this claim is to that ,\ = ,u = 0, X( 1l any finite unramiH.ed is also the certain , ·we can extend this conjecture ber field, any open of "FIFA"). what is the border to be interesting the c::'~m ) VI 1) ( 1. ) f' F Q c::' A ' 11:n or all n >> . Zp-extension of a real num-Let us call this property FIFA due to Boston [1]. The positive answers of this conjecture and Problem 2.2 that is similar to the Galois groups of p-class field towers if k is real. of view, gave the following problem as a strong version of Problem :~LIB. Teal number field, This claim is equivalent to that .A= ,u = 0 and is bounded as ·i-+ oo. If G(koo) is finite, this claim holds The finiteness of G(koo) is to the existence of a extension Kover k such that #X(Koo) = 1 (Le., .\ = IJ· = v = 0 for Kx,)· The abelianp-extenslons K ofQ with trivial X(Koo) are completely characterized 229 by Yamamoto ([25] etc,). It is also a considerable to characterize aU finite _pb)extensions K of Q -with trivial X(K00 ). If G(k00 ) ~ G(kn) for some n ~- 0, ¢is not ) = 1. On the other hand, if¢ is not ) for all n >? 0. Then, under the that k is totally real and holds for p and all subfields o:f L(koo), we can show that G(koo) satisfies FH<'A. by 1 of [8]. The injectivity of ¢ in the real case seems to be considerable as a problem between Greenberg's conjecture and Problem 2.6. For an imaginary k is unramified over the abelianization of Gal(L(kc.r.:;)/k) is PL"•w-•cv is satisfies FIFA -extension k of If this On the other we can find many examples for which ) = i.e., #Im¢ = 1. (The arrow * above depends on the ~ 'Zp (not FIF:A !) but reality of § 3. Explicitly presented excunples ~LL Abelian. , i.e., G(koo) is also a pro-p group. In the case that it is not a trivial whether is abelian or not. The a.u_,w~'"" of G(koo) is equivalent to the As an easiest case, we can shuw the following with nontrivial examples. Proposition 3.1. Let IJ be odd and k a containing and assume that ,u = 0. If A= 1, then G(koo) ~ ~B with some m ~ 0. Put x+ = X(k!,) for the maximal real subfield k+ of k, and let x- be the of X c::: x+ EB x-. Since ~t = 0, x-is a free Zp-module By Leopoldt's Spiegelungssatz ([21] Theorem 10.11), we know that .A(X+) ::; rankx+ ::; rankx- = ). Since ),(X+)+ >.(x-) =.A= 1 our x-we know that G(k;t,) c::: x+ ~ Zjpmz with some m ~ 0 and = L(k!,) is an unramified finite cyclic p-extension of k;t,. Put 230 YASUSH! MIZUSAWA 1(00 = k0 oK;t. Note that K, and that #X ( K;};J = 1. is the Zp-extension of a certain CM-field Kida's formula [9], we know that ,u(.Y(Kxo)) = 0 and J\(X(l(;o)) = 1. vVe can see th.at zu~co) = L(Koo) 0 By the result of Y'amamoto , Kida's formula [9] and we can easily find infinitely rn.any abelian sextic :fields k containing Q( '=3) such that c:::: Zs EB Z/37l in p = 3 case. For odd p and ,I;; = it is a.nnounced Sharifi that 1s abelian if p < 1000 (and there exists p > 1000 such that ) is nonabelian !). for p = 461. for odd p, Okano cl1aracterized an field k \vit.h 11oncyclic abelian as follo~rs: Theorem 3.2 (Okano abelian ·if a'nd F'oT odd p and an if), = 2 and ideal above p. For odd p and u'J""''IS•'"u•:u erfulness of ) are equivalent (cf. [24] fields k with abelian G ( kCY.J) are characterized case is related with the Iwasawa 'I'heormn 3.3 ([13]), 15 (mod32), the other of [25]), we can find a.u.uu""'" e.g., [1 many real 3.2, MetabeHa:n exar.nples in p = 2 case, k, and the pow--in p = 2 case, all imag-Ozaki and author [13]. ·''~'"'u'" P(T). k = Q( '--q) if -1) = = 2 version p = 2 and denote comnmtators k with. ,\ = 1, we can obtain an abelian. presentation of G(koo) which is not necessarily Theorem 3.4 Let p = 2 and k = Q( "=m) be an .,.'f{,."'"''·"u, . .,..,, with positive sqttarefree m = 1 (mod4), J:(+ =a~~·· ,-) ' 1 _ \\,~~ m . " = .. then G(koo) =(a, b I b] = 2N+l ., )pro-2 a =.t where 2N is the order G(Kt,) which is finite cyclic. ON UNRAMIF!ED PRO-p GALOIS GROUPS OVER CYCLOTOMIC Zp-EXTENSIONS -·- A SURVEY 231 Corollary 3JS. X(il(k,x,) ~ Z/2Z 2 :S: i :S: N + 1, and #X(i) (koo) = l N + 2 :S: i. = l, = 0 all i ;?: 2 and } < oo. In Theorem 3A, the G(koo) is nonabelian if and only if N;?: 1, and such cases exist. For example, N = 1 if m = 13 · 29. :Further, we have the following as an example met abelian Theorem 3.6 ([12]). Let p with 2 and k = Q( q2 = 7 we lwve a presentation = (a, b, c b] = a - 2 , [b, c] = , [a, c] = 1 the lwasawa Cm'ollary :5/L all i ;?: 2 and , where C\, Co E Z2 are the coefficients all i ;?: = 2, =0 The Galois group in Theorem :3.6 is 2-adic a Poincare pro-2 group of dimension it tw·e, the Galois groups t'TIJP•JSltiO'n 2.5 and i?ontaine-Mazur conjec-G(kn) are finite since G(koo) is metabelian. Further, action of r on , we can calculate the presenta.tions of for n ;?: 1 under some assumptions as follows. (It is well known that is uuvw:»u. 3J3 ( If z.e., q1 is not . ' n;szaue G(k1) = (a, b, c j b] = [b,c] = c] = a4 = 1 ). = -1 and -·· / r' b " - \'"'' , '-' b] = c] = , [a, c] = = 1) for- all n;?: 2. with q1q2 < one can see that P(T) = +(1+(q!/q2))T+ (1- the numerical computation of Stickelberger elements. one can expect that = 0 4) if ( qi/ q2 ) = 1, but it is not clear yet. Under the stronger assumptions that = -1 and C1 = 0 (mod4), there is another proof of the of G(koo) of Theorem 3.6. It is parallel to the if-part) of Theorem which is based on the calculation of "Gal(L(kn)/Q)" and the decomposition subgroups of some primes. By putting K = '=I, '=ql) and 232 YASUSHI MIZUSAWA F = '=I, '=ql), one can see that Gal(L(Kn)/ F) has a presentation which is very similar to the presentation of of Theorem 3.3. Finally, we remark that there are infinitely rna11y real in p = 2 case [1. 1]\ "' I. References [1] Boston, N., Galois p-groups unrair1ified at p - A survey, 416, Primes inner. Math. pp. 31-40 [2] M.P. Cambridge, 1999. The Iwasawa invariant /-"p vanishes for abelian number D'·''''u"M' curves, modv.lar· Internat. [.5] Fujii, S., On a no. 1, 55-61. S.and no. 1, 211-222. field towers, I, Tokyo J. J. [7] L On the class field tower, Izv. Akad. Nauk SSSR Ser. , 261-272. [8] On the Iwasawa invariants of 1~ 263-284. [9] Kida, Y., l-extensions of Clvi-fi.elds and cydotornic no. 4, 519-528. real number J. Afum.ber IVJ:azur, B. and no. 2, 179-330. Class flelds of abelian extensions of Q!, fnvent 12 .LVl:tzt!Sc<w•a. Y., On the maximal umamified Dro,-:L-E;xt of l£2 .. extensions of certain real no. 1, 93-107. i1elds II, Acta Arith., 119 On the over the extension of an uu.u5~u'"< [13] Y, and Abelian 2-class field towers over the quadratic fields, 'Tir,,nr'i.nt Analogies between numbers and (Japanese) 58 (English) to appear in Sugaku Expositions, AMS. [15] Schmidt, A. and Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschajten, 323, Springer-Verlag, Berlin, 2000. [16] K., Abelian p-class field towers over the cyclotomic Zp-extensions of irnaginary quadratic fields, Acta 125 no. 4, 363-381. [17] Ozaki, Non-abelian Iwasawa. (.Japanese) philosopheTs in nnmber- 2001), RHviS Kokyilroku, 1256 (2002), 25-37. [18] Ozaki, Non-abelian lwasawa theory of Zp-extensions, J, Reine ll.ngeVJ. Math., 602 59-94. ON UNRAM!F!ED PRO-p GALOIS GROUPS OVER CYCLOTOMIC 233 [19] R. T,, Massey and ideal class groups, J. Reine 603 1-33. [20] R. T., Galois groups of unramified pro-p Extensions Global Fields and pro-:JJ Grottps, Oberwolfach vol. 3 (2006), issue 2, 1473-1475. L. Introduction to Cyclotomic Fields, second edition, Graduate Texts in voL 83, Springer, 1997. The Iwasawa conjecture for totally real no. 3, 493-540. Reine 440 129-156. V/ingberg, On the F'ontaine-Mazm (2002), no. 3, 341-354. for Ann. Math. 131 number field, J. Compositio 131 ''-H""""'""'"' of Iwasawa invariants of absolutely abelian p-extensions, no. 4, 365-371. 

On unramified pro-$p$ Galois groups over cyclotomic $mathbb{Z}_p$-extensions - A survey (Proceedings of the Symposium on Algebraic Number theory and Related Topics)

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On unramified pro-$p$ Galois groups over cyclotomic $\mathbb{Z}_p$-extensions - A survey (Proceedings of the Symposium on Algebraic Number theory and Related Topics)

On unramified pro- $p$ Galois groups over cyclotomic $\mathbb{Z}_p$ -extensions - A survey (Proceedings of the Symposium on Algebraic Number theory and Related Topics)

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On unramified pro-ppp Galois groups over cyclotomic Zp\mathbb{Z}_pZp​-extensions - A survey (Proceedings of the Symposium on Algebraic Number theory and Related Topics)

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On unramified pro- $p$ Galois groups over cyclotomic $\mathbb{Z}_p$ -extensions - A survey (Proceedings of the Symposium on Algebraic Number theory and Related Topics)