Recent Progress in Determining P-Class Field Towers

Recent Progress in Determining p-Class Field Towers

Conference: 1st International Colloquium ANCI 2016 Place: Facult´ePolydisciplinaire Taza, USMBA Venue: Taza, Morocco Date: November 11 – 12, 2016 Author: Daniel C. Mayer (Graz, Austria) Aﬃliation: Austrian Science Fund (FWF)

Keynote within the frame of the international scientiﬁc research project Towers of p-Class Fields over Algebraic Number Fields supported by the Austrian Science Fund (FWF): P 26008-N25

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Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

INTRODUCTION.

• The key for determining the Galois group (∞) (∞) G := Gp F = Gal(Fp /F ) (∞) of the unramified Hilbert p-class field tower Fp , i.e. the maximal unramified pro-p extension, of an algebraic number field F is the Artin pattern AP(G) combined with bounds for the relation rank d2G. G is briefly called the p-tower group of F .

• My principal goal is to draw the attention of the audience to the striking novelty of three-stage towers of p-class ﬁelds over quadratic, cubic and quartic number ﬁelds F , discovered by myself in the past four years, partially in cooperation with M. R. Bush (WLU, VA) and M. F. Newman (ANU, ACT).

• This lecture can be downloaded from http://www.algebra.at/ANCI2016DCM.pdf

• It is an updated and compact version of my article [9] D. C. Mayer, Recent progress in determining p-class ﬁeld towers, Gulf J. Math. (Dubai, UAE), arXiv: 1605.09617v1 [math.NT] 31 May 2016. 3

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

1. The Hilbert p-class field tower

Assumptions: p . . . a prime number, F... an algebraic number field, ClpF := SylpClF ... the p-class group of F . (n) Definition. For n ≥ 0, the nth Hilbert p-class field Fp is the maximal unramified Galois extension of F with group n (n) Gp F := Gal Fp /F n having derived length dl(Gp F ) ≤ n and order a power of p. (∞) The Hilbert p-class field tower Fp over F is the maximal unramified pro-p extension of F and has the Galois group ∞ (∞) n Gp F := Gal Fp /F ' ←−lim Gp F. n≥0

(1) (2) (n) (∞) F ≤ Fp ≤ Fp ≤ ··· ≤ Fp ≤ ··· ≤ Fp

1 GpF

2 GpF 2nd p-class group

n Gp F nth p-class group

∞ G := Gp F p-tower group 4

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 n The nth stage Gp F of the p-class tower arises as the nth derived quotient G/G(n) of the p-tower group G, for n ≥ 1. In particular, for n = 2, the second p-class group 2 (2) M := GpF ' G/G can be viewed as a two-stage approximation of G.

(1) (2) (n) (∞) F ≤ Fp ≤ Fp ≤ ··· ≤ Fp ≤ ··· ≤ Fp

G/G0

2 00 GpF ' G/G 2nd p-class group

n (n) Gp F ' G/G nth p-class group

∞ G = Gp F p-tower group

Deﬁnition. The derived length dl(G) of the p-tower group ∞ G = Gp F is called the length `pF of the p-class tower of F . (∞) When do we consider the p-class tower Fp as “known”?

Is it suﬃcient to determine its length `pF ? No, not at all! The length alone is a poor amount of information. According to my conviction, we are not done before we can give an explicit pro-p presentation of the p-tower group ∞ (∞) G = Gp F = Gal Fp /F . 5

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 An equivalent inductive definition of the p-class tower: (∞) S (i) The tower Fp := i≥0 Fp arises recursively from the base (0) field Fp := F by the successive construction of maximal (1) (n) (n−1) abelian unramified p-extensions Fp := Fp , n ≥ 1. p

By the Artin reciprocity law of class ﬁeld theory [1], the (n) (n−1) relative Galois group Gal Fp /Fp is isomorphic to the (n−1) (abelian!) p-class group ClpFp , for each n ≥ 1. In particular, for n = 1, we have the well-known fact that the ﬁrst p-class group is isomorphic to the ordinary p-class group 1 (1) GpF = Gal Fp /F ' ClpF. In particular, for n = 2, the second p-class group (2) M = Gal Fp /F has an abelian commutator subgroup 0 (2) (1) (1) M = Gal Fp /Fp ' ClpFp and is therefore metabelian. The abelianization is 0 (1) M/M ' Gal Fp /F ' ClpF.

Similarly, the abelianization of the p-tower group G is 0 (∞) (∞) (1) (1) G/G = Gal Fp /F /Gal Fp /Fp ' Gal Fp /F ' ClpF and consequently G is a ﬁnitely generated pro-p group. 6

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

2. Infinite p-class towers

Let G be a finitely generated pro-p group with generator rank 1 2 d1 := dimFp H (G, Fp) and relation rank d2 := dimFp H (G, Fp). Theorem (Golod, Shafarevich, 1964 [2]). 2 d1−1 If G is finite, then d2 > 2 . Theorem (Refinement by Vinberg, Gaschütz,1965 [3]). 2 2 (d1) (d1) #G < ∞ ⇒ d2 > 4 , resp. d2 ≤ 4 ⇒ #G = ∞. Let 1 → R → F → G → 1 be a presentation of G with a free pro-p group F such that d1F = d1G, and suppose that R ⊂ Fn for a term of the Zassenhaus filtration (Fn)n≥1 of F . Theorem (Refinement by Koch, Venkov, 1975 [4]). n (d1) n−1 If G is finite, then d2 > nn · (n − 1) . Corollary (Koch, Venkov, 1975 [4]). Let p ≥ 3 be odd, then √ a complex quadratic field F = Q( d) with p-class rank %p ≥ 3 has an infinite p-class tower with `pF = ∞. Proof. Since F has Dirichlet unit rank r = 0, the p-tower ∞ group G = Gp F has a balanced presentation with d2 = d1 and R ⊂ F3. Thus, a sufficient condition for `pF = ∞ is 3 q √ (d1) 2 27 3 d1 ≤ 33 · 2 , i.e. %p = d1 ≥ 4 = 2 3 ≈ 2.598. Example. d = −4 447 704 ⇒ Cl3F ' (3, 3, 3) ⇒ `3F = ∞. However, ∞ we are far from knowing a pro-3 presentation of G = G3 F . We only have the lower bound #M ≥ 317 for M = G/G00. 7

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

3. The relation rank of the p-tower group

Theorem (I. R. Shafarevich, 1964 [5]). Let p ≥ 2 be a prime number, and K be an algebraic number field with signature (r1, r2) and torsion free Dirichlet unit rank r = r1 + r2 − 1, ζ a primitive pth root of unity, ∞ ∞ G = Gp (K) = Gal(Fp (K)|K) the Galois group of the ∞ maximal unramified pro-p extension Fp (K) of K, 1 d1 = dimFp H (G, Fp) the generator rank of G, 2 d2 = dimFp H (G, Fp) the relation rank of G. Then ( 0 if ζ 6∈ K, d ≤ d ≤ d + r + θ, where θ = 1 2 1 1 if ζ ∈ K. √ Corollary. K = Q( d) quadratic field with discriminant d, ∞ ∞ G = Gp (K) = Gal(Fp (K)|K) the p-class tower group of K. Then  d = d if (d < 0 and p ≥ 3),  2 1 d1 ≤ d2 ≤ d1 + 1 if either (d < 0 and p = 2) or (d > 0 and p ≥ 3),  d1 ≤ d2 ≤ d1 + 2 if (d > 0 and p = 2).

[5] I. R. Shafarevich, Extensions with prescribed ramiﬁcation points, Publ. Math., Inst. Hautes Etudes´ Sci. 18 (1964), 71–95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128–149. 8

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

4. Leaving the realm of class field theory p . . . a prime number, F... an algebraic number ﬁeld, E/F . . . an unramiﬁed abelian p-extension, ClpE... the p-class group of E, TF,E : ClpF → ClpE, a ·PF 7→ (aOE) ·PE ... the transfer of p-classes from F to E.

We translate TF,E from number theory to group theory by entering the system of metabelian unramiﬁed extensions [6]:

System of abelian System of metabelian unramiﬁed extensions unramiﬁed extensions

(1) (1) (2) F ≤ E ≤ Fp ≤ Ep ≤ Fp

Galois correspondence l l l l l

M D H D M0 D H0 D 1 k k k k (2) (2) (2) (1) (2) (1) G Fp /F G Fp /E G Fp /Fp G Fp /Ep

abelian quotient (1) G Ep /E 0 ' ClpE ' H/H 9

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

(2) 0 (2) (1) Let H := Gal(Fp /E), then H := Gal(Fp /Ep ), and the p-class transfer TF,E is connected with the Artin transfer TM,H by two applications of the Artin reciprocity map:

TF,E

ClpF −→ ClpE Artin isomorphism l /// l Artin isomorphism M/M0 −→ H/H0

TM,H

In particular, we have 0 isomorphic domains, ClpF ' M/M , isomorphic kernels, ker TF,E ' ker TM,H, 0 isomorphic targets, ClpE ' H/H . Definition. Recall that the Artin transfer [6] from a group G to a subgroup H ≤ G with finite index n := (G : H) is defined with the aid of the permutation π ∈ Sn of a transversal S˙ n G = i=1 ì · H induced by the action of an element x ∈ G: n 0 0 0 Y −1 0 TG,H : G/G → H/H , x · G 7→ `π(i)xì · H , i=1 where xì · H = `π(i) · H, for each 1 ≤ i ≤ n. [6] E. Artin, Idealklassen in Oberkörpern und allgemeines Rezi- prozitätsgesetz, Abh. Math. Sem. Univ. Hamburg 7 (1929), 46–51. 10

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

5. The Artin pattern

Assumptions: p . . . a prime number, F... an algebraic number field, G ... a pro-p group or a finite p-group, X... a placeholder, either X = F or X = G. Definition. We call ( Cl F if X = F, τ X := p 0 G/G0 if X = G, bottom, resp. top, layer of abelian quotient invariants of X. t Suppose that #τ0X = p for some integer t ≥ 0, and let 0 ≤ n ≤ t be an integer. Definition. The finite set ( (1) n {F ≤ E ≤ Fp | [E : F ] = p } if X = F, LyrnX := 0 n {G ≤ H E G | (G : H) = p } if X = G, is called the nth layer of abelian unramified p-extensions, resp. of intermediate normal subgroups, of X.

Deﬁnition. For any 0 ≤ n ≤ t and Y ∈ LyrnX, the mapping TX,Y : τ0X → τ0Y , ( x ·PX 7→ (xOY ) ·PY if X = F, 0 Qn −1 0 x · X 7→ i=1 `π(i)x`i · Y if X = G, is called the p-class transfer, resp. the Artin transfer [6], from X to Y . 11

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Deﬁnition. We call

τn(X) := (τ0Y )Y ∈LyrnX, for 0 ≤ n ≤ t, the components of the multi-layered Transfer Target Type (TTT) τ(X) := [τ0X; ... ; τtX],

κn(X) := (ker TX,Y )Y ∈LyrnX, for 0 ≤ n ≤ t, the components of the multi-layered Transfer Kernel Type (TKT) κ(X) := [κ0X; ... ; κtX]. The pair AP(X) := (τ(X), κ(X)) is called the abelian Artin pattern of X. Definition. The index-p abelianization data (IPAD) of X is a first order approximation of the multi-layered TTT, (1) τ X := [τ0X; τ1X]. A generalization to non-abelian unramified extensions is given recursively by the iterated IPAD of order n ≥ 2, (n) (n−1) τ X := [τ0X;(τ Y )Y ∈Lyr1X], (0) formally supplemented by τ X := τ0X.

Definition. The cover of a finite metabelian p-group M is defined as the set cov(M) := {G | #G < ∞, G/G00 ' M}, and the total cover of M arises by dropping the finiteness condition: 00 covtotM := {G | G/G ' M}. 12

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Theorem. (Uniformity of the Artin pattern on the total cover) If M is a ﬁnite metabelian p-group, then

(∀G ∈ covtotM) AP(G) = AP(M). That is, any pro-p group G shares a common abelian Artin pattern with its metabelianization M = G/G00. Corollary. (Uniformity of the Artin pattern of all higher p-class groups) n All higher p-class groups Gp F of a number ﬁeld F share a 2 common abelian Artin pattern with M := GpF , n (∀n ≥ 2) AP(Gp F ) = AP(M).

n n 00 2 n 2 Proof. Gp F/ Gp F ' GpF , and thus Gp F ∈ covtotGpF , for any n ≥ 2. Theorem. 2 (Coincidence of the Artin pattern of a field F and of GpF ) Each number field F shares a common abelian Artin pattern 2 with its second p-class group M := GpF , AP(F ) = AP(M). Corollary. 2 (Coincidence of the IPAD of a field F and of GpF ) Each number field F shares a common IPAD with its second 2 p-class group GpF , (1) (1) 2 τ F = τ GpF.

Proof. AP(F ) = (τ(F ), κ(F )) and τ(F ) = [τ0F ; ... ; τtF ] (1) contains τ F = [τ0F ; τ1F ]. 13

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Iterated IPADs of increasing order approximate the complete ∞ information on the p-tower group G = Gp F . Successive Approximation Theorem. (Coincidence of the IPAD of order m of a ﬁeld F and of all n groups Gp F with n ≥ m + 1) For any order m ≥ 0, the number ﬁeld F and all higher p-class n groups Gp F with n ≥ m + 1 share a common iterated IPAD of order m, (m) (m) n (∀n ≥ m + 1) τ F = τ Gp F.

Corollary. ∞ (Coincidence of all iterated IPADs of a ﬁeld F and of Gp F ) The number ﬁeld F shares all iterated IPADs with its p-tower ∞ group G = Gp F , (m) (m) ∞ (∀m ≥ 0) τ F = τ Gp F.

The length of the p-class tower can be determined by repeated applications of the following criterion. Stage Separation Conjecture. (Proving length bigger than m by means of IPAD of order m) (m) (m) m (∀m ≥ 0) `pF > m ⇐⇒ τ F > τ Gp F. That is, the iterated IPAD of order m indicates when the p- class tower has more than m stages.

Proof. of “⇐=” by contraposition: `pF ≤ m =⇒ (m) (m+1) (m) (m) m+1 (m) m Fp = Fp =⇒ τ F = τ Gp F = τ Gp F . 14

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Our aim is to give an explicit pro-p presentation of the p-tower group of F , ∞ (∞) G = Gp F = Gal Fp /F . For this purpose, we proceed in several steps: (1) We compute the p-class groups ClpE and the p-capitulation ker TF,E of all unramified cyclic extensions E/F of relative degree p. (2) With the aid of Artin’s reciprocity law, we interpret this number theoretic information (τ1F, κ1F ) as group theoretic invariants (τ1M, κ1M) of the metabelian second p-class group of F , 2 (2) M = GpF = Gal Fp /F . (3) By a search for the assigned invariants in the descendant tree T R with root R ' ClpF we find a finite batch of contestants for M, using Thm. M as break-off condition. (4) We construct small members of the cover cov(M) of M which satisfy the Shafarevich inequality d2 ≤ d1 + r + θ for the relation rank d2 = d2(G) in dependence on the generator rank d1 = d1(G), the Dirichlet unit rank r, and the invariant θ. Theorem M. (Monotony of the Artin pattern on trees) Let T R be the descendant tree with a finite non-trivial p- group as its root R > 1 and let G → πG be a directed edge of the tree. Then the abelian Artin pattern AP = (τ, κ) satisfies the following monotonicity relations (in componentwise sense) τ(G) ≥ τ(πG), κ(G) ≤ κ(πG). 15

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

6. Contestants for the second p-class group

τ0 ... finite abelian p-group with generator rank d ≥ 2, pd−1 τ1 ... family (τ1(i))1≤i≤n of n := p−1 abelian type invariants. 2 Definition. By Cntp(τ0, τ1) we denote the set of all (isomorphism classes of) finite metabelian p-groups M such that 0 0 τ0M = M/M ' τ0 and τ1M = (H/H )H∈Lyr1M ' τ1. Conjecture. (Finiteness of the batch of contestants) 2 Cntp(τ0, τ1) is a finite set. 2 Remark. Note that Cntp(τ0, τ1) = ∅, when τ1 is malformed. 2 Theorem 1. p = 3, τ0 ' (3, 3) =⇒ #Cnt3(τ0, τ1) < ∞. 2 Theorem 2. p = 2, τ0 ' (2, 2, 2) =⇒ #Cnt2(τ0, τ1) < ∞. The general Conjecture holds if the following Principle is true. Polarization Principle. There exist a few components of a non-malformed family τ1 which determine the nilpotency class c := cl(M) and the coclass r := cc(M) of a finite metabelian 0 p-group M with τ1M = (H/H )H∈Lyr1M ' τ1. Proof of Theorem 1, resp. 2, is based on Theorem 3, resp. 4.

Theorem 3. (Bipolarization) p = 3, and τ0 ' (3, 3) =⇒ c−k r+1 #τ1(1) = 3 with 0 ≤ k ≤ 1, and #τ1(2) = 3 . Theorem 4. (Unipolarization with independent factors) c r−1 p = 2, and τ0 ' (2, 2, 2) =⇒ τ1(1) ' (3 , 3 ). 16

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

7. The first 3-class towers of length 3

Notation. Logarithmic type invariants of abelian 3-groups, for instance (21) ˆ=(9, 3), (32) ˆ=(27, 9), and (43) ˆ=(81, 27). √ Let F = Q( d) be an imaginary quadratic number ﬁeld, with 3-class group Cl3F ' (3, 3), and E1,...,E4 be the unramiﬁed cyclic cubic extensions of F . Theorem. Suppose the capitulation of 3-classes of F in E1,...,E4 is of type κ1F ∼ (1, 2, 3, 1) (called type E.8). Assume further that the 3-class groups of E1,...,E4 are of type τ1F ∼ [T1, 21, 21, 21], where T1 ∈ {32, 43, 54}.

Then the length of the 3-class tower of F is `3F = 3. Proof. We employ the p-group generation algorithm for search- ing the Artin pattern AP(F ) = (τ1F, κ1F ) among the descen- dants of the root R := C3 × C3 = h9, 2i in the tree T R. After two steps, h9, 2i ← h27, 3i ← h243, 8i, we find the next 2 root U5 := h243, 8i of the unique relevant coclass tree T U5, using the assigned TKT E.8, κ3 = (1231), and its scaffold TKT c.21, κ0 = (0231). Finally, the first component T1 = τ1(1) ∈ {32, 43, 54} of the TTT provides the break-off condition, according to Theorem M, and we get M ' h2187, 304i = h729, 54i − #1; 4 for the ground state T1 = (32), M ' h729, 54i−#1; 3−#1; 1−#1; 2 for the 1st excited state T1 = (43), and M ' h729, 54i − 3 #1; 3(−#1; 1) − #1; 2 for the 2nd excited state T1 = (54), where h729, 54i − #1; 3 = h2187, 303i. The situation is visualized by the next figure. 17

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

order 3n τ1(1) = h8i 243 35 (21) Q ¡ Q ¡ t Q branch Q B(5) ¡ Q ¡ +540 365 Q h53i h55i h54iQ h52i 6 2 ¡ Q 729 3 (2 ) bifurcation from ¡@Q G(3, 2) to G(3, 3)Q Q Q h294i t t ¡ t @Q t Q h295i B(6) h296i ¡ @Q Q h297i h302i Qh307i h301i Q h298i h300i h309i h306ih 304i¡h 303i @ hQ308i h305i Q h299i ∗ ∗22 Q ∗2 Q ∗6 7 ¡ @@ Q Q 2 187 3 (32) Q Q ¡@Q A@Q A 6 t t ¡ t @Q tA@Q r A -9 748 -34 867 Q B(7) Q ¡ @ Q A @ Q A ¡ +1 001@ 957 Q A @ Q #4 A Q ∗2Q ∗2 8 2 ¡ @@ Q A @@ Q A 6 561 3 (3 ) Q Q 6 ¡@Q A@Q A t t ¡ t @Q tA@Q r r A Q B(8) Q -17 131 ¡ @ Q A @ Q A depth 3 ¡ @ Q A @ Q A 2∗ Q ∗2 ∗3Q ∗3 ∗2 9 ¡ @@ Q A @@ Q A 19 683 3 (43) period length 2 Q Q ¡@Q A@Q A t t ¡ t @Q tA@Q r r A -297 079 -370 740 Q B(9) Q ¡ @ Q A @ Q A ¡ +116 043@ 324Q A @ Q #4 A Q ∗2Q ∗2 ∗2 10 2 ? ¡ @@ Q A @@ Q A ? 59 049 3 (4 ) Q Q ¡@Q A@Q A t t ¡ t @Q tA@Q r r A Q B(10) Q -819 743 ¡ @ Q A @ Q A ¡ @ Q A @ Q A 2∗ Q ∗2 ∗3Q ∗3 ∗2 11 ¡ @@ Q A @@ Q A 177 147 3 (54) Q Q ¡@Q A@Q A t t ¡ t @Q tA@Q r r A -1 088 808 -4 087 295 Q B(11) Q ¡ @ Q A @ Q A ¡ @ Q A @ Q #4 A Q ∗2Q ∗2 ∗2 12 2 ¡ @@ Q A @@ Q A 531 441 3 (5 ) Q Q ¡@Q A@Q A t t ¡ t @Q tA@Q r r A Q B(12) Q-2 244 399 ¡ @ Q A @ Q A ¡ @ Q A @ Q A 2∗ Q ∗2 ∗3Q ∗3 13 ¡ @@ Q A @@ Q A 1 594 323 3 (65) Q Q ¡@Q A@Q A t t ¡ t @Q tA@Q r r A -11 091 140 -19 027 947 Q B(13) Q ¡ @ Q A @ Q A ¡ @ Q A @ Q #4 A Q ∗2Q ∗2 14 2 ¡ @@ Q A @@ Q A 4 782 969 3 (6 ) Q Q ¡@Q @A Q A t t t t r r ¡ @Q B(14) A@Q -30 224 744 A ¡ @Q A @Q A -94 880 548 Q Q ¡ inﬁnite@ Q A @ Q A 2∗ Q ∗2 ∗3Q ∗3 15 ¡ ?mainline @@ Q A @@ Q A 14 348 907 3 (76) 2 t t T (h243, 8i) t r r TTT TKT E.9 E.8 c.21 c.21 G.16 G.16 G.16 G.16 G.16 κ1 = (2231) (1231) (0231) (0231) (4231)(4231) (4231) (4231) (4231) ? 18

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Proof. (continued) The last figure, showing the second 3-class groups M, was essentially known to J. A. Ascione in 1979, and to B. Nebelung in 1989. In the next three figures, which were unknown until 2012, we present the decisive break-through establishing the first rigorous proof for three-stage towers of 3-class fields. The key ingredient is the discovery of periodic bifurcations in the complete descendant tree T U5 which is of considerably higher 2 complexity than the coclass tree T U5. For the ground state T1 = (32), the first bifurcation yields the cover cov(M) = {M, h729, 54i−#2; 4} of M ' h2187, 304i = h729, 54i − #1; 4. The relation rank d2M = 3 eliminates M as a candidate for the 3-tower group G, according to the Corollary of the Shafarevich Theorem, and we end up getting G ' h729, 54i − #2; 4 = h6561, 622i with a siblings topology 1 2 E → c ← E describing the relative location of M and G. For the 1st excited state T1 = (43), the second bifurcation yields the cover cov(M) = {M, h729, 54i − #2; 3 − #1; 1 − #1; 2, h729, 54i − #2; 3 − #1; 1 − #2; 2} of M ' h729, 54i − #1; 3 − #1; 1 − #1; 2. The relation rank d2 = 3 eliminates M and h729, 54i − #2; 3 − #1; 1 − #1; 2 as candidates for the 3-tower group G, according to Shafarevich, and we get the unique G ' h729, 54i − #2; 3 − #1; 1 − #2; 2 with an 1 1 2 2 1 2 advanced fork topology E → c → c ← c ← c ← E describing the relative location of M and G. Similarly, the 2nd excited state T1 = (54) yields an advanced 1 1 4 2 1 2 2 fork topology E → c → c ← c ← c ← E. 19

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Order h8i Symmetric topology symbol (ground state): 243 5 3 Leaf Leaf t z }| !{ Fork z }|! { 1 2 E z}|{c E h54i → ← 729 36 ¡ACS@ ¡ tCAS@ 1st bifurcation Transfer kernel types: ¡ C AS@ ¡ 1;C 3A S@ E.8: κ3 = (1231), c.21: κ0 = (0231) 2 187 37 ¡ C A S @ 1; 6HYH HYH1; 4HYH t 1;Ht 2 Ht Ht C A S @ Minimal discriminant: H H HC A S @ HH HH HH −34 867 H HC AH S @ 1;H 1 H H 2; 3 8 HC HA HS @@ 6 561 3 ¡ 2; 6 2; 4 ¡ t 2; 2 ¡ ¡ 1; 1 1; 1 9 ¡ 19 683 3 1; 6 1; 2 ¡A@SC t 1;t 4 t t ¡ CAS@ 2nd bifurcation ¡ C AS@ 1; 1 ¡ 1;C 1A S@ 10 ¡ 59 049 3 C A S @ ¡ 1; 6 1; 2 C A S @ ¡ t 1; 4 C A S @ ¡ C A S @ ¡ 1; 1 1; 1 2; 1 11 ¡ C A S @@ 177 147 3 1; 6 1; 2 ¡ 2; 6 2; 2 t 1;t 4 t t ¡ 2; 4 ¡ ¡ 1; 1 1; 1 12 ? ¡ 531 441 3 T 2(h243, 8i) 1; 6 1; 2 ¡CA@S ∗ 1; 4 ¡ CAS@ 3rd bifurcation ¡ C AS@ ¡ 1;C 1A S@ 1 594 323 313 ? ¡ C A S @ 3 1; 6 1; 2 T∗ (h729, 54i − #2; 3) 1; 4 C A S @ C A S @ C A S @ C A S @ 2; 1 4 782 969 314 ? C A S @ 4 2; 6 2; 2 T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1) 2; 4

? 5 ? T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1 − #1; 1 − #2; 1)

TKT: κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 20

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Order h8i Symmetric topology symbol (1st excited state): 243 35 Leaf Mainline Trunk Leaf z }| { t z }| !{ ( !)2 Fork z( ! }| ! ){ z }|! { 1 1 2 1 2 E c z}|{c c c E h54i → → ← ← ← 729 36 ¡@ACS ¡ tCA@S 1st bifurcation Transfer kernel types: ¡ C AS@ ¡ 1;C 3 A S@ E.8: κ3 = (1231), c.21: κ0 = (0231) 2 187 37 ¡ C A S@ 1; 6 1; 4 t 1;t 2 t t C A S @ Minimal discriminant: C A S @ −370 740 C A S @ 1; 1 2; 3 8 C A S @@ 6 561 3 ¡ 2; 6 2; 4 ¡ t 2; 2 ¡ ¡ 1; 1 1; 1 9 ¡ 19 683 3 HY HY HY 1; 6 H H1; 2 H ¡A@SC 1; 4 nd t Ht H Ht H Ht H ¡ CAS@ 2 bifurcation H H H ¡ C AS@ HH HH HH 1;H 1 H H ¡ 1;C 1A S@ 10 H H H ¡ 59 049 3 H H H C A S @ ¡ 1; 6 H H1; 2 H t 1;H 4 H H C A S @ ¡ H H HC A S @ ¡ H H H HH HHC HAH S @ ¡ 1; 1 H1; 1 H H 2; 1 11 ¡ HC HA HS @@ 177 147 3 1; 6 1; 2 ¡ 2; 6 2; 2 t 1;t 4 t t ¡ 2; 4 ¡ ¡ 1; 1 1; 1 12 ? ¡ 531 441 3 T 2(h243, 8i) 1; 6 1; 2 ¡CA@S ∗ 1; 4 ¡ CAS@ 3rd bifurcation ¡ C AS@ ¡ 1;C 1A S@ 1 594 323 313 ? ¡ C A S @ 3 1; 6 1; 2 T∗ (h729, 54i − #2; 3) 1; 4 C A S @ C A S @ C A S @ C A S @ 2; 1 4 782 969 314 ? C A S @ 4 2; 6 2; 2 T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1) 2; 4

? 5 ? T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1 − #1; 1 − #2; 1)

TKT: κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 21

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Order h8i Symmetric topology symbol (2nd excited state): 243 35 Leaf Mainline Trunk Leaf z }| { z }| { t z }| !{ ( !)4 Fork ( ! ! )2 z }|! { 1 1 2 1 2 E c z}|{c c c E h54i → → ← ← ← 729 36 ¡@ACS ¡ tCA@S 1st bifurcation Transfer kernel types: ¡ C AS@ ¡ 1;C 3 A S@ E.8: κ3 = (1231), c.21: κ0 = (0231) 2 187 37 ¡ C A S@ 1; 6 1; 4 t 1;t 2 t t C A S @ Minimal discriminant: C A S @ −4 087 295 C A S @ 1; 1 2; 3 8 C A S @@ 6 561 3 ¡ 2; 6 2; 4 ¡ t 2; 2 ¡ ¡ 1; 1 1; 1 9 ¡ 19 683 3 1; 6 1; 2 ¡@ACS t 1;t 4 t t ¡ CA@S 2nd bifurcation ¡ C AS@ 1; 1 ¡ 1;C 1 A S@ 10 ¡ 59 049 3 C A S@ ¡ 1; 6 1; 2 C A S @ ¡ t 1; 4 C A S @ ¡ C A S @ ¡ 1; 1 1; 1 C A S @ 2; 1 177 147 311 ¡ C A S @ 1; 6HYH HYH 1; 2HYH ¡ 2; 6 2; 2 H1; 4 HH HH 2; 4 t tHH t H t H ¡ H H H ¡ H H HH HH HH H ¡ 1; 1 1; 1 12 ? H H H¡ 531 441 3 H H H 2 1; 6 H H1; 2 H ¡CA@S T∗ (h243, 8i) 1;H 4 H H rd H H H ¡ CAS@ 3 bifurcation H H H ¡ C AS@ H H HH HH HH H ¡ 1;C 1A S@ 13 ?H H H ¡ 1 594 323 3 H H H C A S @ 3 1; 6H H 1; 2H T∗ (h729, 54i − #2; 3) H1; 4 H H C A S @ H H H C A S @ HH HH HH H HC HA S @ H HC AH S @ 2; 1 4 782 969 314 ?H C HA HS @ 4 2; 6 2; 2 T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1) 2; 4

? 5 ? T∗ (h729, 54i − #2; 3 − #1; 1 − #2; 1 − #1; 1 − #2; 1)

TKT: κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 κ1 κ2 κ3 κ0 22

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General theorems on three-stage√ towers of 3-class fields over quadratic fields F = Q( d) whose second 3-class group M = 2 G3F belongs to a periodic sequence on a coclass tree. Theorem E. A parametrized infinite family of fork topolo- gies is given uniformly for the states ↑n, n ≥ 0, of any TKT in section E by the symmetric topology symbol Leaf Mainline Trunk Leaf z }| { z }| 2n{ z }| n{ z }| { 1 1 Fork 2 1 2 E c z}|{c c c E → → ← ← ← with scaffold type c and the following invariants: distance d = 4n + 2, weighted distance w = 5n + 3, class increment ∆cl = (2n + 5) − (2n + 5) = 0, coclass increment ∆cc = (n + 3) − 2 = n + 1, logarithmic order increment ∆lo = (3n+8)−(2n+7) = n+1. Theorem c. A parametrized infinite family of fork topolo- gies is given uniformly for the states ↑n, n ≥ 0, of any TKT in section c by the symmetric topology symbol Mainline Path Leaf z }| 2n{ z }| n{ z }| { 1 Fork 2 1 1 c z}|{c c c c → ← ← ← (with identical scaffold type c) and the following invariants: distance d = 4n + 1, weighted distance w = 5n + 1, class increment ∆cl = (2n + 5) − (2n + 4) = 1, coclass increment ∆cc = (n + 2) − 2 = n, logarithmic order increment ∆lo = (3n+7)−(2n+6) = n+1. The ground state ↑0 degenerates to a child topology. 23

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

8. Illuminating 3-class towers of length 3

Notation. Logarithmic type invariants of abelian 3-groups, for instance (321) ˆ=(27, 9, 3), (221) ˆ=(9, 9, 3), and (412) ˆ=(81, 3, 3).

Let F be a number ﬁeld with Cl3 ' (3, 3) and Artin pattern AP(F ) = (τ1F, κ1F ), where κ1F ∼ (4, 2, 3, 1) is of type G.16 and τ1F ∼ [32, 21, 21, 21] indicates the ground state.

Theorem√ 1. (Imaginary quadratic with fork topology) If F = Q( d), d < 0, is imaginary quadratic, and τ (2)F = [(32; 321, (412)3), (21; 321, (31)3)3], then M ' h38, 2048|2058i, and G ' h38, 619|623i − #1; 4 − #2; 1 is a Schur σ-group.

Theorem√ 2. (Real quadratic with child topology) If F = Q( d), d > 0, is real quadratic, and τ (2)F = [(32; 321, (412)3), (21; 321, (21)3)3], resp. τ (2)F = [(32; 321, (312)3), (21; 321, (21)3)3], then M ' h38, 2048|2058i, and G ' M − #1; 1, resp. M − #1; 2 is a strong σ-group. Theorem 3. (Cyclic cubic with fork topology) If F is cyclic cubic, and τ (2)F = [(32; 221, (312)3), (21; 221, (31)3)3], then M ' h37, 301|305i, and G ' h38, 619|623i is a weak σ-group. 24

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 By arrows we denote projections G → M = G/G00 from the 3-tower group G onto its metabelianization M.

h54i Fork, type c.21 6 ¨ 729 3 ¨ @A ¨¨ u A@ ¨ ¨¨ A @ ¨ A @ type¨¨ G.16 A @ 7 ¨¨ 2 187 PiP PiP A @ 3 P P A @ h301i u h305PiPu PP PP PP A @ PP PP PP PAP @ PP A PP @ PP typePP G.16 8 PPAA P@P@ 6 561 3 Z} Z} ¡ 6 Z ¡ 6Z h2048i¡ r h2058Z i ¡r Z h619i h623i ¡ Z¡ Z ¡ ¡Z Z weak σ-groups Z Z ¡ ¡ Z Z 9 ¡ ¡ Z Z 19 683 3 Z Z A A #1; 1 Z #1;Z 4 2 1 2 Z Z A 4 A strong σ-groups Z Z A A Z ZA A Z Z Z AZ A 59 049 10 Z A Z A 3 Z A Z A Z Z Z A Z A Topology Symbols: Z A Z A Z A Z A 1 2 Z A Z A 177 147 311 G.16 c.21 G.16 ZA ZA → ← #2; 1 1 1 1 G.16 ← G.16G .16 ← G.16 Schur σ-groups ? G.16 1 G.16 1 c.21 2 G.16 1 G.16 2 G.16 Order 3n → → ← ← ←

Smallest concrete realizations: √ F = Q(√d) with d = −17 131, F = Q(√d) with d = +8 711 453, F = Q( d) with d = +9 448 265, F cyclic cubic ﬁeld with conductor c = 48 393. 25

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Theorem√ 4. (Real quadratic with fork topology) If F = Q( d), d > 0, is real quadratic, and τ (2)F = [(32; 3221, (3212)3), (32; 3221, (312)3), (13; 3221, (212)3, (13)9)2], then M ' h37, 64i − #2; 57|59 − #1; 1|6, and G ' h37, 64i − #3; 170|180 − #1; 1 − #1; i with i ∈ {7, 8, 10, 12, 14, 15} is a strong σ-group.

h64i Fork, type b.10 7 2 187 3 ¡JB ¡ uBJ ¡ B J ¡ B J ¡ B J 6 561 8 ¡ B J 3 ¡ B J ¡ B J ¡ B J ¡ B J type d.25*¡ B J 9 Pi ¡Pi 19 683 3 PP PP B J #2; 57 u PP u PP B J 59PP PP PP PP B J PP PPB J PP PP PP typeB PP d.25*J 10 PPB PPJ 59 049 3 ¡ HYH ¡ HYH #1; 1¡ u HH 6¡ u HH #3; 170 180 ¡ H¡ H HH HH ¡ ¡ H H ¡ ¡ HH HH 11 ¡ ¡ H H HH HH 177 147 3 H @ H @ #1; 7 5 u 7 5 u #1;HH 1 @ HH1 @ ...... H @ H @ HH HH H @ H @ 15 15 HH@ HH@ 531 441 312 H@ @H #1; i i strong σ-groups Topology Symbol: 1 594 323 313 ∗ 1 ∗ 2 3 ∗ 1 ∗ 1 ∗ d.25 → d.25 → b.10 ← d.25 ← d.25 ← d.25 ? Order 3n √ Smallest concrete realization: F = Q( d) with d = +8 491 713. 26

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

9. The first 5-class tower of length 3

Notation. Logarithmic type invariants of abelian 5-groups, for instance (12) ˆ=(5, 5), (13) ˆ=(5, 5, 5), and (213) ˆ=(25, 5, 5, 5). √ Let F = Q( d) be a real quadratic ﬁeld with 5-class group Cl5F ' (5, 5), and E1,...,E6 be the unramiﬁed cyclic quintic extensions of F .

Theorem 1. Suppose the 5-capitulation of F in E1,...,E6 5 is of type κ1(F ) ∼ (1, 0 ). Then the length of the 5-class tower of F is 3 2 5 (1) `5F = 2, if τ1(F ) ∼ [1 , (1 ) ], 3 2 5 (2) `5F = 3, if τ1(F ) ∼ [21 , (1 ) ].

(For the given TKT κ1 , the TTT τ1 determines the length.) √ Examples. Among the 377 quadratic ﬁelds F = Q( d) 2 with 5-class group Cl5F ' (1 ) and 0 < d < 26 695 193, 5 there are 57 with 5-capitulation of type κ1(F ) ∼ (1, 0 ). 3 2 5 (1) 55 of them have τ1(F ) ∼ [1 , (1 ) ] and thus `5F = 2, 3 2 5 (2) only 2 have τ1(F ) ∼ [21 , (1 ) ] and thus `5F = 3. The discriminants of the former start with 1 167 541, the latter have d ∈ {3 812 377, 19 621 905}. 27

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Corollary 1. Under the assumptions of Theorem 1, ∞ 2 4 (1) G5 F = G5F ' h625, 8i with order 5 , class 3, coclass 1 3 2 5 and relation rank 3, if τ1(F ) ∼ [1 , (1 ) ], 2 6 (2) G5F ' h15625, 635i with order 5 , class 5, coclass 1 and 3 2 5 relation rank 4, if τ1(F ) ∼ [21 , (1 ) ], ∞ 3 (3) G5 F = G5F ' h78125, ni, n ∈ {361, 373, 374, 385, 386}, each with order 57, class 5, coclass 2 and relation rank 3, if 3 2 5 τ1(F ) ∼ [21 , (1 ) ].

Let [213; 14, (212)155], [12; 14, (13)5]a, [12; 14, (21)5]b with a+b = 5 (2) denote the IPAD τ F of second order of F , when `5F = 3. 4 (1) (The common component (1 ) belongs to F5 .) Corollary 2. Under the assumptions of Corollary 1, τ (2)F admits a distinction among the identiﬁers n. (1) n ∈ {361, 373}, if (a, b) = (2, 3), (2) n ∈ {374, 385}, if (a, b) = (0, 5), (3) n = 386, if (a, b) = (1, 4). 28

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 The following ﬁgure shows the siblings topology which de- scribes the relative position of the second 5-class group M = 2 ∞ G5F and√ the 5-tower group G = G5 F of a real quadratic ﬁeld F = Q( d) satisfying the assumptions of Theorem 1. It is a special instance of a fork topology, characterized by a minimal vertex distance 2. πM = πG = h30i Fork, type a.1 5 3 125 5 A uA A A Topology Symbol: type a.2 A 1 2 15 625 56 A a.2 a.1 a.2 M = h635i A → ← u A A A 5∗ A type a.2 78 125 57 A G = h361|373|374|385|386i strong σ-group

? Order 5n 29

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

n 10. Action of Gal(F/Q) on Gp F

When F/Q is a normal extension, then its Galois group Gal(F/Q) acts on the class group ClF n n 0 and thus also on all commutator quotients Gp F/ Gp F ' ClpF of higher p-class groups with p ∈ P and n ∈ N ∪ {∞}. Let p be a prime number and d ≥ 2 be an integer. Definition. (σ-Groups of various degrees) A pro-p group G is called a σ-group of degree d, if it has an automophism σ ∈ Aut(G) of order ord(σ) = d such that Pd−1 i the trace Tσ := i=0 σ ∈ Z[Aut(G)] of σ annihilates the commutator quotient G/G0, that is, d−1 Y xTσ = σi(x) ∈ G0, for all x ∈ G. i=0 If d = 2, then G is simply called a σ-group. Definition. (Strong σ-group) A pro-p group G is called a strong σ-group, if it possesses an automophism σ ∈ Aut(G) of order ord(σ) = 2 such that σ acts as the inversion x 7→ x−1 on the cohomology groups 1 2 H (G, Fp) and H (G, Fp). Theorem. (Absolutely cyclic number fields) If F/Q is a cyclic extension of degree d, then all higher p-class n groups Gp F with n ∈ N ∪ {∞} are σ-groups of degree d. If d = 2, that is, when F/Q is a quadratic field, then the ∞ p-tower group G = Gp F is a strong σ-group. 30

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

10.1. Cyclic cubic fields F with Cl5F ' (5, 5) Proposition 1. (σ-groups of degree 3 with type (5, 5)) The transfer kernel type of a pro-5 group G with abelianization G/G0 ' (5, 5) which is a σ-group of degree 3 is restricted to the following admissible types • (0, 0, 0, 0, 0, 0), • two 3-cycles, • a 6-cycle, • the identity, • three 2-cycles. Corollary 1. (Cyclic cubic fields of type (5, 5)) 2 The second 5-class group G5F of a cyclic cubic field F with 5-class group Cl5F ' (5, 5) is restricted to the isomorphism classes of the following groups • h25, 2i, • h125, 3i, • a descendant of h3125, 3i, • h3125, 9i, • h3125, 12i, • h3125, 14i, • a descendant of h3125, 10i All these finite 5-groups are σ-groups of degree 3. 31

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 Examples 1. In the range 1 < c < 1 000 000 of conductors, there are 481 occurrences of Cl5F ' (5, 5). The dominating 2 part of 463 ﬁelds (96%) has G5F ' h125, 3i with six total transfer kernels κ1(F ) = (0, 0, 0, 0, 0, 0). Exceptions occur for the following 18 conductors only, conﬁrming Corollary 1:

2 No. c Factors G5F κ(F ) 1 66 313 13, 5101 h3125, 12i a 6-cycle 2 68 791 prime h3125, 14i the identity 3 77 971 103, 757 h3125, 14i the identity 4 87 409 7, 12487 h3125, 12i a 6-cycle 5 199 621 prime h3125, 9i two 3-cycles 6 317 853 9, 35317 h3125, 12i a 6-cycle 7 425 257 7, 79, 769 h3125, 14i the identity 8 464 191 7, 13, 5101 h3125, 12i a 6-cycle 9 481 537 7, 68791 h3125, 14i the identity 10 545 797 7, 103, 757 h3125, 14i the identity 11 596 817 9, 13, 5101 h3125, 12i a 6-cycle 12 619 119 9, 68791 h3125, 14i the identity 13 678 303 9, 75367 h3125, 12i a 6-cycle 14 701 739 9, 103, 757 h3125, 14i the identity 15 767 623 prime h3125, 9i two 3-cycles 16 786 681 7, 9, 12487 h3125, 12i a 6-cycle 17 894 283 13, 68791 h3125, 14i the identity 18 909 229 487, 1867 h3125, 14i the identity

The ﬁrst, resp. last, example of the dominating part is c = 6 901 = 67 · 103, resp. c = 96 733 = 7 · 13 · 1063. 32

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

10.2. Cyclic quartic fields F with Cl3F ' (3, 3) Proposition 2. (σ-groups of degree 4 with type (3, 3)) The transfer kernel type of a pro-3 group G with abelianization G/G0 ' (3, 3) which is a σ-group of degree 4 is restricted to the 6 admissible types a.1, b.10, D.5, F.7, G.16, G.19 among the 23 possible types. The remaining 17 types a.2, a.3, c.18, c.21, d.19, d.23, d.25, A.1, D.10, E.6, E.8, E.9, E.14, F.11, F.12, F.13, H.4 are forbidden. Corollary 2. (Cyclic quartic fields of type (3, 3)) 2 The second 3-class group G3F of a cyclic quartic field F with 3-class group Cl3F ' (3, 3) is restricted to the isomorphism classes of h9, 2i, h27, 3i, h243, 7i, a descendant of h243, 9i or a descendant of h243, 3i. All these finite 3-groups are σ-groups of degree 4. Examples 2. A cyclic quartic field has a unique represen- q √ tation F = Q a(d + b d) with d = b2 + c2, a ∈ Z, 2 b, c ∈ N, a and d squarefree and coprime. We have G3F ' h27, 3i, κ(F ) ∼ (0, 0, 0, 0), for (a, b, c, d) = (3, 9, 5, 106), h243, 7i, κ(F ) ∼ (4, 2, 2, 4), for (a, b, c, d) = (−1, 10, 7, 149), h729, 57i, κ(F ) ∼ (2, 1, 4, 3), for (a, b, c, d) = (−3, 10, 1, 101), h729, 37i, κ(F ) ∼ (0, 0, 4, 3), for (a, b, c, d) = (−5, 1, 6, 37), Y −#2; 48, κ(F ) ∼ (1, 2, 4, 3), for (a, b, c, d) = (−7, 5, 4, 41), where h729, 57i is an immediate descendant of h243, 9i, h729, 37i is an immediate descendant of h243, 3i, and Y := h2187, 64i is a descendant of step size 2 of h243, 3i. 33

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

10.3. Cyclic quartic fields F with Cl5F ' (5, 5) Proposition 3. (σ-groups of degree 4 with type (5, 5)) The transfer kernel type of a pro-5 group G with abelianization G/G0 ' (5, 5) which is a σ-group of degree 4 is restricted to the admissible types with two 2-cycles, a 4-cycle, identity, and descendant types of (0, 0, 0, 0, 0, 0), (0, 2, 2, 2, 2, 2), (0, 1, 1, 1, 1, 1). Corollary 3. (Cyclic quartic fields of type (5, 5)) 2 The second 5-class group G5F of a cyclic quartic field F with 5-class group Cl5F ' (5, 5) is restricted to the isomorphism classes of h25, 2i, h3125, 11|14i, a descendant of h125, 3i, a descendant of h3125, 7i or a descendant of h3125, 3i, h3125, 4i, h3125, 5i. All these finite 5-groups are σ-groups of degree 4.

2 Examples 3. We have G5F ' h3125, 11i, κ(F ) a 4-cycle, for (a, b, c, d) = (−457, 2, 1, 5), h3125, 14i, κ(F ) the identity, for (a, b, c, d) = (−581, 2, 1, 5). 34

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

11. The first 3-class tower of length 2

Notation. Logarithmic type invariants of abelian 3-groups, for instance (13) = (111) ˆ=(3, 3, 3) and (21) ˆ=(9, 3).

Let F/Q be an algebraic number ﬁeld with 3-class group Cl3F ' (3, 3), and E1,...,E4 be the unramiﬁed cyclic cubic extensions of F . Theorem. Suppose the capitulation of 3-classes of F in E1,...,E4 is of type κ1F ∼ (2, 2, 4, 1) (called type D.10).

Then the length of the 3-class tower of F is `3F = 2 3 and τ1F ∼ [21, 21, 1 , 21].

(The given TKT κ1 determines the length and the TTT τ1.) √ Examples. Let F = Q( d) be a quadratic field. √ (1) Among the 2 020 imaginary quadratic fields F = Q( d) 6 with 3-class group Cl3F ' (3, 3) and −10 < d < 0, there are 667 with 3-capitulation of type κ1F ∼ (2, 2, 4, 1). With a relative frequency of 33.0%, this type is definitely dominating. √ (2) Among the 2 576 real quadratic fields F = Q( d) with 7 3-class group Cl3F ' (3, 3) and 0 < d < 10 , there are 263 with a second 3-class group of even coclass. Among the latter, there are 93 with 3-capitulation of type κ1F ∼ (2, 2, 4, 1). With a relative frequency of 35.4%, this type is dominating 2 when it is taken with respect to cc(G3F ) ≡ 0 (mod 2). The discriminants of the former start with d = −4 027, those of the latter with d = 422 573. 35

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

Proof. A search for type D.10, κ1 ∼ (2, 2, 4, 1), in the descendant tree T R of the root R := C3 × C3 leads to the 2 unique candidate for M = G3F after two steps with path R = h9, 2i ← h27, 3i ← h243, 5i = M. The group M is a metabelian Schur σ-group and any epimorphism onto M must be an isomorphism [7]. Thus, we have `3F = dl(M) = 2.

Theorem 1. A metabelian p-group M of nilpotency class cl(M) = 3 has a trivial cover cov(M) = {M}. Theorem 2. A capable p-group G of odd nilpotency class cl(G) ≡ 1 (mod 2) cannot be a strong σ-group. Corollary. A capable metabelian p-group M of nilpotency class cl(M) = 3 is forbidden as the second p-class group G2F √ p of any quadratic field F = Q( d). 2 Proof. By Theorem 1, we have cov(M) = {M}. If GpF ∞ 2 were isomorphic to M, then Gp F ' GpF ' M. However, ∞ then Theorem 2 yields the contradiction that Gp F were not a strong σ-group. Examples. 1. The parent πM = h243, 4i of the metabelian 3-group M = h729, 45i, both with TKT H.4, κ1 ∼ (4443), is capable and has cl(πM) = 3. 2 Thus, a quadratic field F cannot have G3F ' πM. 2. The parent πM = h2187, 301i of the metabelian 3-group M = h6561, 2048i, both with TKT G.16, κ1 ∼ (4231), has a finite cover cov(πM) = {πM, h6561, 619i} and relation rank d2(πM) = 4. Since G = h6561, 619i is capable with odd class 2 cl(G) = 5, a quadratic field F cannot have G3F ' πM. 36

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

12. References

[1] E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ. Hamburg 5 (1927), 353–363. [2] E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272 (Russian). English transl.: Am. Math. Soc. Transl., II. Ser., 48 (1965), 91–102. [3] E. B. Vinberg, On a theorem concerning the infinite dimen- sionality of an associative algebra, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 209–214 (Russian). English transl.: Am. Math. Soc. Transl., II. Ser., 82 (1969), 237–242. [4] H. Koch und B. B. Venkov, Uber¨ den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24–25 (1975), 57–67. [5] I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Etudes´ Sci. 18 (1964), 71–95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128–149. [6] E. Artin, Idealklassen in Oberkörpern und allgemeines Rezi- prozitätsgesetz, Abh. Math. Sem. Univ. Hamburg 7 (1929), 46–51. [7] N. Boston and J. Ellenberg, Random pro-p groups, braid groups, and random tame Galois groups, Groups Geom. Dyn. 5 (2011), 265–280. 37

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

[8] D. C. Mayer, p-Capitulation over number fields with p-class rank two, J. Appl. Math. Phys. 4 (2016), no. 7, 1280–1293, DOI 10.4236/jamp.2016.47135. (2nd International Conference on Groups and Algebras 2016, Suzhou, China, 26 July, 2016.) [9] D. C. Mayer, Recent progress in determining p-class field towers, Gulf J. Math. (Dubai, UAE), arXiv: 1605.09617v1 [math.NT] 31 May 2016. [10] D. C. Mayer, Three-stage towers of 5-class fields, arXiv: 1604.06930v1 [math.NT] 23 Apr 2016. [11] D. C. Mayer, Annihilator ideals of two-generated metabelian p-groups, arXiv: 1603.09288v1 [math.GR] 30 Mar 2016. [12] D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66 – 104, DOI 10.4236/apm.2016.62008, Special Issue on Group Theory Research. 38

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

[13] D. C. Mayer, New number fields with known p-class tower, Tatra Mt. Math. Pub. 64 (2015), 21–57, DOI 10.1515/tmmp-2015-0040, Special Issue on Number Theory and Cryptology ‘15. (22nd Czech and Slovak International Conference on Number Theory 2015, LiptovskýJán,Slovakia, 31 August, 2015.) [14] D. C. Mayer, Periodic sequences of p-class tower groups, J. Appl. Math. Phys. 3 (2015), 746–756, DOI 10.4236/jamp.2015.37090. (1st International Conference on Groups and Algebras 2015, Shanghai, China, 21 July 2015.) [15] D. C. Mayer, Index-p abelianization data of p-class tower groups, Adv. Pure Math. 5 (2015), no. 5, 286–313, DOI 10.4236/apm.2015.55029, Special Issue on Number Theory and Cryptography. (29ièmes JournéesArithmétiques2015, University of Debrecen, Hungary, 09 July 2015.) [16] D. C. Mayer, Periodic bifurcations in descendant trees of finite p-groups, Adv. Pure Math. 5 (2015), no. 4, 162–195, DOI 10.4236/apm.2015.54020, Special Issue on Group Theory. 39

Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016

[17] D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor.Nombres Bordeaux 25 (2013), no. 2, 401–456, DOI 10.5802/jtnb.842. (27ièmes JournéesArithmétiques2011, Faculty of Mathematics and Informatics, University of Vilnius, Lithuania, 01 July 2011.) [18] D. C. Mayer, Principalization algorithm via class group structure, J. Théor.Nombres Bordeaux 26 (2014), no. 2, 415–464, DOI 10.5802/jtnb.874. [19] D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (2012), no. 3–4, 467–495, DOI 10.1007/s00605-010-0277-x. [20] D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471–505, DOI 10.1142/S179304211250025X.