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 1 2 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 INTRODUCTION. • The key for determining the Galois group (1) (1) G := Gp F = Gal(Fp =F ) (1) 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. (1) The Hilbert p-class field tower Fp over F is the maximal unramified pro-p extension of F and has the Galois group 1 (1) n Gp F := Gal Fp =F ' lim− Gp F: n≥0 (1) (2) (n) (1) F ≤ Fp ≤ Fp ≤ ··· ≤ Fp ≤ ··· ≤ Fp 1 GpF 2 GpF 2nd p-class group n Gp F nth p-class group 1 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) (1) 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 1 G = Gp F p-tower group Definition. The derived length dl(G) of the p-tower group 1 G = Gp F is called the length `pF of the p-class tower of F . (1) 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 informa- tion. According to my conviction, we are not done before we can give an explicit pro-p presentation of the p-tower group 1 (1) 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: (1) 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) (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, Gasch¨utz,1965 [3]). 2 2 (d1) (d1) #G < 1 ) d2 > 4 , resp. d2 ≤ 4 ) #G = 1. 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 p a complex quadratic field F = Q( d) with p-class rank %p ≥ 3 has an infinite p-class tower with `pF = 1. Proof. Since F has Dirichlet unit rank r = 0, the p-tower 1 group G = Gp F has a balanced presentation with d2 = d1 and R ⊂ F3. Thus, a sufficient condition for `pF = 1 is 3 q p (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 = 1. However, 1 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, 1 1 G = Gp (K) = Gal(Fp (K)jK) the Galois group of the 1 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 62 K; d ≤ d ≤ d + r + θ; where θ = 1 2 1 1 if ζ 2 K: p Corollary. K = Q( d) quadratic field with discriminant d, 1 1 G = Gp (K) = Gal(Fp (K)jK) the p-class tower group of K. Then 8 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 2 Sn of a transversal n S_ G = i=1 `i · H induced by the action of an element x 2 G: n 0 0 0 Y −1 0 TG;H : G=G ! H=H ; x · G 7! `π(i)x`i · H ; i=1 where x`i · H = `π(i) · H, for each 1 ≤ i ≤ n.