p-Capitulation over number fields with p-class rank two

Theoretical foundations of a new algorithm for determining the p-capitulation type kappa(K) of a number field K with p-class rank rho=2 are presented. Since kappa(K) alone is insufficient for identifying the second p-class group G=Gal(F(p,2,K) | K) of K, complementary techniques are developed for finding the nilpotency class and coclass of G. An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(tau(K),kappa(K)) of all 34631 real quadratic fields K=Q(squareroot(d)) with discriminants 0<d<100000000 and 3-class group of type (3,3). The results admit extensive statistics of the second 3-class groups G=Gal(F(3,2,K) | K) and the 3-class field tower groups H=Gal(F(3,K) | K).


Introduction
Let p be a prime number. Suppose that K is an algebraic number field with p-class group Cl p K := Syl p Cl K and p-elementary class group E p K := Cl p K Zp F p . By class field theory [18, Cor. 3.1, p. 838], there exist precisely n := p ̺ −1 p−1 distinct (but not necessarily non-isomorphic) unramified cyclic extensions L i |K, 1 ≤ i ≤ n, of degree p, if K possesses the p-class rank ̺ := dim Fp E p K. For each 1 ≤ i ≤ n, let j Li|K : Cl p K → Cl p L i denote the class extension homomorphism induced by the ideal extension monomorphism [17, § 1, p. 74]. We let U K , resp. U Li , be the group of units of K, resp. L i . Proof. The proof of the inclusion ker j Li|K ≤ E p K was given in [ [15, p. 279], the kernel ker j Li|K cannot be trivial.
Definition 1.1. For each 1 ≤ i ≤ n, the elementary abelian p-group ker j Li|K is called the pcapitulation kernel of L i |K. We speak about total capitulation [9,10] if dim Fp ker j Li|K = ̺, and partial capitulation if 1 ≤ dim Fp ker j Li|K < ̺.
If p ≥ 3 is an odd prime, and K = Q( √ d) is a quadratic field with fundamental discriminant d := d K and p-class rank ̺ ≥ 1, then there arise the following possibilities for the p-capitulation kernel in any of the unramified cyclic relative extensions L i |K of degree p, which are absolutely dihedral extensions L i |Q of degree 2p, according to [19,Prop. 4.1,p. 482].
The organization of this article is the following. In § 2, basic theoretical prerequisites for the new capitulation algorithm are developed. The implementation in Magma [16] consists of a sequence of computational techniques whose actual code is given in § 3. The final § 4 demonstrates the results of an impressive application to the case p = 3, presenting statistics of all 3-capitulation types κ(K), Artin patterns AP(K), and second 3-class groups G = Gal(F 2 3 K|K) of the 34 631 real quadratic fields K = Q( √ d) with discriminants 0 < d < 10 8 and 3-class group of type (3,3), which beats our own records in [19, § 6] and [22, § 6]. Theorems concerning 3-tower groups G = Gal(F ∞ 3 K|K) with derived length 2 ≤ dl(G) ≤ 3 perfect the current state of the art.

Theoretical prerequisites
In this article, we consider algebraic number fields K with p-class rank ̺ = 2, for a given prime number p. As explained in § 1, such a field K has n = p + 1 unramified cyclic extensions L i of relative degree p.
Definition 2.1. By the Artin pattern of K we understand the pair consisting of the family τ (K) of the p-class groups of all extensions L 1 , . . . , L n as its first component (called the transfer target type) and the p-capitulation type κ(K) as its second component (called the transfer kernel type), AP(K) := (τ (K), κ(K)) , τ (K) := (Cl p L i ) 1≤i≤n , κ(K) := ker j Li|K 1≤i≤n .
We know from Proposition 1.1 that each kernel ker j Li|K is a subgroup of the p-elementary class group E p K of K. On the other hand, there exists a unique subgroup S < Cl K of index p such that S = Norm Li|K Cl Li , according to class field theory. Thus we must first get an overview of the connections between subgroups of index p and subgroups of order p of Cl K . Lemma 2.1. Let p be a prime and A be a finite abelian group with positive p-rank and with Sylow An application to the particular case A = Cl K and S = Norm Li|K Cl Li < Cl K shows that Three cases must be distinguished, according to the abelian type of the p-class group Cl p K. We first consider the general situation of a finite abelian group A with type invariants (a 1 , . . . , a n ) having p-rank r p (A) = 2, that is, n ≥ 2, p | a n , p | a n−1 , but gcd(p, a i ) = 1 for i < n − 1. Then the Sylow p-subgroup Syl p A of A is of type (p u , p v ) with integer exponents u ≥ v ≥ 1, and the p-elementary subgroup A p of A is of type (p, p). We select generators x, y of Syl p A = x, y such that ord(x) = p u and ord(y) = p v . Lemma 2.2. Let p be a prime number. Suppose that G is a group and x ∈ G is an element with finite order e := ord(x) divisible by p. Then the power x m with exponent m := e p is an element of order ord(x m ) = p. Proof. Generally, the order of a power x m with exponent m ∈ Z is given by (4) ord(x m ) = ord(x) gcd(m, ord(x)) .
This can be seen as follows. Let d := gcd(m, e), and suppose that m = d · m 0 and e = d · e 0 , then gcd(m 0 , e 0 ) = 1. We have (x m ) e0 = x m0·d·e0 = (x e ) m0 = 1, and thus n := ord(x m ) is a divisor of e 0 . On the other hand, 1 = (x m ) n = x m·n , and thus e = d · e 0 divides m · n = d · m 0 · n. Consequently, e 0 divides m 0 · n, and thus necessarily e 0 divides n, since gcd(m 0 , e 0 ) = 1. This yields n = e 0 , as claimed. Now we apply Lemma 2.2 to the situation where A is a finite abelian group with type invariants (a 1 , . . . , a n ) having p-rank r p (A) = 2, that is, n ≥ 2, p | a n , p | a n−1 . Proof. Let generators of A corresponding to the abelian type invariants (a 1 , . . . , a n ) be (g 1 , . . . , g n ), in particular, the trailing two generators have orders ord(g n−1 ) = a n−1 and ord(g n ) = a n divisible by p. According to Lemma 2.2, the powers g  Proof. According to the assumptions, A p is elementary abelian of rank 2, that is, of type (p, p), and consists of the p 2 elements {w i z j | 0 ≤ i, j ≤ p − 1}, in particular, w 0 z 0 = 1 is the neutral element. A possible selection of generators for the p 2 −1 p−1 = p + 1 cyclic subgroups M i of order p is to take M 1 = z and M i = wz i−2 for 2 ≤ i ≤ p + 1, since the two cycles of powers of wz i and wz j for 1 ≤ i < j ≤ p − 1 meet in the neutral element only.  If u > v = 1, then a subgroup U of index p is either of type (p u ), i.e., cyclic, or of type

Theorem 2.1. (Taussky's conditions A and B)
Let L|K be an unramified cyclic extension of prime degree p of a base field K with p-class rank ̺ = 2. Suppose that S = Norm L|K Cl L < Cl K and U = Norm L|K Cl p L < Cl p K are the subgroups of index p associated with L|K, according to class field theory. Then, we generally have ker j L|K S = ker j L|K U , and in particular: Proof. This is an immediate consequence of Proposition 2.3.
Proof. The proof for the case u = v = 1 was given in [17, p. 79] and [27, Rmk. 5.3, pp. 87-88]. It is the unique case where subgroups of index p coincide with subgroups of order p, and a renumeration of the former enforces a renumeration of the latter, expressed by a single permutation σ ∈ S p+1 and its inverse σ −1 .
If u > v = 1, then the distinguished subgroups U p+1 = N = x p , y ≃ (p u−1 , p) of index p, and V p+1 = x p u−1 of order p, should have the fixed subscript p + 1. The other p subgroups U i , resp. V i , can be renumerated completely independently of each other, which can be expressed by two independent permutations π, ρ ∈ S p . For details, see [27,Rmk. 5.6,p. 89].
In the case u ≥ v > 1, finally, the p + 1 subgroups of index p of Cl p K and the p + 1 subgroups of order p of Cl p K can be renumerated completely independently of each other, which can be expressed by two independent permutations σ, τ ∈ S p+1 .

Computational techniques
In this section, we present the implementation of our new algorithm for determining the Artin pattern AP(K) of a number field K with p-class rank ̺ = 2 in MAGMA [4,5,16], which requires version V2.21-8 or higher. Algorithm 3.1 returns the entire class group C := Cl K of the base field K, together with an invertible mapping mC from classes to representative ideals.  3.1. By using the statement K := QuadraticField(d); the quadratic field K = Q( √ d) is constructed directly. However, the construction by means of a polynomial P (X) ∈ Z[X] executes faster and can easily be generalized to base fields K of higher degree.
For the next algorithm it is important to know that in the MAGMA computational algebra system [16], the composition A×A → A, (x, y) → x+y, of an abelian group A is written additively, and abelian type invariants (a 1 , . . . , a n ) of a finite abelian group A are arranged in non-decreasing order a 1 ≤ . . . ≤ a n .
Given the situation in Proposition 2.1, where A is a finite abelian group having p-rank r p (A) = 2, Algorithm 3.2 defines a natural ordering on the subgroups S of A of index (A : S) = p by means of Proposition 2.2, if the Sylow p-subgroup Syl p A is of type (p, p).  Output: Generators x, y of the p-elementary subgroup A p of A, two indicators, NonCyc for one or more non-cyclic maximal subgroups of Syl p A, Cyc for one or more cyclic maximal subgroups of Syl p A, an ordered sequence seqS of the p+1 subgroups of A of index p, and, if there are only cyclic maximal subgroups of Syl p A, an ordered sequence seqI of numerical identifiers for the elements S of seqS.
Proof. This is precisely the implementation of the Propositions 2.1, 2.2 and 2.3 in MAGMA [16]. The class group (C, mC) in the output of Algorithm 3.1 is used as input for Algorithm 3.2. The resulting sequence seqS of all subgroups of index p in C, together with the pair (C, mC), forms the input of Algorithm 3.3, which determines all unramified cyclic extensions L i |K of relative degree p using the Artin correspondence as described by Fieker [11].  3 is independent of the p-class rank ̺ of the base field K. In order to obtain the adequate coercion of ideals, the sequence seqRelOrd must be used for computing the transfer kernel type κ(K) in Algorithm 3.4. The trailing three lines of Algorithm 3.3 are optional but highly recommended, since the size of all arithmetical invariants, such as polynomial coefficients, is reduced considerably. Either the sequence seqAbsOrd or rather the sequence seqOptAbsOrd should be used for calculating the transfer target type τ (K) in Algorithm 3.5.
Algorithm 3.4. (Transfer kernel type, κ(K)) Input: The prime number p, the ordered sequence seqRelOrd of the relative maximal orders of L i |K, the class group mapping mC of the base field K with p-class rank ̺ = 2, the generators x, y of the p-elementary class group E p of K, and the ordered sequence seqI of numerical identifiers for the p + 1 subgroups S of index p in the class group C of K.  Output: The conditional transfer target type TTT of K, assuming the GRH.
With Algorithms 3.4 and 3.5 we are in the position to determine the Artin pattern AP(K) = (τ (K), κ(K)) of the field K. For pointing out fixed points of the transfer kernel type κ(K) it is useful to define a corresponding weak TKT κ = κ(K) which collects the Taussky conditions A, resp. B, of Theorem 2.1, for each extension L i |K: Proof. This is the implementation of Theorem 2.1 in MAGMA [16].

Interpretation of numerical results
By means of the algorithms in § 3, we have computed the Artin pattern AP(K) = (τ (K), κ(K)) of all 34 631 real quadratic fields K = Q( √ d) with Cl 3 K ≃ (3, 3) in the range 0 < d < 10 8 of fundamental discriminants. The results are presented in the following four tables, arranged by the coclass cc(G) of the second 3-class group G = G 2 3 K. Each table gives the type designation, distinguishing ground states and excited states (↑, ↑ 2 , . . .), the transfer kernel type κ = κ(K), the transfer target type τ = τ (K), the absolute frequency AF, the relative frequency RF, that is the percentage with respect to the total number of occurrences of the fixed coclass, and the minimal discriminant MD [28,Dfn. 5.1]. Additionally to this experimental information, we have identified the group G by means of the strategy of pattern recognition via Artin transfers [29, § 4], and computed the factorized order of its automorphism group Aut(G) and its relation rank Proof. This is Theorem 5.3 in [28].
Inspired by Boston, Bush and Hajir's theory of the statistical distribution of p-class tower groups of complex quadratic fields [6], we expect that, in Table 1 and in view of Theorem 4.1, the asymptotic limit of the relative frequency RF of realizations of a particular group G = G 2 3 K ≃ G = G ∞ 3 K is proportional to the reciprocal of the order #Aut(G) of its automorphism group. In particular, we state the following conjecture about three dominating types, a.3 * , a.3 and a.2.
Theorem 4.2. (Section D) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group G = Gal(F 2 3 K|K) is isomorphic to either of the two Schur σ-groups 3 5 , 5 or 3 5 , 7 has exact length ℓ 3 K = 2, that is, the 3-class tower group G = Gal(F ∞ 3 K|K) is isomorphic to G, and K < F 1 3 K < F 2 3 K = F ∞ 3 K. Proof. This statement has been proved by Scholz and Taussky in [30, § 3, p. 39]. It has been confirmed with different techniques by Brink and Gold in [7,Thm. 7,, and by Heider and Schmithals in [13,Lem. 5,p. 20]. All three proofs were expressed for complex quadratic base fields K, but since the cover [26,Dfn. 5.1,p. 30] of a Schur σ-group G consists of a single element, cov(G) = {G}, the statement is actually valid for any algebraic number field K, in particular also for a real quadratic field K. Table 2 shows the computational results for cc(G) = 2, using the relative identifiers of the ANUPQ package [12] for groups G of order #G ≥ 3 8 , resp. G of order #G ≥ 3 8 . The possibilities for the 3-class tower group G are complete for the TKTs c.18, c.21, E.6, E.8, E.9 and E.14, constituting the cover of the corresponding metabelian group G. For the TKTs c.18 ↑, c.21 ↑, the cover cov(G) is given in [26,Cor. 7.1,p. 38,and Cor. 8.1,p.48], and for E.6 ↑, E.8 ↑, E.9 ↑ and E.14 ↑, it has been determined in [23,Cor 21.3,p. 187]. A selection of densely populated vertices is given for the sporadic TKTs G.19 * and H.4 * , according to [28,. We denote two important branch vertices of depth 1 by N 9,j := 3 7 , 303 − #1; 1 − #1; j for j ∈ {3, 5}.
Whereas the sufficient criterion for ℓ 3 K = 2 in Theorem 4.2 is known since 1934 already, the following statement of 2015 is brand-new and constitutes one of the few sufficient criteria for ℓ 3 K = 3, that is, for the long desired three-stage class field towers [8].
For the groups G of coclass cc(G) ≥ 3, the problem of determining the corresponding 3-class tower group G is considerably harder than for cc(G) ≤ 2, and up to now it is still open.
For the essential difference between the location of the groups G as vertices of coclass trees for the types d. The single occurrence of type H.4 belongs to the irregular variant (i), where Cl 3 F 1 3 K ≃ (9, 9,9,9). This is explained in [19, p. 498] and [22, pp. 454-455]. It is the only case in Table 4 where G is determined uniquely.

Acknowledgements
The author gratefully acknowledges that his research is supported by the Austrian Science Fund (FWF): P 26008-N25.