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Daniel C. Mayer

Publications and source records attributed to Daniel C. Mayer.

53 records · Page 3Linked to original sources
arXiv2016

Recent progress in determining p-class field towers

For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K is considered to be known if a pro-p presentation of the Galois group H = Gal( F(p,infinity,K)/K ) is given. In the last few years, it turned out that the Artin pattern AP(K) = (tau(K),kappa(K)) consisting of targets tau(K) = (Cl(p,L)) and kernels kappa(K) = (ker(J(L/K)) of class extensions J(L/K): Cl(p,K) --> Cl(p,L) to unramified abelian subfields L/K of the Hilbert p-class field F(p,1,K) only suffices for determining the two-stage approximation G = H/H" of H. Additional techniques had to be developed for identifying the group H itself: searching strategies in descendant trees of finite p-groups, iterated and multilayered IPADs of second order, and the cohomological concept of Shafarevich covers involving relation ranks. This enabled the discovery of three-stage towers of p-class fields over quadratic base fields K = Q( squareroot(d) ) for p = 2,3,5. These non-metabelian towers reveal the new phenomenon of various tree topologies expressing the mutual location of the groups H and G.

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arXiv2016

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).

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arXiv2016

Three stage towers of 5-class fields

With K=Q((3812377)^(1/2)) we give the first example of an algebraic number field possessing a 5-class tower of exact length L(5,K)=3. The rigorous proof is conducted by means of the p-group generation algorithm, showing the existence of a unique finite metabelian 5-group G with abelianization [5,5] having the kernels (M(1),G^5) and targets ([25,5,5,5],[5,5]^5) of Artin transfers T(i):G-->M(i)/M(i)' to its six maximal subgroups M(i), prescribed by arithmetical invariants of K. Thus, G must be the second 5-class group G(5,2,K) of the real quadratic field K but cannot be its 5-class tower group G(5,K), since the relation rank d(2,G)=4 is too big. We provide evidence of exactly five non-isomorphic extensions H of G having the required relation rank d(2,H)=3 and derived length dl(H)=3 whose metabelianization H/H'' is isomorphic to G. Consequently, G(5,K) must be one of the five non-metabelian groups H.

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arXiv2016

Annihilator ideals of two-generated metabelian p-groups

For a metabelian p-group G = with two generators x and y, the annihilator A < Z[X,Y] of the main commutator [y,x] of G, as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for G. Furtwängler's isomorphism of the additive group underlying the residue class ring Z[X,Y]/A of Z[X,Y] modulo the annihilator A to the commutator subgroup G' of G admits the calculation of the abelian type of G'.

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arXiv2015

Artin transfer patterns on descendant trees of finite p-groups

Based on a thorough theory of the Artin transfer homomorphism \(T_{G,H}:\,G\to H/H^\prime\) from a group \(G\) to the abelianization \(H/H^\prime\) of a subgroup \(H\le G\) of finite index \(n=(G:H)\), and its connection with the permutation representation \(G\to S_n\) and the monomial representation \(G\to H\wr S_n\) of \(G\), the Artin pattern \(G\mapsto(τ(G),\varkappa(G))\), which consists of families \(τ(G)=(H/H^\prime)_{H\le G}\), resp. \(\varkappa(G)=(\ker(T_{G,H}))_{H\le G}\), of transfer targets, resp. transfer kernels, is defined for the vertices \(G\in\mathcal{T}\) of any descendant tree \(\mathcal{T}\) of finite \(p\)-groups. It is endowed with partial order relations \(τ(π(G))\leτ(G)\) and \(\varkappa(π(G))\ge\varkappa(G)\), which are compatible with the parent-descendant relation \(π(G)<G\) of the edges \(G\toπ(G)\) of the tree \(\mathcal{T}\). The partial order enables termination criteria for the \(p\)-group generation algorithm which can be used for searching and identifying a finite \(p\)-group \(G\), whose Artin pattern \((τ(G),\varkappa(G))\) is known completely or at least partially, by constructing the descendant tree with the abelianization \(G/G^\prime\) of \(G\) as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns \((τ(G),\varkappa(G))\) and explaining the stabilization, resp. polarization, of their components in descendant trees \(\mathcal{T}\) of finite \(p\)-groups.

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arXiv2015

New number fields with known p-class tower

The p-class tower $F_p^\infty(k)$ of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group $G:=Gal(F_p^\infty(k)/k)$, can be identified by an explicit presentation. The main intention of this article is to characterize assigned finite 3-groups uniquely by abelian quotient invariants of subgroups of finite index, and to provide evidence of actual realizations of these groups by 3-tower groups G of real quadratic fields $K=Q(\sqrt{d})$ with 3-capitulation type (0122) or (2034).

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arXiv2015

Periodic sequences of p-class tower groups

Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3), form periodic sequences in the descendant tree of the elementary abelian root C(2)xC(2)xC(2), resp. C(3)xC(3). The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.

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arXiv2015

Index-\(p\) abelianization data of \(p\)-class tower groups

Given a fixed prime number \(p\), the multiplet of abelian type invariants of the \(p\)-class groups of all unramified cyclic degree \(p\) extensions of a number field \(K\) is called its IPAD (index-\(p\) abelianization data). These invariants have proved to be a valuable information for determining the Galois group \(G_p^2\) of the second Hilbert \(p\)-class field and the \(p\)-capitulation type \(\varkappa\) of \(K\). For \(p=3\) and a number field \(K\) with elementary \(p\)-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group \(G_p^\infty\) of the maximal unramified pro-\(p\) extension of \(K\).

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arXiv2015

Periodic bifurcations in descendant trees of finite (p)-groups

Theoretical background and an implementation of the (p)-group generation algorithm by Newman and O'Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite (p)-groups.

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arXiv2014

Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\mathbb{Q}\left(\sqrt{\strut p_1p_2q},\sqrt{\strut -1}\right)$

Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is the Hilbert $2$-class field of ${k}$. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group $G=\mathrm{Gal}(\mathbb{L}/{k})$ of the second Hilbert 2-class field ${k}_2^{(2)}$ of ${k}$. Additionally, the abelian type invariants of the groups $\mathbf{C}l_2(\mathbb{K}_j)$ and $\mathbf{C}l_2(\mathbb{L}_j)$ and the length of the $2$-class tower of ${k}$ are given.

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arXiv2014

Normal lattice of certain metabelian p-groups G with \(G/G^\prime\simeq (p,p)\)

Let p be an odd prime. The lattice of all normal subgroups and the terms of the lower and upper central series are determined for all metabelian p-groups with generator rank d=2 having abelianization of type (p,p) and minimal defect of commutativity k=0. It is shown that many of these groups are realized as Galois groups of second Hilbert p-class fields of an extensive set of quadratic fields which are characterized by principalization types of p-classes.

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arXiv2014

Quadratic p-ring spaces for counting dihedral fields

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(ψ:\,f\mapsto V_p(f)\) from the divisor lattice \(\mathbb{N}\) of positive integers to the lattice S of subspaces of the direct product \(V_p\) of the p-elementary class group \(C/C^p\) and unit group \(U/U^p\) of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group \(Gal(N | \mathbb{Q})\) and sharing a common discriminant \(d_N\) and conductor c over K. The number \(m_p(d,c)\) of these extensions is given by a formula in terms of positions of p-ring spaces in S, whose complexity increases with the dimension of the vector space \(V_p\) over the finite field \(\mathbb{F}_p\), called the modified p-class rank \(σ_p\) of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with \(0\leσ_p\le 1\) only. Here, the results are extended to \(σ_p=2\), underpinned by concrete numerical examples.

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arXiv2014

Transfers of metabelian p-groups

Explicit expressions for the transfers \(V_i\) from a metabelian p-group G of coclass cc(G)=1 to its maximal normal subgroups \(M_i\) \((1\le i\le p+1)\) are derived by means of relations for generators. The expressions for the exceptional case p=2 differ significantly from the standard case of odd primes \(p\ge 3\). In both cases the transfer kernels \(Ker(V_i)\) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group \(Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\) of an algebraic number field K. For certain metabelian 3-groups G with abelianisation \(G/G^{\prime}\) of type (3,3) and of coclass \(cc(G)=r\ge 3\), it is shown that the principalisation type determines the position of G on the coclass graph G(3,r) in the sense of Eick and Leedham-Green.

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arXiv2014

The second p-class group of a number field

For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\) of K are determined by the p-class numbers of the unramified cyclic extensions \(N_i | K\), \(1\le i\le p+1\), of relative degree p. In the case of a quadratic field \(K=\mathbb{Q}(\sqrt{D})\) and an odd prime \(p\ge 3\), the invariants of G are derived from the p-class numbers of the non-Galois subfields \(L_i | \mathbb{Q}\) of absolute degree p of the dihedral fields \(N_i\). As an application, the structure of the automorphism group \(G=Gal(F_3^2(K) | K)\) of the second Hilbert 3-class field \(F_3^2(K)\) is analysed for all quadratic fields K with discriminant \(-10^6<D<10^7\) and 3-class group of type (3,3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3,r), \(1\le r\le 6\), in the sense of Eick and Leedham-Green is investigated.

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arXiv2014

The distribution of second p-class groups on coclass graphs

General concepts and strategies are developed for identifying the isomorphism type of the second p-class group \(G=Gal(F_p^2(K) | K)\), that is the Galois group of the second Hilbert p-class field \(F_p^2(K)\), of a number field K, for a prime p. The isomorphism type determines the position of G on one of the coclass graphs G(p,r), \(r\ge 0\), in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p-class group \(Cl_p(K)\), the position of G is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second p-class groups G for various series of number fields K having p-class groups \(Cl_p(K)\) of fixed type and p in {2,3,5,7}.

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arXiv2014

Principalization algorithm via class group structure

For an algebraic number field K with 3-class group \(Cl_3(K)\) of type (3,3), the structure of the 3-class groups \(Cl_3(N_i)\) of the four unramified cyclic cubic extension fields \(N_i\), \(1\le i\le 4\), of K is calculated with the aid of presentations for the metabelian Galois group \(G_3^2(K)=Gal(F_3^2(K) | K)\) of the second Hilbert 3-class field \(F_3^2(K)\) of K. In the case of a quadratic base field \(K=\mathbb{Q}(\sqrt{D})\) it is shown that the structure of the 3-class groups of the four \(S_3\)-fields \(N_1,\ldots,N_4\) frequently determines the type of principalization of the 3-class group of K in \(N_1,\ldots,N_4\). This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all 4596 quadratic fields K with 3-class group of type (3,3) and discriminant \(-10^6<D<10^7\) to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups \(G_3^2(K)\) on various coclass trees of the coclass graphs G(3,r), \(1\le r\le 6\), in the sense of Eick, Leedham-Green, and Newman.

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arXiv2013

$3$-class field towers of exact length $3$

The $p$-group generation algorithm is used to verify that the Hilbert $3$-class field tower has length $3$ for certain imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_3(K) \cong [3,3]$. Our results provide the first examples of finite $p$-class towers of length $> 2$ for an odd prime $p$.

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