arXiv subjects

Daniel C. Mayer

Publications and source records attributed to Daniel C. Mayer.

At least 37 records · Page 2Linked to original sources

arXiv2020

Construction and classification of p-ring class fields modulo p-admissible conductors

Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet M(K(f))=[(N(c,i))_{1<=i<=m(c)}]_{c|f} of dihedral fields N(c,i) with various conductors c|f having p-multiplicities m(c) over K such that sum_{c|f} m(c)=(p^r(f)-1)/(p-1). The advanced viewpoint of classifying the entire collection M(K(f)), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields, and the actual construction of the multiplet M(K(f)) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.

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arXiv2020

Finite non-metabelian Schur sigma-Galois groups of class field towers

For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many non-metabelian Schur sigma-groups S of order at least 5^14 with unbounded derived length dl(S)>=3 and metabelianizations S/D(D(S)) of fixed order 5^7. These results admit the conclusion that d=-159592 is the first known discriminant of an imaginary quadratic field with 7-class field tower of precise length L=3, and d=-90868 is a discriminant of an imaginary quadratic field with 5-class field tower of length L>=3, whose exact length remains unknown.

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arXiv2020

Schur sigma-groups with abelian quotient invariants (9,3)

By the construction of suitable non-metabelian Schur sigma-groups S of type (9,3) with log order lo(S) = 21 and nilpotency class cl(S) = 9, evidence is provided of a new class of imaginary quadratic fields K with 3-class group Cl(3,K) ~ C(9) * C(3) and punctured principalization type kappa ~ (1,4,4;4) whose 3-class field tower consists of precisely three stages. In contrast, previous examples of three-stage towers were associated with kappa in { (1,1,2;2), (1,1,2;3), (1,1,4;2), (1,2,3;1) }, lo(S) = 9 and cl(S) = 5.

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arXiv2020

Extremal root paths of Schur $σ$-groups and first $3$-class field towers with four stages

An extremal property of finite Schur sigma-groups G is described in terms of their path to the root in the descendant tree of their abelianization G/G'. The phenomenon is illustrated and verified by all known examples of Galois groups G=Gal(F(3,infty,K)/K) of 3-class field towers K=F(3,0,K)<F(3,1,K)<F(3,2,K)<=...<=F(3,infty,K) of imaginary quadratic number fields K=Q(sqrt{d}), d<0, with elementary 3-class group Cl(3,K) of rank two. Such Galois groups must be Schur sigma-groups and the existence of towers with at least four stages is justified by showing the non-existence of suitable Schur sigma-groups G with derived length dl(G)<=3. By means of counter-examples, it is emphasized that real quadratic number fields with the same type of 3-class group reveal a totally different behavior, usually without extremal path.

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arXiv2019

5-Class towers of cyclic quartic fields arising from quintic reflection

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0<d<10^4, respectively -2*10^5<d<0.

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arXiv2019

Harmonically balanced capitulation over quadratic fields of type (9,9)

The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to unramified abelian 3-extensions. First, as an approximation of the group G, its metabelianization M=G/G", which is isomorphic to the Galois group of the second Hilbert 3-class field, is sought by sifting the SmallGroups library with the aid of pattern recognition. In cases with order |M|>3^8, the SmallGroups database must be extended by means of the p-group generation algorithm, which reveals new phenomena of groups with harmonically balanced transfer kernels and trees with periodic trifurcations. Bounds for the relation rank d2(M) of M in dependence on the signature of the quadratic base field admit the decision whether the derived length of G is dl(G)=2 or dl(G)>=3.

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arXiv2019

Principal factors and lattice minima

Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(ζ_3)$ be the cyclotomic field containing a primitive cube root of unity $ζ_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Θ=(θ_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(α)\in\mathcal{P}_{\mathit{k}}^G/\mathcal{P}_{\mathit{k}_0}$ among the lattice minima $Θ=(θ_i)_{i\in\mathbb{Z}}$ of the underlying pure cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of the chain $Θ$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $\mathit{k}$ by means of Voronoi's algorithm.

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arXiv2019

The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields

The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abelian 2-extensions, collected in the Artin pattern. In recent investigations by Benjamin and Snyder, the length of the tower of such fields has turned out to be dependent on the rank of the 2-class group of the first Hilbert 2-class field. Significant progress is achieved by extending the pool of possible metabelian 2-groups of the second Hilbert 2-class field from the SmallGroups database, resp. Hall-Senior classification, with the aid of the p-group generation algorithm, and sifting the pool by means of pattern recognition.

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arXiv2019

Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture

The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N_i)$ is generalized for heterogeneous multiplets of ramified extensions. By application to quadratic subfields K of dihedral fields N of degree 2p with an odd prime p, a conjecture of Scholz concerning the index of subfield units, $(U_N:U_0)$, for ramified extensions N/K with conductor f>1 is verified computationally.

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arXiv2019

$3$-rank of ambiguous class groups of cubic Kummer extensions

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(ζ_3)$ with $d>1$ a cube-free integer and $ζ_3$ a primitive third root of unity. Denote by $C_{k,3}^{(σ)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langleσ\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(σ)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_{k}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_{k}$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.

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arXiv2018

Coarse cohomology types of pure meta cyclic fields

An unexpected behavior of pure septic fields L = Q(D^1/7) with certain prime radicands D congruent to 2 or 4 modulo 7, which split in the cyclotomic field of seventh roots of unity, is proved by direct computation. Whereas the Galois closures N = Q(zeta_7, D^1/7) of these fields contain relative differential principal factorizations in the kernel of the norm of N/L, the class numbers of L and N are not divisible by 7.

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arXiv2018

Differential principal factors and Polya property of pure metacyclic fields

Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(ζ_3,\sqrt[3]{D})$ with $3$ types is generalized to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and pure metacyclic fields $N=\mathbb{Q}(ζ_5,\sqrt[5]{D})$ with $13$ possible types. The classification is based on the Galois cohomology of the unit group $U_N$, viewed as a module over the automorphism group $\mathrm{Gal}(N/K)$ of $N$ over the cyclotomic field $K=\mathbb{Q}(ζ_5)$, by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index $(U_K:N_{N/K}(U_N))$ by the number $\#(\mathcal{P}_{N/K}/\mathcal{P}_K)$ of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different $\mathfrak{D}_{N/K}$. The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units $(U_N:U_0)$. Generalizing criteria for the Polya property of Galois closures $\mathbb{Q}(ζ_3,\sqrt[3]{D})$ of pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ by Leriche and Zantema, we prove that pure metacyclic fields $N=\mathbb{Q}(ζ_5,\sqrt[5]{D})$ of only $1$ type cannot be Polya fields. All theoretical results are underpinned by extensive numerical verifications of the $13$ possible types and their statistical distribution in the range $2\le D<10^3$ of $900$ normalized radicands.

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arXiv2018

Tables of pure quintic fields

By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(ζ_3,\sqrt[3]{D})$ with 3 possible types to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their pure metacyclic normal fields $N=\mathbb{Q}(ζ_5,\sqrt[5]{D})$ with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields $N$ having normalized radicands in the range $2\le D<10^3$. Our classification is based on the Galois cohomology of the unit group $U_N$, viewed as a module over the automorphism group $\mathrm{Gal}(N/K)$ of $N$ over the cyclotomic field $K=\mathbb{Q}(ζ_5)$, by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index $(U_K:N_{N/K}(U_N))$ by the number $\#(\mathcal{P}_{N/K} / \mathcal{P}_K)$ of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different $\mathfrak{D}_{N/K}$. The precise structure of the $\mathbb{F}_5$-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units $(U_N:U_0)$. The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

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arXiv2018

Co-periodicity isomorphisms between forests of finite p-groups

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.

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arXiv2017

Successive approximation of p-class towers

Let F be a number field and p be a prime. In the Successive Approximation Theorem, we prove that, for each positive integer n, finitely many candidates for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower F(p,infinity) over F are determined by abelian type invariants of p-class groups Cl(p,E) of unramified extensions E/F with degree [E:F]=$p^{n-1}$. Illustrated by the most extensive numerical results available currently, the transfer kernels ker(T(F,E)) of the p-class extensions T(F,E):Cl(p,F)-->Cl(p,E) from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S<G of all 3-groups G with coclass cc(G)=1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extensions E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups G(3,infinity,E) of dihedral fields E with degree 6, which could not be realized up to now.

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arXiv2017

Deep transfers of p-class tower groups

Let p be a prime. For any finite p-group G, the deep transfers T(H,G'):H/H' --> G'/G'' from the maximal subgroups H of index (G:H)=p in G to the derived subgroup G' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels kappa_d(G)=(ker(T(H,G')))_{(G:H)=p}. For all finite 3-groups G of coclass cc(G)=1, the family kappa_d(G) is determined explicitly. The results are applied to the Galois groups G=Gal(F_3^\infty/F) of the Hilbert 3-class towers of all real quadratic fields F=Q(d^1/2) with fundamental discriminants d>1, 3-class group Cl_3(F)~C_3*C_3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1<d<5*10^6, and a few exceptional cases are pointed out for 1<d<64*10^6.

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arXiv2017

Modeling rooted in-trees by finite p-groups

The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted in-trees with weighted vertices and edges perfectly admit an abstract formulation of the group theoretic notions of successive extensions, nuclear rank, multifurcation, and step size. Since all graphs in this chapter are infinite and dense, we use methods of pattern recognition and independent component analysis to reduce the complex structure to periodically repeating finite patterns. The method of group cohomology yields subgraph isomorphisms required for proving the periodicity of branches along mainlines. Finally the mainlines are glued together with the aid of infinite limit groups whose finite quotients form the vertices of mainlines. The skeleton of the infinite graph is a countable union of infinite mainlines, connected by periodic bifurcations. Each mainline is the backbone of a minimal subtree consisting of a periodically repeating finite pattern of branches with bounded depth. A second periodicity is caused by isomorphisms between all minimal subtrees which make up the complete infinite graph. Only the members of the first minimal tree are metabelian and the bifurcations, which were unknown up to now, open the long desired door to non-metabelian extensions whose second derived quotients are isomorphic to the metabelian groups. An application of this key result to algebraic number theory solves the problem of p-class field towers of exact length three.

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arXiv2016

Criteria for three-stage towers of p-class fields

Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K), that is the second derived quotient M=G/G". The family tau(1,K) of abelian type invariants of the p-class groups Cl(p,L) of all unramified cyclic extensions L/K of degree p is called the index-p abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ((tau(1,K),tau(2,K)) includes data on unramified abelian extensions L/K of degree p^2 and enables sharper bounds for the order of M in the case Cl(p,K)=(p,p,p), where current implementations of the p-group generation algorithm fail to produce explicit contestants for M, due to memory limitations. The iterated IPAD of second order tau^(2)(K) contains information on non-abelian unramified extensions L/K of degree p^2, or even p^3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(squareroot(d)) with Cl(p,K)=(p,p) possessing a p-class field tower of exact length L(p,K)=3 as a striking novelty.

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