Class Field Towers Research Articles

Asymptotic growth patterns for class field towers

Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a \mathbb{Z}_{p} -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain \mathbb{Z}_{p} -extensions in which a given prime above p is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for p -primary class numbers of the number fields in a \mathbb{Z}_{p} -tower.

Infinite Hilbert class field tower

Infinite Hilbert class field tower

On the Metacyclicity of the Hilbert 2-Class Field Tower of Imaginary

Let d be a square-free integer <0. In this paper, we will determine all imaginary quadratic number fields Q(√d) that have a metacyclic Hilbert 2-class field tower. Finally, we will numerically validate our theoretical results.

On metabelian 2-class field towers over -extensions of real quadratic fields

AbstractWe give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$ -extensions have metabelian Galois groups of abelian invariants $[2,2,2]$ . We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.

English

Let $$\mathbb{k} = \mathbb{Q} \left( {\sqrt 2 ,\; \sqrt d } \right)$$ be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and $$\mathbb{k}_2^{\left( 2 \right)}$$ its second Hilbert 2-class field. Denote by $$G = {\rm{Gal}}\left( {\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}} \right)$$ the Galois group of $${\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}}$$ . The purpose of this note is to investigate the Hilbert 2-class field tower of $$\mathbb{k}$$ and then deduce the structure of G.

Infinite class field towers of number fields of prime power discriminant

For every prime number p, we show the existence of a solvable number field L ramified only at {p,∞} whose p-Hilbert Class field tower is infinite.

Units and2-class field towers of some multiquadratic number fields

In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$.

Growth of Fine Selmer Groups in Infinite Towers

AbstractIn this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple $\mathbb{Z}_{p}$-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the $\unicode[STIX]{x1D707}=0$ conjecture for cyclotomic $\mathbb{Z}_{p}$-extensions. We show that in certain non-cyclotomic $\mathbb{Z}_{p}$-towers, the $\unicode[STIX]{x1D707}$-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified $p$-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-$p$-adic analytic.

The narrow 2-class field tower of some real quadratic number fields

The narrow 2-class field tower of some real quadratic number fields

On Hilbert class field tower for some quartic number fields

We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck,2 is isomorphic to Z∕2Z×Z∕2Z.

On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions

Abstract We study the structure of the Galois group of the maximal unramified 2-extension of some family of number fields of large degree. Especially, we show that for each positive integer n, there exist infinitely many number fields with large degree, for which the defined Galois group is quaternion of order 2 n .

On the 2-class field towers of some imaginary quartic cyclic number fields

On the 2-class field towers of some imaginary quartic cyclic number fields

English

english

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

Let$k$be an imaginary quadratic field with$\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert$2$-class field tower is at least$2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J.40(1) (1998), 63–69) calculated the density of$k$where the length of the tower is$1$; that is, the maximal unramified$2$-extension is a$V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

On the second 2-class group [formula omitted] of some imaginary quartic cyclic number field K

Let H8 denote the quaternion group of order 8 and put G=Z/2Z×H8. Let K be some imaginary quartic cyclic number field whose 2-class group is of type (2,2,2). In this paper, we prove in particular that G is realizable over K, i.e., G≃Gal(K2(2)/K) where K2(2) is the second Hilbert 2-class field of K. Then we study the capitulation problem of the 2-ideal classes of K in the fourteen intermediate unramified extensions between K and its first Hilbert 2-class field. Additionally, these fourteen unramified extensions are constructed, and the abelian type invariants of their 2-class groups and the length of the 2-class field tower of K are given.

A remark on the capitulation problem

We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K. The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower.

On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields

Inspired by the Odlyzko root discriminant and Golod–Shafarevich p-group bounds, Martinet (1978) asked whether an imaginary quadratic number field K/Q must always have an infinite Hilbert 2-class field tower when the class group of K has 2-rank 4, or equivalently when the discriminant of K has 5 prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite 2-towers for many K in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but still many cases remain open, especially when the class group of K has small 4-rank.Recently, Benjamin (2015) made partial progress on several of these open matrices when the class group of K has 4-rank 1 or 2. In this paper, we partially address many open cases when the 4-rank is 0 or 2, affirmatively answering some questions of Benjamin. We then investigate barriers to our methods and ask an extension question (of independent interest) in this direction. Finally, we suggest places where speculative refinements of Golod–Shafarevich or group classification methods might overcome the “near miss” inadequacies in current methods.

Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

Let p1 ≡ p2 ≡ -q ≡ 1 ( mod 4) be primes such that [Formula: see text] and [Formula: see text]. Put [Formula: see text] and d = p1p2q, then the bicyclic biquadratic field [Formula: see text] has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.

A positive proportion of some quadratic number fields with infinite Hilbert 2-class field tower

We show that a positive proportion of real and imaginary quadratic number fields with 2-class rank equal to 2 have 4-rank equal to 1 or 2 and infinite Hilbert 2-class field tower.

On the 2-class field tower conjecture for imaginary quadratic number fields with 2-class group of rank 4

We demonstrate the existence of infinitely many new imaginary quadratic number fields k with 2-class group Ck,2 of rank 4 such that k has infinite 2-class field tower. In particular, we demonstrate the existence of new fields k as above when the 4-rank of the class group Ck is equal to 1 or 2, and infinitely many new fields k in the case that the 4-rank of Ck is equal to 1, exactly three negative prime discriminants divide the discriminant dk of k, and dk is not congruent to 4 mod 8. This lends support to the conjecture that all imaginary quadratic number fields k with Ck,2 of rank 4 have infinite 2-class field tower.