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.     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

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.     

Densities for some real quadratic fields with infinite Hilbert 2-class field towers

Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers.     

On 2-ciass field towers of some imaginary quadratic number fields

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class fields are finite.     

A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers

This paper shows that a positive proportion of the imaginary quadratic fields with 2-class rank equal to 3 have 4-class rank equal to zero and infinite Hilbert 2-class field towers. Received: 14 January 2003     

Imaginary quadratic fields with Cl2(k)?(2,2,2)

We characterize those imaginary quadratic number fields, k, with 2-class group of type (2,2,2) and with the 2-rank of the class group of its Hilbert 2-class field equal to 2. We then compute the length of the 2-class field tower of k.     

Class field towers of imaginary quadratic fields

In their 1934 paper, Scholz and Taussky defined the notion of capitulation type for imaginary quadratic fields whose ideal class group has a Sylow 3-subgroup which is elementary abelian of order 3². For one particular capitulation type (type D) they prove that the 3-class field tower of the quadratic field has length 2. They briefly indicate how a similar result can be shown to hold for capitulation type E. In this paper we give a simpler proof of their type D result and we construct a group theoretic counterexample to their type E assertion.     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

Application of a character sum estimate to a 2-class number density

An asymptotic formula is obtained for the number of imaginary quadratic number fields with 2-class number equal to 2, from which one can then obtain a type of density result for the 2-class number. The solution of this problem leads to an interesting question about a character sum over primes.     

Cyclicity of the unramified Iwasawa module

For a number field k and a prime number p, let k _?? be the cyclotomic Z _p -extension of k with finite layers k _n . We study the finiteness of the Galois group X _?? over k _?? of the maximal abelian unramified p-extension of k _?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k _n . As a consequence, we determine the complete list of real quadratic number fields for which X _?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.     

On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.     

The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

Note on 2-rational fields

We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi-Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42-76] and, at this occasion, we bring into focus some classical technics of abelian ?-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ?-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey's paper in such a way using a result of Herfort-Zalesskii. This Note is mainly an attempt of clarification about ?-rationality.     

Avoiding abelian squares in partial words

Erd?s raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n.     

The logic of pseudo-S-integers

Let {n _i } be a sequence of natural numbers and let {p _i } be a listing of rational primes. Then an abelian groupG={x ∈ √| ord _pi x ≥ −n _i } is called a group of pseudo-integers. We investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. We show that, while the addition problem of any recursive group of pseudo-integers is decidable, the Diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over ℤ in several variables has solutions in the group. This result is obtained by giving a Diophantine definition of ℤ over these groups. The proof is based on the strong Hasse norm principal. The research for this paper has been partially supported by NSA grant MDA904-96-1-0019.     

Exposed 2-homogeneous polynomials on Hilbert spaces

We show that every extreme point of the unit ball of 2-homogene- ous polynomials on a separable real Hilbert space is its exposed point and that the unit ball of 2-homogeneous polynomials on a non-separable real Hilbert space contains no exposed points. We also show that the unit ball of 2-homogeneous polynomials on any infinite dimensional real Hilbert space contains no strongly exposed points.

     

Principal ideal theorems in the genus field for absolutely abelian extensions

This paper has the following contents. 1°. In an abelian extension field K over the rational number field, any ambiguous ideal is a principal ideal in the genus field in the wide sense. 2°. A number theoretical proof of the following. In a cyclic extension field K over the rational number field, any ambiguous class ideal is a principal ideal in the genus field in the wide sense.     

A generalization of maillet and demyanenko determinants for the Cyclotomic Zp-Extension

LetK be an imaginary abelian number field. By means of a generalization of Maillet and Demyanenko determinants we give a relative class number formula for an intermediate field of the cyclotomic ℤp-extension ofK. The degree of the generalized determinant is a half of the degree ofK over ℚ.