On imaginary abelian number fields of type (2, 2,..., 2) with one class in each genus

We show that there exist only finitely many imaginary abelian number fields of type (2,2,...,2) with one class in each genus. Moreover, if the Generalized Riemann Hypothesis is true, we have exactly 301 such fields, whose degrees are less than or equal to 2³. Finally we give the table of those 301 fields.     

Remarks on unit indices of imaginary abelian number fields

Let K be an imaginary abelian number field. We determine the unit index of K of certain type whose conductor is a product of two or three prime powers. As a consequence we construct some counterexamples to Satz 29 in Hasse's monograph [2].     

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.     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

A new proof of the gradient estimate for graphs of prescribed mean curvature

The aim of this paper is to give a new proof of the gradient estimate for graphs of prescribed mean curvatureH=H(x,y,z). Similarly as in [2] where the caseH=H(x,y) is studied, we introduce conformal parameters for the surface. Then we employ the differential equation for the unit normal of the surface derived in [3] Satz 1. By this method, which is contained in [4] Satz 4, we prove the following     

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.     

A note on pépin’s counter examples to the hasse principle for curves of genus 1

In a series of articles published in the C.R. Paris more than a century ago, T. PéPIN announced a list of “theorems” concerning the solvability of diophantine equations of the type ax⁴ +by ⁴ = z². In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

Generalization of Kummer's criterion for divisibility of class numbers

We give a necessary and sufficient condition for the relative class number of an imaginary field contained in Q(e^2πi/p?) to be divisible by p. We also give a sufficient condition for the class number of a real field contained in Q(e^2πi/p?) not to be divisible by p.     

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.     

Kombinatorische Beweise eines Satzes von Birman und Hilden und eines Satzes von Magnus mit der Theorie der automorphen Mengen und der zugehörigen Zopfgruppenoperationen

We deal with the well-known operation ofARTIN?S Braid GroupB _n on the free groupF _n by automorphisms, and give a proof for a theorem ofBIRMAN/HILDEN (here Satz B) by showing, that the images of the generators ofF _n underB _n are of a special form (Satz C). The theory ofBRIESKORN?S Automorphic Sets comes in here. With similar methods we give a proof of a theorem of Magnus saying thatB _n operates on a certain polynomial ring effectively by automorphisms (here Satz 9.2).     

A 2-adic automorphy lifting theorem for unitary groups over CM fields

We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for \({{\mathrm{GL}}}_2\).     

Homogeneous random fields and statistical mechanics

We illustrate the connection between homogeneous perturbations of homogeneous Gaussian random fields over Rⁿ or Zⁿ, with values in R^m, and classical as well as quantum statistical mechanics. In particular we construct homogeneous non-Gaussian random fields as weak limits of perturbed Gaussian random fields and study the infinite volume limit of correlation functions for a classical continuous gas of particles with inner degrees of freedom. We also exhibit the relation between quantum statistical mechanics of lattice systems (anharmonic crystals) at temperature β^?1 and homogeneous random fields over Zⁿ × S_β, where S_β is the circle of length β, which then provides a connection also with classical statistical mechanics. We obtain the infinite volume limit of real and imaginary times Green's functions and establish its properties. We also give similar results for the Gibbs state of the correspondent classical lattice systems and show that it is the limit as h → 0 of the quantum statistical Gibbs state.     

Powerful Necessary Conditions for Class Number Problems

We give a necessary condition for the ideal class group of a CM-field to be of exponent at most two. This condition enables us to drastically reduce the amount of relative class number computation for determination of the CM - fields of some types (e. g. the imaginary cyclic non -quadratic number fields of 2 - power degrees) whose ideal class groups are of exponents at most two. We also give a necessary condition for some quartic non - CM - fields to have class number one.     

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ?/2? ⊕ ?/2? if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.     

Geometrical solution of some cubic equations with non-real roots

A general cubic equation ax ³ + bx ² + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given.     

On the singular values of Weber modular functions

The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of Gross-Zagier for the level 1 case.

     

Similarity of a Triangular Operator to a Diagonal Operator

We give sufficient conditions under which the non-self-adjoint operator A = G + iV ^1/2 JV ^1/2 (with a well-defined imaginary part) is similar to a self-adjoint one. We also give sufficient conditions (these conditions become necessary in the dissipative case) under which the triangular operator is similar to a self-adjoint one. Bibliography: 34 titles.     

Globally generated vector bundles on a smooth quadric surface

We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c1(2,2)in terms of the indices of the bundles,and extend the result to arbitrary higher rank case.We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.     

The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K ₂ O _F the tame kernel of F. In this article, we determine all possible values of r ₄(K ₂ O _F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r ₄(K ₂ O _F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K ₂ O _F for infinitely many imaginary quadratic number fields F.