Extensions Of Imaginary Quadratic Fields Research Articles

Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields and imaginary quadratic fields with coprime discriminants. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.

Finite submodules of Iwasawa modules for multi-variable extensions

For multi-variable extensions of number fields, we study the maximal finite submodules of classical Iwasawa modules. Our principal objects are the unramified Iwasawa modules for two-variable extensions of imaginary quadratic fields. On the one hand, we obtain a novel way to show a certain upper bound, or even the vanishing, of the maximal finite submodules. On the other hand, we obtain a sufficient condition for the non-vanishing, and then construct numerical examples for which the non-vanishing holds.

Eisenstein cocycles over imaginary quadratic fields and special values of L-functions

We generalize Sczech's Eisenstein cocycle for GL(n) over totally real extensions of Q to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L-functions.

On the Schertz Conjecture

AbstractSchertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.

Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $\xi$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We consider an infinite family of octic fields $K_{c}=M\left( \xi\right)$ with the ring of integers $\mathbb{Z}_{K_{c}}.$ Our goal is to determine all generators of relative power integral basis of $\mathcal{O=}\mathbb{Z}_{M}\left[ \xi\right]$ over $\mathbb{Z}_{M}.$ We show that our problem reduces to solving the system of relative Pellian equations \[ cV^{2}-\left( c+2\right) U^{2}=-2\mu, cZ^{2}-\left( c-2\right) U^{2}=2\mu, \] where $\mu$ is an unit in $\mathbb{Z}_{M}$. We solve the system completely and find that all non-equivalent generators of power integral basis of $\mathcal{O}$ over $\mathbb{Z}_{M}$ are given by $\alpha=\xi,$ $2\xi-2c\xi ^{2}+\xi^{3}$ for $\left\vert c\right\vert \geq159108$ and $|c|\leq200$.

Integrality of Stickelberger elements attached to unramified extensions of imaginary quadratic fields

In this paper, we prove the integrality of Stickelberger elements attached to finite (not necessarily abelian) unramified extensions of imaginary quadratic fields. As an application of this result, we describe the relation between non-commutative Fitting invariants of some arithmetic modules and Stickelberger elements. Consequently, we prove two types of annihilation results for ray class groups.

On a new type of -adic regulator for algebraic number fields (the -adic regulator without logarithms)

For an algebraic number field such that a prime splits completely in , we define a regulator that characterizes the subgroup of universal norms from the cyclotomic -extension of in the completed group of -units of , where consists of all prime divisors of . We prove that the inequality follows from the -adic Schanuel conjecture and holds for some Abelian extensions of imaginary quadratic fields. We study the connection between the regulator and the feeble conjecture on the -adic regulator, and define analogues of the Gross regulator.

Calculating Power Integral bases by Solving Relative Thue Equations

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

CONSTRUCTION OF THE FIRST LAYER OF ANTI-CYCLOTOMIC EXTENSION

In this paper, using a theorem of Brink for prime decomposition of the anti-cyclotomic extension, we explicitly construct the first layer of the anti-cyclotomic ${\mathbb Z}_3$-extension of imaginary quadratic fields.

On the two-variables main conjecture for extensions of imaginary quadratic fields

Let $p$ be a prime number at least 5, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two conjugate primes. Let $k_\infty$ be the unique ${\boldsymbol Z}_p^2$-extension of $k$, and let $K_\infty$ be a finite extension of $k_\infty$, abelian over $k$. We prove that in $K_\infty$, the characteristic ideal of the projective limit of the $p$-class group coincides with the characteristic ideal of the projective limit of units modulo elliptic units. Our approach is based on Euler systems, which were first used in this context by Rubin.

ANTI-CYCLOTOMIC EXTENSION AND HILBERT CLASS FIELD

In this paper, we show how to construct the first layer <TEX>$k^{\alpha}_{1}$</TEX> of anti-cyclotomic <TEX>${\mathbb{{Z}}}_{3}$</TEX>-extension of imaginary quadratic fields <TEX>$k(=\;{\mathbb{{Q}}}(\sqrt{-d}))$</TEX> when the Sylow subgroup of class group of k is 3-elementary, and give an example. This example is different from the one we obtained before in the sense that when we write <TEX>$k^{\alpha}_{1}\;=\;k({\eta}),{\eta}$</TEX> is obtained from non-units of <TEX>${\mathbb{{Q}}}({\sqrt{3d}})$</TEX>.

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

For the cyclotomic \({\mathbb Z_2}\)-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 \({\mathbb {Z}_2}\)-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.

Special units and ideal class groups of extensions of imaginary quadratic fields

Let K be an imaginary quadratic field, and let F be an abelian extension of K, containing the Hilbert class field of K. We fix a rational prime p &gt; 2 which does not divide the number of roots of unity in the Hilbert class field of K. Also, we assume that the prime p does not divide the order of the Galois group G:=Gal(F/K). Let AF be the ideal class group of F, and EF be the group of global units of F. The purpose of this paper is to study the Galois module structures of AF and EF.

On the first layer of anti-cyclotomic $\mathbf {Z}_{p}$-extension of imaginary quadratic fields

In this paper, we give an explicit description of the first layer of anti-cyclotomic $\mathbf{Z}_{p}$-extension of imaginary quadratic fields.

Abelian p-class field towers over the cyclotomic Zp-extensions of imaginary quadratic fields

Abelian p-class field towers over the cyclotomic Z_p-extensions of imaginary quadratic fields

Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

Let k = Q ( − D ) be an imaginary quadratic number field with ring of integers Z k and let k ( α ) be the cubic extension of k generated by the polynomial f t ( x ) = x 3 − ( t − 1 ) x 2 − ( t + 2 ) x − 1 with t ∈ Z k . In the present paper we characterize all elements γ ∈ Z k [ α ] with norms satisfying | N k ( α ) / k | ≤ | 2 t + 1 | for | t | ≥ 14 . This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals.

Defining polynomial of the first layer of anti-cyclotomic $\mathbf {Z}_3$-extension of imaginary quadratic fields of class number 1

In this paper, we explicitly compute defining polynomials of the first layer of anti-cyclotomic $\mathbf{Z}_3$-extension of imaginary quadratic fields of class number 1.

Chinburg's Third Invariant for Abelian Extensions of Imaginary Quadratic Fields

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions N/K of quadratic imaginary fields K with group G. For abelian groups G, the conjecture specialises to the statement that Chinburg's third invariant Ω(N/K, 3)=0 ∈ Cl(Z[G]). We prove (Theorem 1) that if K has class number 1 (and some technical conditions on the conductor of N/K and its roots of unity hold) then Ω(N/K, 3) lies in the kernel group D(Z[G]). For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that Ω(N/K, 3)=0. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order MG ∈ Q[G]. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated Z[G]-modules, which is isomorphic to Cl(MG). For Theorem 2, the fact that the class number of N is coprime to [N:K] in such genus extensions N/K allows one to lift these results to Cl(Z[G]), after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining Ω (N/K, 3) has the correct extension class (which is related to the fundamental class of N/K). 1991 Mathematics Subject Classification: 11R33.

The Exponent 2-Class-Group Problem for Non-Galois Over Q Quartic Fields That Are Quadratic Extensions of Imaginary Quadratic Fields

Focusing on a particular case, we will show that one can explicitly determine the quartic fields K that have ideal class groups of exponent ≤ 2, provided that K/ Q is not normal, provided that K is a quadratic extension of a fixed imaginary quadratic number field, and provided that the regulator of K is not too large compared with the discriminant of K.

Chinburg's Third Invariant in the Factorisability Defect Class Group

AbstractChinburg's third invariant Ω(N/K, 3) ∊ C1(Z[Γ]) of a Galois extension N/K of number fields with group Γ is closely related to the Galois structure of unit groups and ideal class groups, and deep unsolved problems such as Stark's conjecture.We give a formula for Ω(N/K, 3) modulo D(ZΓ) in the factorisability defect class group, reminiscent of analytic class number formulas. Specialising to the case of an absolutely abelian, real field N, we give a natural conjecture in terms of Hecke factorisations which implies the vanishing of the invariant in the defect class group.We prove this conjecture when N has prime-power conductor using Euler systems of cyclotomic units, Ramachandra units and Hecke factorisation. This supports a general conjecture of Chinburg, which in our situation specialises to the statement that Ω(N/K, 3) = 0 for such extensions.We also develop a slightly extended version of Euler systems of units for general abelian extensions, which will be applied to abelian extensions of imaginary quadratic fields elsewhere