Imaginary quadratic fields with isomorphic abelian Galois groups

@inproceedings{Angelakis2012ImaginaryQF,
  title={Imaginary quadratic fields with isomorphic abelian Galois groups},
  author={Athanasios Angelakis and Peter Stevenhagen},
  year={2012}
}
In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct… 

Figures and Tables from this paper

A remark On Abelianized Absolute Galois Group of Imaginary Quadratic Fields
The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we
On Abelianized Absolute Galois Group of Global Function Fields
The main purpose of this paper is to describe the abelian part $\mathcal G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as pro-finite group. We will show that the
Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves
The present thesis focuses on two questions that are not obviously related. Namely,(1)What does the absolute abelian Galois group AK of an imaginary quadratic number field K look like, as a
Hecke algebra isomorphisms and adelic points on algebraic groups
Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$,
Adelic point groups of elliptic curves
We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois
L-series and isomorphisms of number fields
  • H. Smit
  • Mathematics
    Journal of Number Theory
  • 2019
On p-rationality of number fields. Applications – PARI/GP programs
— LetK be a number field. We prove that its ray class group modulo p2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short and very fast PARI Programs (Sections
L-functions of Genus Two Abelian Coverings of Elliptic Curves over Finite Fields
Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian
Reconstructing global fields from dynamics in the abelianized Galois group
We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to
...
...

References

SHOWING 1-10 OF 10 REFERENCES
Galois group of the maximal abelian extension over an algebraic number field
The aim of the present work is to determine the Galois group of the maximal abelian extension Ω A over an algebraic number field Ω of finite degree, which we fix once for all. Let Z be a continuous
Class numbers of imaginary quadratic fields
TLDR
A modification of the Goldfeld-Oesterle work, which used an elliptic curve L-function with an order 3 zero at the central critical point to instead consider Dirichlet L-functions with low-height zeros near the real line, which agrees with the prediction of the Generalised Riemann Hypothesis.
Chebotarëv and his density theorem
The Russian mathematician Nikolăı Grigor′evich Chebotarëv (1894–1947) is famous for his density theorem in algebraic number theory. His centenary was commemorated on June 15, 1994, at the University
Class Field Theory: From Theory to Practice
Preface.- Introduction to Global Class Field Theory.- Basic Tools and Notations.- Reciprocity Maps - Existence Theorems.- Abelian Extensions with Restricted Ramification.- Invariant Classes Formulas
Kennzeichnung derp-adischen und der endlichen algebraischen Zahlkörper
Number theory, Classics in Mathematics, Springer-Verlag, Berlin, 2002, Reprint of the 1980 English edition, Edited and with a preface
  • 1980
Class field theory, AMS Chelsea, Providence, RI, 2009, reprinted with corrections from the 1967 original
  • MR 2009k:11001
  • 2009
Class field theory, AMS Chelsea Publishing
  • Providence, RI,
  • 2009