An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Veronika Trnková
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Relative Class ◽  
Relative Class Number ◽  
Stickelberger Ideal ◽  
Imaginary Field ◽  
Divisibility Properties

AbstractWe consider certain extension of the Stickelberger ideal of the compositum of a bicyclic field and a quadratic imaginary field, obtained by adding new annihilators to the Stickelberger ideal. We compute the index of this extension, from which we get some divisibility properties for the relative class number of the compositum.

scholarly journals Formulae for the relative class number of an imaginary abelian field in the form of a determinant

2001 ◽  
Vol 163 ◽  
pp. 167-191 ◽  
Author(s):  
Radan Kučera
Keyword(s):  
Class Number ◽  
Relative Class ◽  
Abelian Field ◽  
Relative Class Number ◽  
Stickelberger Ideal ◽  
Cyclotomic Units

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.

A certain eta-quotient and the class number of an imaginary quadratic field

2014 ◽  
Vol 10 (06) ◽  
pp. 1485-1499
Author(s):  
Takeshi Ogasawara
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Theta Series ◽  
Imaginary Quadratic Field

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

Another look at real quadratic fields of relative class number 1

2014 ◽  
Vol 163 (4) ◽  
pp. 371-377 ◽  
Author(s):  
Debopam Chakraborty ◽  
Anupam Saikia
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Relative Class ◽  
Relative Class Number ◽  
Real Quadratic

scholarly journals On Dirichletʼs conjecture on relative class number one

2012 ◽  
Vol 132 (7) ◽  
pp. 1398-1403 ◽  
Author(s):  
Amanda Furness ◽  
Adam E. Parker
Keyword(s):  
Class Number ◽  
Relative Class ◽  
Relative Class Number

scholarly journals Frobenius fields for Drinfeld modules of rank 2

2008 ◽  
Vol 144 (4) ◽  
pp. 827-848 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Chantal David
Keyword(s):  
Quadratic Field ◽  
Galois Representations ◽  
Rational Functions ◽  
Galois Representation ◽  
Field Extension ◽  
Imaginary Quadratic Field ◽  
Drinfeld Module ◽  
Rank 2 ◽  
Imaginary Field ◽  
Square Sieve

AbstractLet ϕ be a Drinfeld module of rank 2 over the field of rational functions $F=\mathbb {F}_q(T)$, with $\mathrm {End}_{\bar {F}}(\phi ) = \mathbb {F}_q[T]$. Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime $\mathfrak {p}$ of good reduction for ϕ, let $\pi _{\mathfrak {p}}(\phi )$ be a root of the characteristic polynomial of the Frobenius endomorphism of ϕ over the finite field $\mathbb {F}_q[T] / \mathfrak {p}$. Let Πϕ(K;d) be the number of primes $\mathfrak {p}$ of degree d such that the field extension $F(\pi _{\mathfrak {p}}(\phi ))$ is the fixed imaginary quadratic field K. We present upper bounds for Πϕ(K;d) obtained by two different approaches, inspired by similar ones for elliptic curves. The first approach, inspired by the work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the Drinfeld module ϕ. The second approach, inspired by the work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained with the first method are better, but depend on the fixed quadratic imaginary field K. In our application of the second approach, we improve the results of Cojocaru, Murty and Fouvry by considering projective Galois representations.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

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