Let
H
λ
4
be the Hecke group
x
,
y
:
x
2
=
y
4
=
1
and, for a square-free positive integer
n
, consider the subset
ℚ
∗
−
n
=
a
+
−
n
/
c
|
a
,
b
=
a
2
+
n
/
c
∈
ℤ
,
c
∈
2
ℤ
of the quadratic imaginary number field
ℚ
−
n
. Following a line of research in the relevant literature, we study the properties of the action of
H
λ
4
on
ℚ
∗
−
n
. In particular, we calculate the number of orbits arising from this action for every such
n
. Some illustrative examples are also given.
Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.
Abstract
We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation
\[{x^2} - d{y^2} = - 1\]
.
Abstract
We develop methods for constructing explicit generators, modulo torsion, of the
$K_3$
-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic
$3$
-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite
$K_3$
-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for
$ K_3 $
of any field, predict the precise power of
$2$
that should occur in the Lichtenbaum conjecture at
$ -1 $
and prove that this prediction is valid for all abelian number fields.
The Hecke Group
H
λ
4
Acting on Imaginary Quadratic Number Fields
