1. Introduction
Let
K be a finite extension of
and let
denote the collection of normalized absolute values on
K. Let
denote the additive group of adeles over
K and let
denote the additive group of
K viewed as a subgroup of
. Let
denote the
content map defined as
for
. We have the following classical result due to J. W. S. Cassels [
1](Lemma, p. 66).
Proposition 1.1 (J. W. S. Cassels) There is a constant depending only on the field K with the following property: Let be an adele for which . Then there exists a non-zero element for which , .
Let
denote the set of all positive constants for which Proposition 1.1 holds. Then
is a non-empty set of real numbers that is bounded below by 0. Thus
exists. We define
to be the
content bound for K. In the case that
K is a real quadratic field extension there is a known upper bound for
due to S. Lang [
2](Chapter V, §1, Theorem 0).
Proposition 1.2 (S. Lang) Let d be a positive square-free integer and let be a quadratic extension.
(i) If , then ,
(ii) if , then .
In this paper we construct a new upper bound for in the case that K is a real quadratic extension with class number one. We prove the following proposition.
Proposition 1.3 Let K be a real quadratic extension with class number one. Let f be a fundamental unit of K with . Then .
It is of interest to compare our new bound with Lang’s bound for various extensions with class number one. For example, if , then the fundamental unit . Since , in this case Lang’s bound is better. On the other hand, if , then the fundamental unit . Since , the new bound of Proposition 1.3 is better. Overall, our new bound is better than Lang’s in many instances.
For the convenience of the reader we begin with a review of some preliminary material (§2, §3.) In §4 we prove the formula for our new bound and in §5 we compare our new bound on with Lang’s bound for some real quadratic extensions K.
2. Absolute Values
Let K be a finite extension of with ring of integers R. An absolute value on K is a function that satisfies
(i) if and only if ,
(ii) ,
(iii) there exists a constant M so that whenever .
The trivial absolute value is defined as and for .
Two absolute values
and
on
K are
equivalent if there exist
so that
. Thus, the absolute values on
K can be partitioned into equivalence classes. It is well-known that up to equivalence the non-trivial absolute values on
consist of
Here
is the ordinary absolute value on
restricted to
, and for a rational prime
p,
is the
p-adic absolute value defined as
and
for
,
,
.
Let
η be an absolute value on
K. Then
η determines a topology on
K where the basic open sets are of the form
,
,
, with
The topology thus described is the
η-topology on K. Let
denote the completion of
K with respect to the
η-topology. In a natural way the absolute value
η on
K extends to a unique absolute value on
, which we also denote by
η, cf. [
3](Chapter XII, §2). In the case
,
, the completion
is the set of real numbers
. If
,
, then the completion
is the field of
p-adic rationals,
. If
L is a finite extension of the completion
, then the absolute value
η on
extends uniquely to an absolute value
on
L and
L is complete with respect to the
-topology [
3](Chapter XII, Proposition 2.5).
If
K is a finite extension of
of degree
N, then each absolute value on
extends to a finite number (
) of absolute values
η on
K[
1](Chapter II, Theorem, p. 57). To see how the ordinary absolute value
extends to
K, let
for some
, and let
. Let
denote the factorization of
over
into irreducible polynomials. Note that
. For each
i,
, there exists an embedding
,
, where
is a root of
. One defines an absolute value
on
K by setting
where
is the unique extension of
to
. The collection
is the set of extensions of
to
K.
The
p-adic absolute value
extends to
K in the following manner. Let
be the unique factorization of
into prime ideals
of
R. Each
corresponds to an extension
of
to
K as follows. Put
. For
,
, let
be the integer
for which
,
. Now let
,
,
. One then puts
The collection
is the set of extensions of
to
K. Since
, there are at most
N extensions.
The extensions η of are the Archimedean absolute values on K. The extensions of η of are the non-Archimedean (or discrete) absolute values on K. Absolute values η on K obtained as extensions constitute all of the absolute values on K (up to equivalence.)
If
is Archimedean and corresponds to a real embedding
, then the
local degree , and we define the
normalized absolute value to be
If
is Archimedean and corresponds to a complex embedding
, then the local degree
, and we define the
normalized absolute value as
If
is a discrete extension of
corresponding to the prime ideal
, the local degree is
where
is the residue class field degree. In this case the
normalized absolute value is given as
where
.
If ν is the normalized absolute value obtained from η, then the ν-topology on K is equal to the η-topology on K since ν and η are equivalent absolute values. In what follows we let denote the set of normalized absolute values on K; denotes the completion of K with respect to the ν-topology. For ν discrete, we let denote the ring of integers in . The absolute value ν extends to an absolute value on (also denoted by ν.) We consider to be endowed with the ν-topology.
3. The Adele Ring
Let
K be a finite extension of
and let
denote the set of normalized absolute values on
K. For each discrete
ν,
is a compact open subset of
. The
adele ring over K is the topological ring that is the restricted product of the completions
with respect to the collection
, together with the restricted product topology on the completions
with respect to the collection
. This means that
consists of those vectors
for which
for all but finitely many
ν. The ring structure of
is given component-wise:
We write
for the adele
. A basis for the topology on
consists of open sets of the form
where
is open in
for all
ν and
for all but finitely many
ν.
Let denote the additive group of the adele ring and let denote the additive group of K.
Proposition 3.1 Let , . Then .
Proof. For two proofs, see [
1](Chapter II, Theorem, p. 60 and p. 66). ⋄
Proposition 3.2 embeds into through the map .
Proof. Let and write , where , . Since there are only a finite number of prime divisors of c, for all but a finite number of ν. Thus with for all is an adele of K. It is easy to show that the map is an injection of groups . ⋄
With these preliminaries in mind, we now give two upper bounds for the content bound in the case that K is a real quadratic extension with class number one.
4. Two Bounds for
Let d be a square-free positive integer, let denote the real quadratic extension with ring of integers R. Let be the content bound for K. We recall some number-theoretic facts about K. If then and if then . The discriminant if , and if . If , then the only rational primes that ramify are divisors of d. If , the rational primes that ramify are 2 and the divisors of d.
The set of normalized absolute values on
K is computed as follows. The Archimedean absolute value
on
extends to two normalized absolute values,
,
, defined as follows. For
,
and
The discrete absolute values on
extend to
K in the following manner. If
, then
for some prime ideal
P of
R. Thus
extends to one normalized absolute value
ν on
K. On the other hand, if
and
, then
for
P prime, and so,
p remains prime in
R. In this case,
extends to one normalized absolute value
ν on
K. If
and
, then
for
prime and so,
extends to two normalized absolute values
.
Let
denote the set of normalized absolute values on
K, and let
be the additive group of adeles. There is a known bound for
due to S. Lang [
2].
Proposition 4.1 (S. Lang) Let d be a positive square-free integer and let be a quadratic extension.
(i) If , then ,
(ii) if , then .
Proof. For a proof see [
2](Chapter V, §1, Theorem 0). ⋄
To prove our formula for a new bound on
, we need some lemmas regarding units in
R. The units group of
R is
where
h is a fundamental unit in
R. Note that
.
Lemma 4.2 There exists a fundamental unit f in R with .
Proof. Let h be a fundamental unit. We consider first the case . If , then we set and condition is satisfied. Else, assume that . Then implies that . Of course, is a fundamental unit and so we set . If , then is a fundamental unit and as shown above we may take . ⋄
Lemma 4.3 If h is a fundamental unit of R, then .
Proof. Let
be the norm map defined as
for
. The norm map restricts to give a map
. Now suppose that
h is a fundamental unit with inverse
. Then
and
are in
. Moreover,
yields
Consequently,
, and thus
, or
. ⋄
We now give the new bound on .
Proposition 4.4 Let d be a square-free positive integer, let and assume that K has class number one. Let be a fundamental unit in R. Then .
Proof. We show that if
is an adele in
with
, then there exists
,
, so that
for all
. For
ν discrete, let
denote the completion of
K with respect to the
ν-topology, and let
denote the ring of integers in
. We have
where
is a unit in
,
, and where
is a uniformizing parameter for
. Since
is an adele,
for all but a finite number of
ν, and since
,
for all but a finite number of
ν. Let
denote the collection of discrete
ν for which
, listed so that
are those
with
and
are the
for which
.
For
, let
denote the ideal of
R corresponding to the discrete normalized absolute value
. Then the ideal
is principal and generated by an element
. Moreover, the ideal
is principal and generated by an element
.
Let
. We can assume without loss of generality that
. For all
ν discrete,
. Thus
Since
, there exists an integer
k for which
Put
. Then
. Now from Inequality (2),
So,
Observe that
thus
. For
ν discrete,
is a unit in
. Thus
for all
ν discrete. Thus
b is as required. ⋄
5. A Comparison of Bounds
Let
d be a square-free positive integer, let
and let
be the content bound for
K. In
Table 1 below we compare Lang’s bound on
(Proposition 4.1) with the new bound obtained in Proposition 4.4 for all
for which
K has class number one. The values of
d were obtained from [
4](A003172). The values of the fundamental unit
f were computed using an algorithm based on [
5](Theorem 11.5) as implemented in [
6]. We conclude that our new bound is better than Lang’s bound in 17 out of 22 of the cases.
Of course, the fundamental unit f has been proven to be a bound for only in the case that K has class number one. It would be of interest to extend Proposition 4.4 to the case where K has class number greater than one.
Table 1.
Lang’s bound compared with the fundamental unit.
Table 1.
Lang’s bound compared with the fundamental unit.
d | Lang’s bound (L) | new bound (f) | L vs. f |
---|
2 | 32 | | |
3 | 48 | | |
5 | | | |
6 | 96 | | |
7 | 112 | | |
11 | 176 | | |
13 | | | |
14 | 224 | | |
17 | | | |
19 | 304 | | |
21 | | | |
22 | 352 | | |
23 | 368 | | |
29 | | | |
31 | 496 | | |
33 | | | |
37 | | | |
38 | 608 | | |
41 | | | |
43 | 688 | | |
46 | 732 | | |
47 | 752 | | |