On the Content Bound for Real Quadratic Field Extensions

Let K be a finite extension of Q and let S = {ν} denote the collection of normalized absolute values on K. Let V + K denote the additive group of adeles over K and let c : V + K → R≥0 denote the content map defined as c({aν}) = ∏ ν∈S ν(aν) for {aν} ∈ V + K . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if {aν} ∈ V + K with c({aν}) > c, then there exists a non-zero element b ∈ K for which ν(b) ≤ ν(aν), ∀ν ∈ S. Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.


Introduction
Let K be a finite extension of Q and let S = {ν} denote the collection of normalized absolute values on K. Let V + K denote the additive group of adeles over K and let K + denote the additive group of K viewed as a subgroup of V + K .Let c : V + K → R ≥0 denote the content map defined as c({a ν }) = ν∈S ν(a ν ) for {a ν } ∈ V + K .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 c > 0 depending only on the field K with the following property: Let {a ν } ∈ V + K be an adele for which c({a ν }) > c.Then there exists a non-zero element b ∈ K + ⊆ V + K for which ν(b) ≤ ν(a ν ), ∀ν ∈ S.
Let {c} denote the set of all positive constants for which Proposition 1.1 holds.Then {c} is a non-empty set of real numbers that is bounded below by 0. Thus inf({c}) exists.We define c K = inf({c}) 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 c K 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 In this paper we construct a new upper bound for c K in the case that K is a real quadratic extension with class number one.We prove the following proposition.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 c K with Lang's bound for some real quadratic extensions K.

Absolute Values
Let K be a finite extension of Q with ring of integers R.An absolute value on K is a function The trivial absolute value is defined as η(0) = 0 and η(x) = 1 for x = 0. Two absolute values η 1 and η 2 on K are equivalent if there exist r ∈ R >0 so that η 1 (x) = (η 2 (x)) r , ∀x ∈ K. 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 Here | | ∞ is the ordinary absolute value on R restricted to Q, and for a rational prime p, | | p is the p-adic absolute value defined as |0| p = 0 and Let η be an absolute value on K. Then η determines a topology on K where the basic open sets are of the form U x, , x ∈ K, > 0, with The topology thus described is the η-topology on K. Let K η 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 K η , which we also denote by η, cf.[3](Chapter XII, §2).In the case If L is a finite extension of the completion K η , then the absolute value η on K η extends uniquely to an absolute value η * on L and L is complete with respect to the η * -topology [3](Chapter XII, Proposition 2.5).
If K is a finite extension of Q of degree N , then each absolute value on Q extends to a finite number (≤ N ) of absolute values η on K [1](Chapter II, Theorem, p. 57).To see how the ordinary absolute value | | ∞ extends to K, let K = Q(α) for some α ∈ C, and let p(x) = irr(α; Q).Let p(x) = g i=1 p i (x) denote the factorization of p(x) over R into irreducible polynomials.Note that g ≤ N .For each i, 1 ≤ i ≤ g, there exists an embedding λ i : K → R(α i ), α → α i , where α i is a root of p i (x).One defines an absolute value η i on K by setting The p-adic absolute value | | p extends to K in the following manner.Let (p) = P e 1 1 P e 2 2 • • • P eg g be the unique factorization of (p) into prime ideals P i of R. Each P i corresponds to an extension η i of | | p to K as follows.Put η i (0) = 0.For r ∈ R, r = 0, let t r be the integer t r ≥ 0 for which (r) ⊆ P tr i , (r) ⊆ P tr+1 i .Now let x = r/s ∈ K, r = 0, s = 0.One then puts R] = 1, and we define the normalized absolute value to be If η i is Archimedean and corresponds to a complex embedding λ i , then the local degree d η i = 2, and we define the normalized absolute value as is the residue class field degree.In this case the normalized absolute value is given as 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 S = {ν} denote the set of normalized absolute values on K; K ν denotes the completion of K with respect to the ν-topology.For ν discrete, we let R ν denote the ring of integers in K ν .The absolute value ν extends to an absolute value on K ν (also denoted by ν.)We consider K ν to be endowed with the ν-topology.

The Adele Ring
Let K be a finite extension of Q and let S = {ν} denote the set of normalized absolute values on K.For each discrete ν, R ν is a compact open subset of K ν .The adele ring V K over K is the topological ring that is the restricted product of the completions K ν with respect to the collection {R ν : ν discrete}, together with the restricted product topology on the completions K ν with respect to the collection {R ν : ν discrete}.This means that V K consists of those vectors {. . ., a ν , . . .} ∈ ν∈S K ν for which a ν ∈ R ν for all but finitely many ν.The ring structure of V K is given component-wise: where U ν is open in K ν for all ν and U ν = R ν for all but finitely many ν.
Let V + K denote the additive group of the adele ring V K and let K + denote the additive group of K.

A Comparison of Bounds
Let d be a square-free positive integer, let K = Q( √ d) and let c K be the content bound for K.In Table 1 below we compare Lang's bound on c K (Proposition 4.1) with the new bound obtained in Proposition 4.4 for all d < 50 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
Proposition 1.3 Let K be a real quadratic extension with class number one.Let f be a fundamental unit of K with f > 1.Then c K ≤ f .