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.