On the Content Bound for Real Quadratic Field Extensions

Abstract: Let K be a finite extension of Q and let S = {ν} denote the collection of K normalized absolute values on K. Let V⁺_K denote the additive group of adeles over K and let K ≥0   c : V + → R denote the content map defined as c({aν }) = Q K   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({a_ν })  > c, then there exists a non-zero element b  ∈ K for which ν (b) ≤ ν (aν ), ∀ν  ∈ S. Let c_K 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 c_K due to S. Lang. The purpose of this paper is to construct a new upper bound for c_K 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.