%0 Journal Article
%T On the Content Bound for Real Quadratic Field Extensions
%A Robert G. Underwood
%J Axioms
%D 2013
%I MDPI AG
%R 10.3390/axioms2010001
%X Let K be a finite extension of Q and let S = { ¦Í} denote the collection of K normalized absolute values on K. Let V + K denote the additive group of adeles over K and let K ¡Ý0 £¿ c : V + ¡ú R denote the content map defined as c({a¦Í }) = Q K £¿ ¦Í ¡ÊS ¦Í (a¦Í ) for {a¦Í } ¡Ê£¿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 {a¦Í } ¡Ê V + K with c({a ¦Í }) £¿> c, then there exists a non-zero element b £¿¡Ê K for which ¦Í (b) ¡Ü ¦Í (a¦Í ), £¿¦Í £¿¡Ê 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.
%K adele group
%K content map
%K real quadratic extension
%U http://www.mdpi.com/2075-1680/2/1/1