 
Summary: A BANACH SPACE
DETERMINED BY THE WEIL HEIGHT
DANIEL ALLCOCK AND JEFFREY D. VAALER
Abstract. The absolute logarithmic Weil height is well defined on the quo
tient group Q
×
/ Tor Q
×
and induces a metric topology in this group. We
show that the completion of this metric space is a Banach space over the field
R of real numbers. We further show that this Banach space is isometrically
isomorphic to a codimension one subspace of L1(Y, B, ), where Y is a certain
totally disconnected, locally compact space, B is the algebra of Borel subsets
of Y , and is a certain measure satisfying an invariance property with respect
to the absolute Galois group Aut(Q/Q).
706, May 29, 2008
1. Introduction
Let k be an algebraic number field of degree d over Q, v a place of k and kv the
completion of k at v. We select two absolute values from the place v. The first is
denoted by v and defined as follows:
