On Humbert-Minkowski’s constant for a number field

Abstract

We use Humbert's reduction theory to introduce an obstruction for the unimodularity of minimal vectors of positive definite quadratic forms over totally real number fields. Using this obstruction we obtain an inequality relating the values of a generalized Hermite constant for such fields, which over the field of rational numbers leads to a well-known result of Mordell. Let K/Q be a totally real number field of degree m with ring of integers OK and real embeddings 1, , am K K R. For n > 1 let Pn,K C R 2mn(n+1) be the space of all m-tuples S = (Si, , Sm), where Si is an n x n symmetric positive definite real matrix. We call such tuples Humbert forms. The unimodular group GL(n, OK) acts on Pn,K as follows. For S = (Si, . Sm) E Pn,K and U E GL(n, OK), (1) S[U] := (S1 [U()] * Sm[U( M)]) where 0) denotes the ith conjugate ai (U) of U, and A[B] =BtAB, whenever this product of matrices is defined. Two Humbert-forms S, S' E Pn,K are called equivalent (S S') if S' = S[U] for some U E GL(n, OK). In [H] a fundamental domain for this action, say RK C Pn,K , is constructed. We will call the elements of RK Humbert reduced forms. For S E Pn,K we define its determinant and minimum as m m detS = f~detSi and m(S) = Min{fJSi[u(')] o 0 7& u E On} i=l i=l Of course if S S', then detS = detS' and m(S) = m(S'). In particular for any S E Pn,K the number -YK (S) = Mr(S) (detS) 1/n depends only on the class [S] of S. Received by the editors January 18, 1996 and, in revised form, June 13, 1996. 1991 Mathematics Subject Classification. Primary 11E12, 11H50; Secondary 11R29, 15A48.