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Multiplicative lattices in global fields. (English) Zbl 0739.11024

Let \(L\) be an \(N\)-dimensional lattice. Denote \(\hbox{det}(L)\) the volume of the fundamental domain and \(d(L)\) the length of the shortest non-zero lattice vector of \(L\). The density exponent for \(L\) is defined by \[ \lambda(L)=-(1/N)\log_ 2\left({d(L)^ N V_ N \over 2^ N \hbox{det}(L)}\right), \] where \(V_ N\) denotes the volume of the \(N\)- dimensional unit sphere. The aim of this paper is to construct families of lattices \(\{L_ N\}\) for which \(\lim_{N\to\infty}\lambda(\{L_ N\})\) is small, i.e. for which the points of \(\{L_ N\}\) are the centers of dense sphere packing.
The following types of lattices are treated: principal and congruence lattices associated to divisors over curves over a finite field and principal and congruence lattices associated to \(S\)-unit groups of towers of algebraic number fields.
The best asymptotic density exponent \(\lambda(\{L_ N\})\leq 1.39\) is proved for certain congruence lattices in the function field case.

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
11R37 Class field theory
11R58 Arithmetic theory of algebraic function fields
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References:

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