Stability of Arakelov Bundles and Tensor Products without Global Sections

This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that $E \otimes F$ has at least a certain slope, but no global sections. It is motivated by an analogous theorem of Faltings for vector bundles over algebraic curves and contains the Minkowski-Hlawka theorem on sphere packings as a special case. The proof uses an adelic version of Siegel's mean value formula.

2000 Mathematics Subject Classification: Primary 14G40; Secondary 11H31, 11R56.

Keywords and Phrases: Arakelov bundle, arithmetic curve, tensor product, lattice sphere packing, mean value formula, Minkowski-Hlawka theorem

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