DOCUMENTA MATHEMATICA, Vol. 8 (2003), 115-123

Norbert Hoffmann

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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