Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

Abstract

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” Ext ˆ X i ( F , G ) we introduce are extensions by groups of analytic types of the usual extension groups Ext X i ( F , G ) attached to O X -modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group Ext ˆ X 1 ( F , G ) – the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X – and we especially consider the case when X is an “arithmetic curve”, namely the spectrum Spec O K of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers. Namely, for any two hermitian vector bundles F ¯ and G ¯ over X : = Spec O K , we attach a logarithmic size s F ¯ , G ¯ ( α ) to any element α of Ext ˆ X 1 ( F , G ) , and we give an upper bound on s F ¯ , G ¯ ( α ) in terms of slope invariants of F ¯ and G ¯ . We further illustrate this notion by relating the sizes of restrictions to points in P 1 ( Z ) of the universal extension over P Z 1 to the geometry of PSL 2 ( Z ) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K ′ ): when the base field K is Q , we establish that the size, which cannot increase under base change, is actually invariant when the field K ′ is an abelian extension of K, or when F ¯ ∨ ⊗ G ¯ is a direct sum of root lattices and of lattices of Voronoi's first kind. The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.