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

TL;DR

This paper introduces and studies arithmetic extension groups in Arakelov geometry, focusing on hermitian vector bundles over arithmetic schemes, especially arithmetic curves, linking to lattice theory and the geometry of numbers.

Contribution

It defines arithmetic extension groups involving hermitian vector bundles and explores their properties on arithmetic schemes, particularly arithmetic curves.

Findings

Characterization of arithmetic extension groups via admissible short exact sequences

Connections established between arithmetic extensions and lattice theory

Insights into the geometry of numbers through the study of arithmetic extensions

Abstract

We define and investigate extension groups in the context of Arakelov geometry. The 'arithmetic extension groups' we introduce are extensions by groups of analytic types of the usual extension groups attached to $\O_X$-modules over an arithmetic scheme $X$. In this paper, we focus on the first arithmetic extension group - 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.