Theta invariants of euclidean lattices and infinite-dimensional hermitian vector bundles over arithmetic curves

Abstract

In this monograph, we lay some foundations of a theory of infinite dimensional Euclidean lattices - and more generally, of infinite dimensional Hermitian vector bundles over some "arithmetic curve" $\mathrm{Spec}{\mathcal{O}}_K$ attached to the ring of integers ${\mathcal{O}}_K$ of some number field $K$ - with a view towards applications to transcendence theory and Diophantine geometry. In the first chapters of this monograph, we study the properties of the invariant $h_{\theta }^0(\overline{E})$ attached to some Euclidean lattice $\overline{E}:=(E,\Vert .\Vert )$, defined by the expression $$h_{\theta }^0(\overline{E}):=\log \sum _{v\in E}{e}^{-\pi \Vert v{\Vert }^2},$$ and, more generally, attached to some finite rank Hermitian vector bundle $\overline{E}$ over an arithmetic curve. Then we construct categories of infinite dimensional Hermitian vector bundles and we show that it is possible to associate generalized $\theta$-invariants to these objects, so that they satisfy suitable subadditivity and summability properties. In the last chapter, we present a first application of this formalism to Diophantine geometry: we show how it allows one to establish some algebraicity criterion à la Chudnovsky concerning formal curves over number fields embedded in some projective space, by arguments that are direct counterparts of classical algebraization proofs in complex analytic and formal geometry.

arXiv comment284 pages. Revised and enlarged version. The final chapter presents in detail an application of the formalism of infinite dimensional vector bundles to Diophantine geometry, namely to the derivation of algebraicity criteria concerning germs of formal curves in algebraic varieties over number fields