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

TL;DR

This paper develops a foundational theory of infinite-dimensional Euclidean lattices and Hermitian vector bundles over arithmetic curves, introducing new invariants and applying them to algebraicity criteria in Diophantine geometry.

Contribution

It introduces a formalism for generalized theta-invariants of infinite-dimensional Hermitian bundles and demonstrates their application to algebraicity criteria in number theory.

Findings

Defined the invariant h^0_θ for Euclidean lattices and Hermitian bundles.

Constructed categories of infinite-dimensional Hermitian vector bundles.

Applied the formalism to establish algebraicity criteria for formal 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" ${\rm Spec}\,\mathcal{O}_K$ 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 $h^0_\theta(\overline{E})$ attached to some Euclidean lattice $\overline{E}:= (E, \Vert.\Vert)$, defined by the expression $$h^0_\theta(\overline{E}) := \log \sum_{v \in E} e^{- \pi \Vert v \Vert^2},$$ 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 \`a 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.