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.
Categories: math.NT