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In this monograph, we lay some foundations of a theory of infinitedimensional Euclidean lattices - and more generally, of infinite dimensionalHermitian vector bundles over some "arithmetic curve" ${\rmSpec}\,\mathcal{O}_K$ attached to the ring of integers $\mathcal{O}_K$ of somenumber field $K$ - with a view towards applications to transcendence theory andDiophantine geometry.  In the first chapters of this monograph, we study the properties of theinvariant $h^0_\theta(\overline{E})$ 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 bundlesand we show that it is possible to associate generalized $\theta$-invariants tothese objects, so that they satisfy suitable subadditivity and summabilityproperties.  In the last chapter, we present a first application of this formalism toDiophantine geometry: we show how it allows one to establish some algebraicitycriterion \`a la Chudnovsky concerning formal curves over number fieldsembedded in some projective space, by arguments that are direct counterparts ofclassical algebraization proofs in complex analytic and formal geometry.

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