2015/12/30 by Bost, Jean-Benoît · 1 citation
#11G35 #11H06 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1512.08946
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 OK attached to the ring of integers OK 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 h0θ(E) attached to some Euclidean lattice E:= (E, \Vert.\Vert), defined by the expression h0θ(E) := log ∑v ∈ E e- π\Vert v \Vert2, and, more generally, attached to some finite rank Hermitian vector bundle 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 θ-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.