In this paper we extend the arithmetic Grothendieck-Riemann-Roch Theorem to projective morphisms between arithmetic varieties that are not necessarily smooth over the complex numbers. The main ingredient of this extension is the theory of generalized holomorphic analytic torsion classes previously developed by the authors.
Let be a curve over a field with a rational point . We define a canonical cycle . Suppose that is a number field and that has semi-stable reduction over the integers of with fiber components non-singular. We construct a regular model of and show that the height pairing is well defined where and are correspondences. The paper ends with a brief discussion of heights and -functions in the case that is a modular curve.
On donne une nouvelle démonstration directe du théorème de Hilbert-Samuel arithmétique et on déduit un critère numérique pour l’existence de sections d’un fibré en droite sur une variété arithmétique de norme sup inférieure à un.
Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of -rational points of a curve of genus under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.