An introduction by Burgos-Gil
Lang: Introduction to Arakelov theory
http://www.ncatlab.org/nlab/show/Arakelov+geometry
http://mathoverflow.net/questions/78460/learning-arakelov-geometry
Manin-Panchiskin has some material
Faltings: Calculus on arithmetic surfaces
Gillet in Arcata: Intro to higher-dimensional Arakelov theory
Soulé, Christophe Hermitian vector bundles on arithmetic varieties. Algebraic geometry—Santa Cruz 1995, 383–419, Proc. Sympos. Pure Math., 62, Part 1. This looks excellent, I haven’t read it but noted that on p 397 there are references to several attempts to develop adelic intersection theory.
I think Neukirch ANT works out things for Pic-hat of the ring of integers in a number field.
MR1087394 (92d:14016) Gillet, Henri(1-ILCC); Soulé, Christophe(F-IHES) Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. No. 72 (1990), 93–174 (1991). (Long review)
MR1724892 (2000m:14030) Künnemann, Klaus(D-KOLN); Maillot, Vincent(F-ENS-MI) Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire. (French)
A course syllabus for a course by Gillet
Vojta: Applications of arithmetic algebraic geometry to Diophantine approximations (1993)
One could maybe look at Bismut, at least the reviews
arXiv:0909.3680 On Volumes of Arithmetic Line Bundles II from arXiv Front: math.AG by Xinyi Yuan For a hermitian line bundle over an arithmetic variety, we construct a convex continuous function on the Okounkov body associated to the generic fibre of the line bundle. The integration of the continuous function gives the growth of the Euler characteristic of the hermitian line bundle. It is the global version of the recent work of Nystrom.
arXiv:1010.1599 Toward Dirichlet’s unit theorem on arithmetic varieties from arXiv Front: math.NT by Atsushi Moriwaki In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet’s unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.
arXiv:1102.2063 Hermitian structures on the derived category of coherent sheaves from arXiv Front: math.AG by José Ignacio Burgos Gil, Gerard Freixas i Montplet, Razvan Litcanu The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. To this end, we first define hermitian structures on the objects of the bounded derived category of coherent sheaves on a smooth complex variety. Secondly we extend the theory of Bott-Chern classes to these hermitian structures. Finally we introduce the category whose morphisms are projective morphisms with a hermitian structure on the relative tangent complex.
nLab page on Arakelov theory