Recently I asked Faltings for some references for a self-study of Arakelov theory beyond Lang’s Arakelov theory.

He suggested the following (all revolving around the arithmetic Riemann-Roch theorem):

1. Papers by Gillet and Soule on arithmetic intersection theory

2. Papers by Bismut, Gillet and Soule on determinant of cohomology of an arithmetic variety (towards Riemann-Roch for this determinant.)

It seems that these papers are summarized in Soule’s book. Also there is a nice book by Faltings that modesty prevented him from referring it to me: Lectures on the arithmetic Riemann-Roch theorem.

Speaking of Arakelov theory, two final comments:

(1) The best place to get into the right frame of mind for these type of questions in an elementary setting is Neukirch’s book Algebraic number theory

(2) Our fellow student here Nikolai Durov has recently reworked the foundations of the entire theory from the point of view of generalized rings (including exotic objects like the field with one element $\mathbb{F}_1$). His thesis is available here.