Here is the entry of Lang's book in Springer Catalogue
Lang, S., Yale University, New Haven, CT, USA
Introduction to Arakelov Theory
1988. X, 187 pp. 8 figs.
ISBN 3-540-96793-1
Arakelov introduced a component at infinity in arithmetic considerations,
thus giving rise to global theorems similar to those of the theory of
surfaces, but in an arithmetic context over the ring of integers of a
number field. The book gives an introduction to this theory, including
the analogues of the Hodge Index Theorem, the Arakelov adjunction formula,
and the Faltings Riemann-Roch theorem. The book is intended for second
year graduate students and researchers in the field who want a systematic
introduction to the subject. The residue theorem, which forms the basis
for the adjunction formula, is proved by a direct method due to Kunz and
Waldi. The Faltings Riemann-Roch theorem is proved without assumptions
of semistability. An effort has been made to include all necessary details,
and as complete references as possible, especially to needed facts of
analysis for Green's functions and the Faltings metrics.
Contents: Foreword.- Metrics and Chern Forms.- Green's Functions on
Riemann Surfaces.- Intersections on an Arithmetic Surface.- Hodge Index
Theorem and the Adjunction Formula.- The Faltings Riemann-Roch Theorem.
- Faltings Volumes on Cohomology.- Diophantine Inequalities and Arakelov
Theory.- References.- Freqently Used Symbols.- Index.