A good text book on arithmetic surfaces is:

Qing Liu, Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts 
in Mathematics 6, Oxford Science Publications 2002

A good text book on Riemann surfaces is:

Otto Forster, Lectures on Riemann surfaces, Graduate Texts in 
Mathematics 81, Springer-Verlag New York Berlin 1993

For those who want to get deeper into the subject matter of the lectures:

Serge Lang, Introduction to Arakelov theory, Springer-Verlag New York 
Berlin 1988

Laurent Moret-Bailly, Hauteurs et classes de Chern sur les surfaces 
arithme'tiques. In: Se'minaire sur les pinceaux des courbes elliptiques, 
Aste'risque 183 (1990), 37--58.

A.N. Parshin, The Bogomolov-Miyaoka-Yau inequality for arithmetic surfaces 
and its applications. In: Se'minaire de The'orie des nombres, Paris 
1986-87. Progress in Mathematics 75, Birkhauser Verlag 1989.

Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984), 
387--424.

P. Vojta, Diophantine inequalities and Arakelov theory. Appendix to Serge 
Lang's book mentioned above.


I will try to discuss the subject matter of these texts with 
emphasis on "explicit examples" involving for instance  elliptic curves.