I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read to learn about the geometry that in turn inspired the ideas in Arakelov theory?
Best Answer
To get a nice overview of how and why Arakelov theory started you could read the introduction to R. de Jong's Ph.D. thesis on
http://www.math.leidenuniv.nl/~rdejong/publications/
I remember that being very helpful to me.
To avoid too many complex analytic difficulties you should stick to the case of arithmetic surfaces (i.e. integral regular flat projective 2-dimensional $\mathbf{Z}$-schemes). The complex analysis involved is all "Riemann surfaces theory". An elementary and thorough treatment of this is given in P. Bruin's master's thesis
http://www.math.leidenuniv.nl/~pbruin/
Arakelov theory provides an intersection pairing on an arithmetic surface $X$. The idea is to add vertical divisors on $X$ above the "points at infinity" on Spec $\mathbf{Z}$ (or Spec $O_K$ ). There will be two contributions: finite and infinite. To get a good understanding of the finite contributions I recommend reading Chapter 8.3 and 9.1 of Q. Liu's book.
I remember that after reading these texts the article by Faltings was much more readible to me. I also enjoyed the very nice asterisk by Szpiro on the subject, Séminaire sur les pinceaux de courbes de genre au moins deux (all in French, last page has an English abstract).
Here's some advice on what you shouldn't read when you just start. I wouldn't start immediately reading the papers by Gillet and Soulé (unless you really want too). The complex analysis is very involved. The paper by Bost "Potential theory and Lefschetz theorems for arithmetic surfaces" introduces the most general intersection theory (based upon $L^2_1$ Green functions) and should also be left for later reading in my opinion.
To learn Arakelov theory the proofs don't really help me understand the statements for they are based upon moduli space arguments usually (e.g. the proof of the Noether formula). Therefore, I would also recommend you skip most of the proofs on a first reading.
What did help is seeing how Arakelov theory gets applied. I recommend the recent book by Couveignes, Edixhoven, et al. available here
http://arxiv.org/abs/math/0605244