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?

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It might not immediately help to read Faltings' theorem where Arakelov theory is used. It might also cause further confusions. But I thought it shows its strength. – Banquiz Feb 7 '11 at 7:39

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. I think it's called something like "pinceaux de courbes de genre deux". (Not all texts in this exposé are in French, I believe.)

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

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To call X an 'arithmetic surface' shouldn't be required also that fibers over SpecZ have dim 1 ? – Qfwfq Feb 7 '11 at 9:57
+1 for giving your advice on what not to read: this is very helpful and rarely done. – Georges Elencwajg Feb 7 '11 at 12:57
@unknowngoogle: Of course. That condition is equivalent to the scheme being 2-dimensional (by flatness). – Ariyan Javanpeykar Feb 7 '11 at 14:51
I wasn't looking to learn Arakelov theory, but I read the introduction to R de Jong's thesis anyway and am quite intrigued now. I want to learn more. – Matt May 14 '11 at 16:46
That's great! :D – Ariyan Javanpeykar May 14 '11 at 16:51

You can try to read this wonderful paper by Burgos. It is in Catalan, though. If you are familiar with Spanish or French, you can manage to read it.

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The journal site is here, the article apparently free available: iecat.net/pperiodiques/BV_pub_periodiques2.asp?CollectionID={9910A5B5‌​-F371-4143-88D0-C0A26489D242} – Thomas Riepe Feb 7 '11 at 8:41