I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes some aspects of an analogy between function fields and algebraic number fields.
This led me to google for a while and I ended up reading the Wikipedia entry for Global Field. And this is where my question comes from. In the last sentence of that entry there's the following passage, which I find really interesting:
It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.
Unfortunately, being as dramatic as it is, the example mentioned does not tell me anything because not even the Wikipedia entry on Arakelov Theory is somehow close to give even a small hint as to what it is about.
So I would like to ask for some insight and/or examples that illustrate why it is said to be easier to work with function fields than with algebraic number fields and then try to develop parallel techniques for the number field case.
Thank you very much for any help.