Why is it "easier" to work with function fields than with algebraic number fields? 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.
 A: In number theory one is interested in counting different objects which have arithmetic significance. When working over function fields, this problem can usually be translated into counting points of an algebraic variety over a finite field. One then has a host of techniques available to study such numbers, most importantly, l-adic cohomology. For example, orbital integrals are about counting certain arithmetically significant finite set. Over global fields, this problems translates to point-counts on variants of affine Springer fibres. This is the starting point for Ngo's proof of the Fundamental Lemma. 
A: One answer is that we can take formal derivatives. For example, Fermat's last theorem is rather difficult but the function field version is a straightforward consequence of the Mason-Stothers theorem, whose elementary proof crucially relies on the ability to take formal derivatives of polynomials.
There is no obvious way to extend this construction to integers in a way that preserves its good properties. If there were, then the abc conjecture (of which Mason-Stothers is the function field version) would be trivial, which it's not. There is a thing called the arithmetic derivative, but it is of course not linear, and it doesn't seem to me to be very easy to prove anything with it.
The problem is that if we want to think of $\mathbb{Z}$ as being analogous to a function field, then the "field" that it's a function field over is the field with one element, so if a reasonable notion of formal derivative exists here it needs not to be $\mathbb{Z}$-linear, but to be $\mathbb{F}_1$-linear, whatever that means... if we understood what that meant, perhaps we could construct the "correct" version of the arithmetic derivative and presumably prove the abc conjecture.

Arakelov theory addresses another difference between function fields and number fields, which is the existence of Archimedean places. Over a function field all places are non-Archimedean and I understand this makes various things easier, but I don't know much about this so someone else should chime in here.
A: Here is one aspect of the difference in Arakelov theory: In the number field case we have a "naive" Riemann-Roch formula:
$$
\chi(\alpha)=-\log \textrm{Vol}(\alpha)
$$
where $\alpha$ is an Arakelov divisor on $K$. Using the explicit formula for $\textrm{Vol}(\alpha)$ we can re-write this as
$$
\chi(\alpha)-\chi(O_{K})=\deg(\alpha)
$$
And using Poisson summation formula, we can define $h^{0}(\alpha), h^{1}(\alpha)$ such that
$$
h^{0}(\alpha)=\log(\sum_{f\in I}e^{-\pi|f|_{\alpha}^{2}}), h^{1}(\alpha)=h^{0}(K-\alpha), h^{0}-h^{1}=\deg(\alpha)-\frac{1}{2}|\Delta|
$$
But the corresponding construction in the functional field case is radically different. As we know from Hodge theorem:
$$
H^{1}(X, L)\cong \ker \Delta^{0,1}(L)
$$
and in particular its number is an non-negative integer. Formally we do have an  analgous McKean-Singer formula:
$$
\chi(X,L)=Tr(e^{-D^{*}D})-Tr(e^{-DD^{*}})
$$
But the analogy is not perfect: While both formulas are related to the trace of heat kernel coming from a Dirac operator $D$, there seems to be no good way for us to interpret $h^{0}(\alpha)$ as the dimension of the kernel of some linear operator because of the presence of the $\log$ term in the front. Similarly other approaches of bridiging the Riemann-Roch in number field case and functional field case collapses: There is no good analog of derived functor cohomology that defines $H^{0}, H^{1}$ in the number field case. The best analogy we have for Cech cohomology is theory of ghost spaces, and it is not even a group. On the other hand, translating ghost space machinery to function field case is also not easy. Thus the correspondence of most common cohomology theories (de Rham, Cech, derived functor, singular/betti) between the two worlds breaks down quite badly. The functional field case is "infinitely better" because of certain level of smoothness than the number field.  
Rather, the analogy comes from elsewhere: Recall that we have the Riemann singularity theorem:
Let $C$ be a smooth curve of genus $g$, then for every effective divisor of degree $g-1$:
$$
\textrm{multi}_{\mu(D)+K}(\Theta)=h^{0}(C,O(D))
$$
Van Der Geer and Rene Schoof viewed this as the their motivation for the Riemann-Roch in the number field case (see page 7, for example). Personally I think what is really remarkable is that the analogy persists even in arithemetic surface case, that we have the Faltings volume on the determinant of cohomology defined by the pull-back of the metric of the line bundle $O_{J}(-\Theta)$. While there is no higher dimensional analog of this fact in Arakelov theory (we have to replace Faltings volume by Quillen metric using analytic torsion), this nevertheless suggests a very deep and sutble role played by $\theta$ functions both in the number field and functional field case. 
A: Let's consider an example to see why function fields are easier:
Let $q$ be a prime, and consider the global function field, $\mathbb F_q(T)$.  An ideal $\mathfrak a$ of $\mathbb F_q[T]$ is just the principal ideal $\mathfrak a=(f)=(T^d+a_{d-1}T^{d-1}+\dots+a_0)$.  The norm $N\mathfrak a=q^d$, and you can see that there are exactly $q^d$ ideals of norm $q^d$.
Then the zeta function over this field is $$\zeta_{\mathbb F_q[T]}(s)=\sum_{\mathfrak a \neq 0}N\mathfrak a^{-s}=\sum_{d=0}^\infty q^d(q^d)^{-s}=1/(1-q^{1-s})$$
That's a very simple expression for the zeta function.  Note that it has no zeros, so it trivially satisfies the Riemann Hypothesis.
