Neukirch's motivation for $p$-adic numbers I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows:
"The idea originated from the observation made in the last chapter that the numbers $f\in\mathbb{Z}$ may be viewed in analogy with the polynomials $f(z)\in\mathbb{C}[z]$ as functions on the space $X$ of prime numbers in $\mathbb{Z}$, associating to them their "value" at the point $p\in X$, i.e., the element
$\begin{equation}
f(p):=f \mod p
\end{equation}$
in the residue class field $\kappa(p)=\mathbb{Z}/p\mathbb{Z}$.
This point of view suggests the further question: whether not only the "value" of the integer $f\in\mathbb{Z}$ at $p$, but also the higher derivatives of $f$ can be reasonably defined."
My question is: If we define the higher derivatives of $f\in\mathbb{Z}$ at $p$ as the coefficientes of its $p$-adic expansion (as he does later on), can we give these derivatives an interpretation somehow analogous to the analytical one that goes with the $f(z)\in\mathbb{C}[z]$ polynomials? 
I'm asking this because if there is no interpretation other than "coefficientes of the expansion", it seems like a very artificial analogy to me. I mean, if not, why would he even mention derivatives instead of just saying something like: "we expand functions this way, we expand integers this way and it's kind of similar."
Any explanations and/or motivations are welcome.
Thanks in advance.
 A: "My question is: If we define the higher derivatives of f∈Z at p as the coefficientes of its p-adic expansion (as he does later on), can we give these derivatives an interpretation somehow analogous to the analytical one that goes with the f(z)∈C[z] polynomials?"
Answer: At the following link
https://mathoverflow.net/questions/55244/why-must-nilpotent-elements-be-allowed-in-modern-algebraic-geometry/378811#378811
you will find an elementary construction of a "Taylor morphism"
T1. $T^l: E \rightarrow J^l(E)$
for any commutative ring $A$ and any left $A$-module $E$. If $A:=k[x_1,..,x_n]$
with $k$ a field of characteristic zero, you get a map
T2. $T^l: A \rightarrow J^l(A):=A\otimes_k A/I^{l+1}$
defined by $T^l(f(x_1,..,x_n)):=f(y_1,..,y_n)$. Let $dx_i:=y_i-x_i$. You may prove that there is an isomorphism
T3. $J^l(A)\cong A\{dx_1^{l_1}\cdots dx_n^{l_n}: \sum_i l_i\leq l\}$.
Hence the left $A$-module $J^l(A)$ is a free $A$-module on the set $\{dx_1^{l_1}\cdots dx_n^{l_n}: \sum_i l_i\leq l\}$. In this situation it follows the
element $T^l(f(x_1,..,x_n))$ is the Taylor expansion of the polynomial $f$ in the variables $x_i$.
Comment: "I think his point is that if one takes X=A1C as a variety with sheaf OX, then for any point p∈X one has that OX,pˆ=C[[z−p]] and the map C[z]=OX,x→OX,pˆ is just taking f∈C[z] and expanding it as a Taylor series at p. Similarly, one may view the p-adic case as taking the scheme X=Z, some global section f∈Z=OX(X) and mapping it to OX,pˆ=Zp."
Answer: The fiber of the $A$-module $J^l(A)$ at a $k$-rational point $\mathfrak{m}$ is the local ring $A/\mathfrak{m}^{l+1}$, hence this construction may be related to the construction mentioned in the comments above.
In Neukirch, Prop 2.6 page 114 the following isomorphism is constructed:
There is for every non-zero prime $p$ a canonical isomorphism
$T_p: \mathbb{Z}_{p}\cong \mathbb{Z}[[x]]/(x-p)$
and a canonical injective map $i_p: \mathbb{Z} \rightarrow \mathbb{Z}_p$.
There is a canonical map $T: \mathbb{Z}\rightarrow \mathbb{Z}[[x]]$ and we recover the map $i_p$ by passing to the quotient $P(p): \mathbb{Z}[[x]] \rightarrow \mathbb{Z}[[x]]/(x-p)$.
$i_p:=P(p)\circ T: \mathbb{Z}\rightarrow \mathbb{Z}[[x]]/(x-p)\cong \mathbb{Z}_p$.
Hence all the injections $i_p$ can be constructed from one "global" map $T$.
We may pass to "spectra" $C:=Spec(\mathbb{Z}[[x]])$ and $S:=Spec(\mathbb{Z})$
and the canonical map
$\pi: C \rightarrow S$.
The ideal $I(p):=(x-p) \subseteq \mathbb{Z}[[x]]$ corresponds to a closed subscheme $C_p:=Z(x-p)\subseteq C$ and we get a canonical morphism $\pi_p: C_p\rightarrow S$. There is a corresponding map of structure sheaves
$\pi_p^{\#}: \mathcal{O}_S \rightarrow (\pi_p)_*\mathcal{O}_{C_p}$
whose global sections is the canonical map $i_p:\mathbb{Z}\rightarrow \mathbb{Z}_p$. Hence all maps $i_p$ may be given a "geometric" construction using the map $\pi$: You restrict the map $\pi$ to the closed subscheme $C_p$ and pass to global sections.
