TITLE: How and why does Grothendieck's work provide tools to attack problems in number theory?
QUESTION [66 upvotes]: This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing.
I have always been fascinated with Grothendieck and the way he did mathematics.
I've heard Mochizuki's work on the abc conjecture heavily involves Grothendick-ean algebro-geometric ideas (it kind of seems like everything does). I also know Grothendieck's work has been crucial in the development of tools and subsequent proof of the Weil conjectures. Moreover, I've generally seen Grothendieck's name pop up in all kinds of number-theoretic contexts.
From what I've read about Grothendieck himself, it seems fair to say he wasn't very interested in numbers at all, at least not in the sense of solving specific problems like the abc conjecture. It also seems safe to say that Grothendieck's mathematics was not generally developed for the purpose of solving specific problems. And yet, his highly abstract ideas find use in one of the seemingly least "functorial" branches of math. Why is this?
Knowing very little mathematics myself, I tried to read about Grothendieck's part in the proof of the Weil conjectures. The term "algebraic geometry over the integers" (page 15, third paragraph) caught my eye, and I have (falsely?) concluded that in a very rough sense, the generality in which Grothendieck worked gave algebraic geometry the flexibility needed to "work" over the integers.
I have not been able to find anything as accessible about Mochizuki's work, and I can understand absolutely nothing from the discussions on MO.
I also don't know any number theory, I'm just curious how such abstract mathematics can produce (even merely interact with) such incredibly concrete, specific results.
So I guess my questions are:


How do Grothendieck's ideas manifest in number theory?
Why is this possible? Is it "just" about brilliantly applying algebro-geometric ideas to the integers?
Exactly which ideas of Grothendieck's ideas appear in these number-theoretic contexts?
Am I missing the point entirely and asking the wrong questions because I don't know anything concrete?

REPLY [6 votes]: A somewhat belated answer to this question:
Firstly, this earlier answer to a very similar question might be of interest.
Secondly, the spirit of the question in large part seems to be: numbers and number theory are concrete, whereas Grothendieckian geometry is abstract and structural, so how can the two interact successfully?
To my mind, a big part of the answer is that number theory (at least, algebraic number theory) is actually incredibly structural (despite first appearances) and so a highly structured theory like Grothendieck's is exactly the right kind of tool for investigating it.  And we don't have to go to Grothendieckian levels of theory to illustrate this.  Much simpler examples will already convey the flavour:

Example:  It is a structural result from algebra that a finite subgroup
of the multiplicative group of a field is necessarily cyclic.  (It follows
by a counting argument using the fact that $x^n = 1$ has at most $n$
solutions in a field.)  
It is an even easier structural result from algebra that a finite integral
domain is a field.  (Multiplication by a non-zero divisor is injective,
thus on a finite set it must also be surjective.)
The notion of prime number is essentially defined so that $p$ is prime
if and only if the ring $\mathbb Z/p$ of integers mod $p$ is an integral domain.  
This residue ring is also finite.  (This follows e.g. from the Euclidean 
algorithm.)
Putting all these together, we see that if $p$ is prime, then the multiplicative group of non-zero residues mod $p$ is cyclic.  This is the famous result about the existence of "primitive roots" mod $p$, proved now by structural means.

Example:  In a cyclic group of even order, the unique element of order $2$ has a square root if and only if the order of the group is divisible by $4$.  (This could be a problem on a first exam, or first homework, in group theory.)
Applying this to the non-zero residues mod $p$ (which by the previous example is cyclic!), we get one of the supplementary laws to quadratic reciprocity:  $-1$ is a quadratic residue modulo an odd prime $p$ if and only if $p \equiv 1 \bmod 4$.

Example (a more involved one now):
Consider the problem of solving $x^2 - p y^2 = -1$, for an odd prime $p$.  The previous result shows that a necessary condition is that $p \equiv 1
\bmod 4$.  
This is also sufficient!  
This can be proved by considering the extension $\mathbb Q(\sqrt{p})$,
various short exact sequences that arise out of its ideal theory,
and then computing some group cohomology for the cyclic group of order $2$
(the Galois group of this field over $\mathbb Q$).  See e.g. this exercise sheet for 
a guide to the details.

The uses of Grothendieck's ideas in number theory are very similar in spirit to the examples just described.  The topic is highly structured, and once you begin to see what the structures are and how they interact, it is natural to try to use structural tools to investigate them.  (Note also that my examples built one on the other.  This is characteristic of structural areas of mathematics such as algebraic number theory or Grothendieck's algebraic geometry, and the fact that they are made up of many pieces connected together in extended and sometimes elaborate series of steps gives them something of a monumental character.  It also adds to the mystery and mystique, and can sometimes make it hard to see how the individual steps are often quite natural and straightforward.  Hopefully my simple examples can help you penetrate some of the mystery.)