How and why does Grothendieck's work provide tools to attack problems in number theory?

Question

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:

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

Answer 1

Let's first get this out of the way: your question is impossible to answer precisely in less than ten thousand pages at least.
Semi-technical answers are to be found in thousands of articles on the Web, in Wikipedia and other sources.
In a very soft nutshell:

At the end of the 19th century arithmeticians like Kronecker, Dedekind, Weber realized that the algebra underlying number fields and algebraic curves present strong similarities.
Today we would attribute this to the role of Dedekind domains in both of them.
An even more important common theme shared by arithmetic and geometry is the existence of a zeta function:
The Riemann zeta function $\zeta(s)=\sum \frac {1}{n^s}$ was generalized by Dedekind to number fields, then by Emil Artin to curves and by Weil to varieties of arbitrary dimensions.
The Riemann hypothesis for those zeta functions was proved by Hasse for elliptic curves and by Weil for curves of arbitrary genus.

In 1949 Weil wrote a ground-breaking article introducing his celebrated conjectures on zeta functions for algebraic varieties of arbitrary dimension, together with a road map for solving them, provided one could generalize topological methods like those of Lefschetz to an algebro-geometric context.
Grothendieck's greatest contribution was to invent just that generalization : étale cohomology, which was based on his grandiose anterior re-creation of the tools of algebraic geometry, his scheme theory which needed thousands of pages for its development.
Armed with the deadly weapon of étale cohomology Grothendieck and Deligne proved all the Weil conjectures.
[However: one of these conjectures-rationality of the zeta function-had already been proved by Dwork with more classical tools.]
Later, Faltings, Wiles and others could victoriously defeat some of the hardest problems in arithmetic using Grothendieck's (and his school's) techniques.

Answer 2

Let me try and add to Georges already great answer.

Let me, perhaps at risk of my own peril, try and summarize the very basic idea in one sentence:

Solution sets $S$ of equations ought to have intrinsic geometry which informs us of the nature of $S$.

Now, while this is nowadays a commonplace ideology, let me point out one tiny bit of subtlety in my above statement. Namely, solution sets of what type of polynomials, and solutions where? Classically one would interpret this sentence as being shorthand for something like:

Solution sets $S$ of sufficiently nice equations over the real (complex) numbers have intrinsic geometry which informs us about $S$.

Now, this is totally believable. If one interprets 'sufficiently nice' correctly then these solutions sets will be, say, real (complex) manifolds which, of course, have intrinsic structure, and yes, this structure tells us something about $S$.

That said, one of the basic tenets of number theory is that, sometimes, it's more interesting to consider solutions of equations not over the real or complex number but over objects with much richer arithmetic theory. Perhaps this means looking for solutions over a 'non-geometric' field like $\mathbb{Q}$ or $\mathbb{F}_q$, or perhaps, even over a ring like $\mathbb{Z}[i]$.

Secondly, in number theory we are often times interested in types of equations which are not sufficiently nice. This is a geometric term and, ostensibly, number theory is non-geometric, so such requirements seem unnatural.

Thus, reevaluating my first statement one then may begin to balk—such equations over such general rings have no right to have geometric structure. Where is such a geometric structure coming from? Certainly not from the underlying rings—the ring $\mathbb{Z}[i]$ doesn't carry sufficiently rich geometric structure to be able to talk about curve theory. What should it mean to consider the cotangent bundle of a set of solutions to an equation over $\mathbb{Z}[i]$? What should it mean to take its singular cohomology? What is a 'compact Lie group' over $\mathbb{Z}[i]$, and should it have a structure theory?

The realization of algebraic geometers (including Grothendieck) was that the equations themselves had intrinsic geometry. Then, any geometry on the solutions sets over some given ring are just derivative of the geometry of the underlying equations. This also does away with the fear of doing geometry over something ungeometric such as, say, $\mathbb{F}_q$, since it's the equations themselves providing the geometry.

Or, thought about in a more Grothendieck-ian (like Dickensian?) way the geometry comes from the collection of the solution sets of the polynomials over all rings, and how these sets vary with the solution ring. Said differently, if $X(R)$ denotes the solution set of the polynomials in $S$ over a ring $R$, it's not the set $X(R)$ that has a geometry but the functor $X$ itself that does.

Grothendieck and co.'s great innovation was realizing how to put this philosophy on firm footing. One needs to backup such a brash statement that the equation $x^n+y^n=z^n$ has intrinsic geometry, and a geometry sufficiently rich to be able to say something interesting about its set of solutions in some ring $R$. And, as Georges indicates it requires an extremely formidable amount of technical machinery to do this.

Now that one has this all out of the way all of your questions fall sort of neatly in line:

1. By applying geometric intuition/tools, amongst the most powerful and easily intuited amongst all of a mathematicians toolbox, to something which (recycling my above phrase) ostensibly has no right to be amenable to such techniques (e.g. the equation $x^n+y^n=z^n$ over $\mathbb{Q}$).
2. I think it's a little bit of black magic. One might go to bed every night dreaming of the intrinsic geometry of equations, but without deep technical understanding of the subject and a shifting of ideas (the Grothendieck/Yoneda type philosophy) this seems doomed. This is what makes it so beautiful and exciting.
3. All of them. When talking to a fellow number theorist I can say the words fundamental group, cohomology (of a constant sheaf!), cotangent bundle, smooth, Lefschetz fixed point formula,... and I needn't specify if I am talking about an elliptic curve over $\mathbb{F}_p$ or a smooth surface over $\mathbb{C}$. The utility of this geometric language is not to be understimated. Even though it's a very, very simple instance, this recent answer of mine gives a great example of how geometric language makes purely number theoretic questions fall in line with there geometric counterparts.
4. No, I don't think this is the case. But, without diving headfirst into the subject I don't think you can appreciate the fluidity and power that Grothendieck's uniform language brings to much of algebraic geometry, whether over $\mathbb{Z}$ or $\mathbb{C}$.

Answer 3

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:

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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.

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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 $p \equiv 1 \bmod 4$.

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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.

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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.)