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

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

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 ζ(s)=1ns 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.

Attribution
Source : Link , Question Author : Arrow , Answer Author : Georges Elencwajg

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