Simplest 'proof of concept' that scheme theory can solve significant concrete questions in number theory

Question

In this MO answer, Felipe Voloch makes the claim

Actually, this is my only serious complaint about Hartshorne. He doesn't do any Number Theory and $\operatorname{Spec}\mathbb Z$ is where schemes really shine.

Matt Emerton writes an amazing answer discussing the use of scheme theory in Mazur's theorem on torsion points of elliptic curves, but that result is of course highly highly nontrivial. EDIT: Actually, maybe a small piece of Mazur's proof can work for my question: Emerton sketches the proof that the elliptic curve with equation $$y^2 +y = x^3 - x^2 - 10 x - 20$$ (the explicit equation whose projectivization is $X_0(11)$) has only finitely many rational solutions. Reading through Emerton's sketch, it actually does seem somewhat reasonable to present after developing a decent (but not overwhelming) amount of scheme theory and algebraic number theory.

More generally, many answers "selling"/"hocking" scheme theory extol its theoretical niceties, so my line of questioning can be summarized as:

What is the simplest non-trivial explicit application of this scheme theoretic thinking for rings? Like given this commutative unital ring $R$, cooking up all this data $R_\frak p$ (localizations) and forming this sheaf and this topological space, gluing a bunch of things together... what did I win? What "simple sounding" concrete problem in previously-known language can I now solve with this new language?

This MO answer gives a nice list, but I still think those are both too non-trivial, and/or not "simple sounding" enough. It would be really nice to have some toy Diophantine problem, that is not too hard, but gives a chance for the scheme theory of $\operatorname{Spec} \mathbb Z$ to shine, as Voloch envisions.

Or if such a problem does not exist, what's the simplest possible clue that schemes can tell us deeper information about arithmetic? This beautiful answer by Alex Youcis seems to say that the "win" provided by scheme theory is the bridge between arithmetic (for a given equation) and the intrinsic geometry of that equation (related to the actual geometry of that equation over say $\mathbb C$). If this is truly where the "wins" are, what's the simplest slice of the bridge that can be constructed, to say something non-trivial about some arithmetic problem?

Or if there really is no such simple material, how would you augment the standard intro AG curriculum (like Hartshorne) to highlight more of the true "shine" of schemes? What would be a good "capstone" theorem, to advertise at the beginning of such a class and then to reward students with at the end of a semester of theory-building?

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As an example for something else, I recently came across this MSE question that I think would serve as a nice concrete "reward" for understanding lots of topics in more classical algebraic geometry, like projective varieties, blow-up, etc. But it is unclear if there is enough "scheme flavor" to really show schemes "shining".

Just for funzies (please don't take this more seriously than that!), I asked ChatGPT and it had a suggestion about the unit equation in certain rings of integers.

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EDIT: Alex Youcis pointed out in a comment that for the question "what sort of questions in differential geometry are solved by the definition of a manifold?" the answer is "none, but manifolds are the language in which deep results in differential geometry are written in."

I claim my question is instead more analogous to: "what's the easiest simple-sounding/simply-phrased questions can be solved by taking the perspective of manifolds & answering abstract manifold questions". And I have an answer: in this beautiful video, 3blue1brown shows how the inscribed rectangle problem can be phrased naturally as a topological question about a certain surface. It is immediately clear that having a language to talk about this surface rigorously, plus one small intuitive lemma (1 slide at minute 22:30) directly solve the problem.

My goal is to do something similar for scheme theory as 3blue1brown has done for topology; of course, it will be harder and might take 4 hours instead of 30 minutes, but the pedagogical spirit is the same.

Answer 1

Serre's How to use finite fields for problems concerning infinite fields has some nice examples. Here are the two he starts the paper with:

Theorem 1.1: Let $\sigma : \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial automorphism satisfying $\sigma^2 = 1$. Then $\sigma$ has a fixed point.

Theorem 1.2: More generally, let $P$ be a finite $p$-group for some prime $p$ and let $\pi : P \to \text{Aut}(\mathbb{C}^n)$ be an action of $P$ on $\mathbb{C}^n$. Then $P$ has a fixed point.

The proof is by reduction to the case of a finite field! I will just reproduce it more or less exactly:

Proof. We prove Theorem 1.2. First observe that the corresponding statement is true over a finite field $\mathbb{F}_q$ where $p \nmid q$, since $\mathbb{F}_q^n$ has $q^n$ elements, which is not divisible by $p$. So in this case the result follows from the $p$-group fixed point theorem (a finite $p$-group acting on a finite set of size not divisible by $p$ has a fixed point).

Next, let $\pi$ be an action of $P$ on $\mathbb{C}^n$ by polynomial automorphisms. Suppose by contradiction that the action of $P$ does not have a fixed point. Write $\pi_i(g)$ for the $i^{th}$ coefficient of $\pi(g) : \mathbb{C}^n \to \mathbb{C}^n$. Then the system of polynomial equations

$$x_i - \pi_i(g), 1 \le i \le n, g \in P$$

does not have a solution over $\mathbb{C}$. By the Nullstellensatz, it follows that these polynomials generate the unit ideal in $\mathbb{C}[x_1, \dots x_n]$, hence that there exist polynomials $Q_i(g)$ such that

$$\sum_{i, g} (x_i - \pi_i(g)) Q_i(g) = 1.$$

Now we employ "spreading out": there are only finitely many polynomials involved here with finitely many coefficients, so the coefficients of the polynomials $\pi_i(g)$ and $Q_i(g)$ generate a finitely generated subring $R \subset \mathbb{C}$ of $\mathbb{C}$, to which we can also add $\frac{1}{p}$. By Zariski's lemma, every maximal ideal $m$ of $R$ has the property that $R/m$ is a finite field $\mathbb{F}_q$, and since $\frac{1}{p} \in R$, $p \nmid q$. So reducing the polynomials $\pi_i(g), Q_i(g) \bmod m$ we conclude that the induced action of $P$ on $\mathbb{F}_q^n$ also has no fixed points; contradiction. $\Box$

(The use of contradiction should be removable but I'm not sure what the cleanest way to do it is; we could use ultraproducts but that would lose the scheme theory flavor.)

The point (as I understand it) is not so much that this is a useful result but to demonstrate spreading out as a technique. Of course this doesn't use the full strength of scheme theory - only affine schemes are needed and there are no nilpotents - but in order to reduce from $\mathbb{C}$ to a finite field we must work over a base ring $R$ that is not a field, so that it admits morphisms to both $\mathbb{C}$ and a finite field; so this argument cannot be performed solely in the language of varieties, even though the conclusion of the theorem is stated over $\mathbb{C}$ (which can be replaced by any algebraically closed field of characteristic $\neq p$).

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I'm not going to be able to find this, but Allen Knutson made the point once (edit: here on Burt Totaro's blog, and also here in more detail on MO) that schemes generalize the classical theory of affine varieties in three different directions:

• Not-necessarily-affine,
• Working over a base that is not a field, e.g. $\mathbb{Z}$,
• Allowing nilpotents.

Each of these can be motivated separately and any given application may not require all of them. In this case we only needed the second one. Personally I have managed to completely avoid the theory of non-affine schemes but I find affine schemes very conceptually useful for organizing things, e.g. thinking about the functor of points of the affine group scheme $GL_n$, which is relevant to questions like "what is the determinant, really?"