Pila or Tsimerman?

# Who deserves more credit for Andre-Oort in recent years?

More to the point, and an unanswered question from half a dozen threads - literally WTF should anyone care, except that PS said it's important?

Do you mean you don't care about the fully general AO statement?

Or you even don't care about AO for moduli of abelian varieties? (If this option, are you just from some different area or are you trolling?)

The fame of Andre-Oort is hilarious. It's famous only because PS and JT have a big base of fanboys. Ask them why it's important and you'll get responses like nocv above "but don't you even care about AO for moduli of abelian varieties" or some vague comment like "Andre-Oort is about the fundamental structure of Shimura varieties, which themselves are useful as moduli spaces of various objects (elliptic curves, K3 surfaces, abelian varieties, etc.)" which, upon the casting of any doubt, is met with "don't be a prick lol". Literally quoting from past threads here

You can pose this question about any result in any field. Without specifying what you already do care about, simply asking “why should I care?” is impossible to answer, and is probably not being asked in good faith.

Do you already care about moduli of abelian varieties? Do you care about specific abelian varieties? At least elliptic curves? Do you care about rational points? Do you care about solving diophantine equations? All of these are valid things to either care or not care about, but unless you tell us where to start, how are we supposed to convince you something is interesting? If you don’t care about any of these, you probably won’t ever think Andre-Oort is interesting, and that’s okay. Nobody is obligated to like any specific thing, nor are the experts on any specific thing obligated to convince complete outsiders the value of their work if they don’t even know about the basic foundational objects of their field.

You can pose this question about any result in any field. Without specifying what you already do care about, simply asking “why should I care?” is impossible to answer, and is probably not being asked in good faith.

Do you already care about moduli of abelian varieties? Do you care about specific abelian varieties? At least elliptic curves? Do you care about rational points? Do you care about solving diophantine equations? All of these are valid things to either care or not care about, but unless you tell us where to start, how are we supposed to convince you something is interesting? If you don’t care about any of these, you probably won’t ever think Andre-Oort is interesting, and that’s okay. Nobody is obligated to like any specific thing, nor are the experts on any specific thing obligated to convince complete outsiders the value of their work if they don’t even know about the basic foundational objects of their field.

There is merit to your argument. Having said that, mathematics has gotten more silo-ed over time, and there is a general perception (whether misguided or not) among those who work in the concrete fields that those who work in the cluster G abstract fields no longer spend any time evangelizing about the importance of their work to the rest of us.

It doesn't help that, at the graduate level, a lot of cluster G folks tend to be exactly the folks who used to be annoying undergrads who love namedropping abstractions. You know the type I mean -- the one who is taking their first graduate course in algebra and loves to talk about schemes and categories.

If we react to people questioning the importance of any given work with confrontation, it's unlikely to help. To answer your question, I do care about solving Diophantine equations, and I care about elliptic curves insofar as I agree there's something interesting there and someone should be studying (though it's not for me). If you can explain why I should care about Andre-Oort I would be happy to hear it.

[If this makes a difference -- I work in the analytic part of number theory; I try not to think about cluster G stuff unless I need a bound for a character sum that comes from algebraic geometry. Even then, I blackbox the algebraic geometry as much as possible.]

If we react to people questioning the importance of any given work with confrontation, it's unlikely to help. To answer your question, I do care about solving Diophantine equations, and I care about elliptic curves insofar as I agree there's something interesting there and someone should be studying (though it's not for me). If you can explain why I should care about Andre-Oort I would be happy to hear it.

[If this makes a difference -- I work in the analytic part of number theory; I try not to think about cluster G stuff unless I need a bound for a character sum that comes from algebraic geometry. Even then, I blackbox the algebraic geometry as much as possible.]

I'll give it a try.

You probably care about rational solutions of polynomial equations, and the reason is that they are hard to come by; they look like accidents because the field $\mathbb{Q}$ is not algebraically closed (otherwise it would be too easy to find solutions). Well, not quite: some varieties ``obviously'' have many rational points, such as abelian varieties of positive rank. The Bombieri--Lang conjecture says that in a sense that's it.

But it turns out that you can ask similar questions about other sets of algebraic numbers that are far from being all of the algebraic closure of $\mathbb{Q}$. For instance, solving polynomial equations using complex $n$-th roots of 1 (this case is connected to another conjecture called the Mordell--Lang conjecture --material for another day).

Another source of interesting algebraic numbers are the j-invariants of CM elliptic curves. This is a set of algebraic numbers of unbounded degree which is highly relevant in class field theory. Then the question is: can we describe the solutions of a Diophantine equation if the solutions are required to be numbers of this sort?

In the simplest case, imagine you have an equation f(x,y)=0 and you want to solve it using CM j-invariants. Geometrically, this leads to the question ``what kind of curves can have infinitely many points with both coordinates a CM j-invariant?" The answer is surprising: first, we know exactly what are these curves, and secondly, they essentially come from graphs of Hecke operators.

But f(x,y)=0 is far from general. What Pila did is the general case of systems of diophantine equations $f({x}_{1},...,{x}_{n})=0$ and he described when such a system has ``a lot'' of solutions in CM j-invariants. This is very general and it was achieved with a mix of different tools such as algebraic geometry, transcendence theory, arithmetic (of course) and even some model theory.

Andre--Oort for moduli of abelian varieties is the natural generalization of the above situation. The j-invariant of an elliptic curve is nothing but its corresponding point in the moduli space of elliptic curves. Then one is led to look at the moduli of CM abelian varieties and to study algebraic relations between them --these relations are the more general Diophantine equations.

I hope this clarifies a bit the situation and it makes the problem look more natural.

If we react to people questioning the importance of any given work with confrontation, it's unlikely to help. To answer your question, I do care about solving Diophantine equations, and I care about elliptic curves insofar as I agree there's something interesting there and someone should be studying (though it's not for me). If you can explain why I should care about Andre-Oort I would be happy to hear it.

[If this makes a difference -- I work in the analytic part of number theory; I try not to think about cluster G stuff unless I need a bound for a character sum that comes from algebraic geometry. Even then, I blackbox the algebraic geometry as much as possible.]

I'll give it a try.

You probably care about rational solutions of polynomial equations, and the reason is that they are hard to come by; they look like accidents because the field $\mathbb{Q}$ is not algebraically closed (otherwise it would be too easy to find solutions). Well, not quite: some varieties ``obviously'' have many rational points, such as abelian varieties of positive rank. The Bombieri--Lang conjecture says that in a sense that's it.

But it turns out that you can ask similar questions about other sets of algebraic numbers that are far from being all of the algebraic closure of $\mathbb{Q}$. For instance, solving polynomial equations using complex $n$-th roots of 1 (this case is connected to another conjecture called the Mordell--Lang conjecture --material for another day).

Another source of interesting algebraic numbers are the j-invariants of CM elliptic curves. This is a set of algebraic numbers of unbounded degree which is highly relevant in class field theory. Then the question is: can we describe the solutions of a Diophantine equation if the solutions are required to be numbers of this sort?

In the simplest case, imagine you have an equation f(x,y)=0 and you want to solve it using CM j-invariants. Geometrically, this leads to the question ``what kind of curves can have infinitely many points with both coordinates a CM j-invariant?" The answer is surprising: first, we know exactly what are these curves, and secondly, they essentially come from graphs of Hecke operators.

But f(x,y)=0 is far from general. What Pila did is the general case of systems of diophantine equations $f({x}_{1},...,{x}_{n})=0$ and he described when such a system has ``a lot'' of solutions in CM j-invariants. This is very general and it was achieved with a mix of different tools such as algebraic geometry, transcendence theory, arithmetic (of course) and even some model theory.

Andre--Oort for moduli of abelian varieties is the natural generalization of the above situation. The j-invariant of an elliptic curve is nothing but its corresponding point in the moduli space of elliptic curves. Then one is led to look at the moduli of CM abelian varieties and to study algebraic relations between them --these relations are the more general Diophantine equations.

I hope this clarifies a bit the situation and it makes the problem look more natural.

This is a good answer!

Just to add:

the strategy of the proof is very cool and has been used in numerous diophantine finiteness results (e.g. new proofs of Manin-Mumford, etc.).

AO has numerous consequences that might naturally come up in concrete situations - e.g. how many curves of genus g such that the Jacobian has CM?

The transcendental portion of the proof is itself of independent interest and has now been attacked via various perspectives.

I forgot to answer the initial question: certainly Pila. He came up with the whole strategy and made it work in a very general case. After that it was clear that there was a path, and it was only a matter of time and technical power to reach the final solution.

But there were many twists in the story (e.g. the improved Pila-Wilkie type bounds being used in the Galois theory portion of the proof, for instance) - there were loads of interesting developments from the whole program.

I'll give it a try.

You probably care about rational solutions of polynomial equations, and the reason is that they are hard to come by; they look like accidents because the field $\mathbb{Q}$ is not algebraically closed (otherwise it would be too easy to find solutions). Well, not quite: some varieties ``obviously'' have many rational points, such as abelian varieties of positive rank. The Bombieri--Lang conjecture says that in a sense that's it.

But it turns out that you can ask similar questions about other sets of algebraic numbers that are far from being all of the algebraic closure of $\mathbb{Q}$. For instance, solving polynomial equations using complex $n$-th roots of 1 (this case is connected to another conjecture called the Mordell--Lang conjecture --material for another day).

Another source of interesting algebraic numbers are the j-invariants of CM elliptic curves. This is a set of algebraic numbers of unbounded degree which is highly relevant in class field theory. Then the question is: can we describe the solutions of a Diophantine equation if the solutions are required to be numbers of this sort?

In the simplest case, imagine you have an equation f(x,y)=0 and you want to solve it using CM j-invariants. Geometrically, this leads to the question ``what kind of curves can have infinitely many points with both coordinates a CM j-invariant?" The answer is surprising: first, we know exactly what are these curves, and secondly, they essentially come from graphs of Hecke operators.

But f(x,y)=0 is far from general. What Pila did is the general case of systems of diophantine equations $f({x}_{1},...,{x}_{n})=0$ and he described when such a system has ``a lot'' of solutions in CM j-invariants. This is very general and it was achieved with a mix of different tools such as algebraic geometry, transcendence theory, arithmetic (of course) and even some model theory.

Andre--Oort for moduli of abelian varieties is the natural generalization of the above situation. The j-invariant of an elliptic curve is nothing but its corresponding point in the moduli space of elliptic curves. Then one is led to look at the moduli of CM abelian varieties and to study algebraic relations between them --these relations are the more general Diophantine equations.

I hope this clarifies a bit the situation and it makes the problem look more natural.

Thanks for the answer! That certainly is quite interesting, at least to me (I'm NHPR).

It makes me wonder now what QLOH was going on about. They phrased their comment as if they were an arithmetic geometry insider who thought this was just some PS pet conjecture...

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