For example, I don't give a flying f*ck about Navier-Stokes, but you don't see me asking you to prove the importance of Navier-Stokes or other PDE crap in all threads about it.
Maybe you should.
Who deserves more credit for Andre-Oort in recent years?
For example, I don't give a flying f*ck about Navier-Stokes, but you don't see me asking you to prove the importance of Navier-Stokes or other PDE crap in all threads about it.
Maybe you should.
The sad truth is that any random Schrodinger equation solved by separation of variables is still more important than Andree-Oort. Anybody who says otherwise is massively coping.
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 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 . For instance, solving polynomial equations using complex -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 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 among the very best answers I've seen on MJR, and don't take that as a backhanded compliment!
However, sorry to be a bit of a prick, but are the "model theory" techniques fundamental, or just apparent, like a removable singularity, so to speak? I once went to a series of lectures on Manin-Mumford type results in the style of Pila and Tsimmerman, and I came away with the impression that everything could be phrased in standard groidy terms (all the o-minimal structures ultimately could be re-formulated as certain sheaves on real analytic manifolds).
I'm not saying that people who already like model theory shouldn't use it, and it's cool these people are now solving problems that other people care about, but just as groids are guilty of not explaining to others why they should care, model theorists might be guilty of insisting their ideas be formulated in a way that others find bizarre.
However, sorry to be a bit of a prick, but are the "model theory" techniques fundamental, or just apparent, like a removable singularity, so to speak? I once went to a series of lectures on Manin-Mumford type results in the style of Pila and Tsimmerman, and I came away with the impression that everything could be phrased in standard groidy terms (all the o-minimal structures ultimately could be re-formulated as certain sheaves on real analytic manifolds).
I'm not saying that people who already like model theory shouldn't use it, and it's cool these people are now solving problems that other people care about, but just as groids are guilty of not explaining to others why they should care, model theorists might be guilty of insisting their ideas be formulated in a way that others find bizarre.
Hi, lzfc here.
That is a fair question since there have been cases of other problems where a proof first comes using logic, and then a groidy proof comes along and everyone finds it more natural.
In this case the use of o-minimal theory looks essential to me. Let me explain.
First we need to agree on what is o-minimality. In different notions of geometry in mathematics you have a collection of allowed functions and then use them to define geometric sets; I am not claiming that all geometries are like this, but many indeed are. For instance classical algebraic geometry over is about what can be defined over using polynomials. However, if you use polynomials and work over then you get different sets: already in the affine line the sets over defined by polynomials are finite or cofinite, while over they are finite unions of intervals (e.g. is defined as the image of -coordinate projection of the unit circle ) --this is a theorem of Tarski. Studying the sets of defined by polynomials is very interesting and useful in applications, and it leads to ``real algebraic geometry''. Real algebraic geometry has general theorems that are not something you can deduce from complex algebraic geometry; for instance every real-algebraic set has finitely many connected components.
Unfortunately, real algebraic geometry is too limited for some applications. What van den Dries did in the early 80's is to realize that what happens in the affine line is strong enough to get many of the good features of real algebraic geometry, even if you are not just using polynomials.
An o-minimal structure is simply the following: the field of real numbers endowed with additional data (let's say, functions) such that the definable sets of this structure in the affine line are again finite unions of intervals. For instance, the real numbers endowed with the cosine function is not o-minimal: look at the equation in .
The field is o-minimal. Can we add any interesting function to it so that it remains o-minimal? Yes, but this is hard. For instance, a celebrated theorem of Wilkie proves that adding the exponential function gives again an o-minimal structure. Let me not make a survey of results here but is should be clear from these examples ( and ) that the question is subtle and one cannot simply add random nice functions.
Why do we care? because van den Dries and others developed a very strong theory of o-minimal structures; that is, under the very weak assumption that in dimension you are just defining finitely many intervals, you can prove loads of structural results. Baby example: in any o-minimal structure, every definable set in (notjust dimension 1) has at most finitely many connected components. The strongest results one needs in Diophantine applications are (1) cell-decompositions and (2) reparametrizations (both are useful in families). And I insist: the magic here is that the key (and only) working hypothesis is simply that in dimension 1 you are just defining finite unions of intervals!
Back to arithmetic. In appli
(continued) ... In applications such as Manin--Mumford or Andre--Oort one encounters manifolds (with border) and one needs to count its rational points ordered bi height (complexity). The manifolds are not random things; they are often defined by a couple of nice analytic functions on compact sets, and so, the underlying structure is o-minimal. So you have loads of tools to understand the structure of . In particular one can decompose into finitely many pieces such that on each piece it looks like the graph of a function with controlled derivatives. On such a graph one can show that rational points repel each other (after deleting algebraic subsets) and therefore they are rather sparse.
This idea goes back to the work of Bombieri and Pila for curves where the decomposition can be done in an elementary way. The higher dimensional case is due to Pila and Wilkie, and it needs heavy machinery from o-minimal theory.
I hope that this vague description makes the theory look a bit less bizarre. In any case, the method of proof for the point-counting theorems seems to require these cell decompositions under rather weak hypotheses, and o-minimality achieves precisely that.
I really don't get why certain people here classify André-Oort as a cluster G topic: I know the technical terminology could be misleading for an outsider, but Diophantine geometry as a field is actually very problem solving oriented and has a more hands on approach than "arithmetic geometry Scholze-style". The tools come from all over math, with algebraic geometry being just one of them, together with more "analytically" flavored arguments from transcendence theory/dioph. approx., just to name one.
I really don't get why certain people here classify André-Oort as a cluster G topic: I know the technical terminology could be misleading for an outsider, but Diophantine geometry as a field is actually very problem solving oriented and has a more hands on approach than "arithmetic geometry Scholze-style". The tools come from all over math, with algebraic geometry being just one of them, together with more "analytically" flavored arguments from transcendence theory/dioph. approx., just to name one.
For me the reason was sociological rather than mathematical. I had never heard of André-Oort until it was solved, and even then I first heard about it when JSE tweeted about it. The ensuing discussion had a lot of words I didn't understand with a cluster G flavour, so I assumed it must be a result I couldn't understand with my limited exposure.
I don't know if my experience generalises though.
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 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 . For instance, solving polynomial equations using complex -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 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.
I respect this answer and the effort involved in writing it. There should be more of this on the site.
However, I still don't really care about any of this. I don't know why it's so interesting to know that a polynomial has a lot of solutions in j-invariants or related things. Good though your answer is, I think it might to an extent be relying on the closure of the set of interesting topics under intersections, which I'm not sure is valid (diophantine equations are interesting, CM-points are interesting, therefore diophantine equations in CM points must be interesting). Is there some vaguely concrete example which might convince me more?
I respect this answer and the effort involved in writing it. There should be more of this on the site.
However, I still don't really care about any of this. I don't know why it's so interesting to know that a polynomial has a lot of solutions in j-invariants or related things. Good though your answer is, I think it might to an extent be relying on the closure of the set of interesting topics under intersections, which I'm not sure is valid (diophantine equations are interesting, CM-points are interesting, therefore diophantine equations in CM points must be interesting). Is there some vaguely concrete example which might convince me more?
Hi, it's me again. Let me try to explain the content of looking a Diophantine equations on j-invariants.
If you have a CM elliptic curve you might wonder how to produce another one in a non-trivial way (i.e. don't choose a fixed with CM and return the same for all ). A simple answer is given by isogenies: any elliptic curve isogenous to has CM ("isogenous'' is just a groid way to say "you go from one to the other with a change of variables''). At the level of -invariants this isogeny relation is given by the so-called Hecke operators, which are certain multivalued algebraic functions defined in terms of congruence subgroups acting on the upper half plane (i.e. some modular yabadaba.)
But isogenies are unsurprising in this context; after all they are just a change of variables, so of course they respect the property of being CM.
Is there other "sensible'' transformation on elliptic curves (say, over ) with the property that it respects "CM-ness'', at least for an infinite set of CM elliptic curves? By "sensible'' I mean something functorial, and even better: representable. This means that I expect the construction to show up as an algebraic function on the corresponding j invariants.
The answer is negative. Any other algebraic function at the level of j-invariants will be completely incompatible with the property of being CM, except perhaps for finitely many elliptic curves. This is the content of Andre--Oort in the modular surface .
So, for instance, there are only finitely many CM j-invariants such that is also CM because the function is not a Hecke operator.
Of course this is just another point of view on the same discussion for Diophantine equations but I hope this second point of view seems less artificial.
There are, however, completely unrelated applications. For instance and at a more technical level, it is known that the more general version of the Andre--Oort conjecture has applications in transcendence theory, see for instance chapter 6 in "A Panorama of Number Theory or The View from Baker's Garden''.
I should add that it is not just the applications that make the problem interesting, it is also the techniques developed to attack it (e.g. generalizations of Ax's transcendence theorem) as well as the other problems that were solved in the way of finding the final answer (e.g. the average Colmez conjecture).
If this does not go in the direction you expected, please let me know what kind of examples or applications you would consider as more natural or interesting.
I respect this answer and the effort involved in writing it. There should be more of this on the site.
However, I still don't really care about any of this. I don't know why it's so interesting to know that a polynomial has a lot of solutions in j-invariants or related things. Good though your answer is, I think it might to an extent be relying on the closure of the set of interesting topics under intersections, which I'm not sure is valid (diophantine equations are interesting, CM-points are interesting, therefore diophantine equations in CM points must be interesting). Is there some vaguely concrete example which might convince me more?
Hi, it's me again. Let me try to explain the content of looking a Diophantine equations on j-invariants.
If you have a CM elliptic curve you might wonder how to produce another one in a non-trivial way (i.e. don't choose a fixed with CM and return the same for all ). A simple answer is given by isogenies: any elliptic curve isogenous to has CM ("isogenous'' is just a groid way to say "you go from one to the other with a change of variables''). At the level of -invariants this isogeny relation is given by the so-called Hecke operators, which are certain multivalued algebraic functions defined in terms of congruence subgroups acting on the upper half plane (i.e. some modular yabadaba.)
But isogenies are unsurprising in this context; after all they are just a change of variables, so of course they respect the property of being CM.
Is there other "sensible'' transformation on elliptic curves (say, over ) with the property that it respects "CM-ness'', at least for an infinite set of CM elliptic curves? By "sensible'' I mean something functorial, and even better: representable. This means that I expect the construction to show up as an algebraic function on the corresponding j invariants.
The answer is negative. Any other algebraic function at the level of j-invariants will be completely incompatible with the property of being CM, except perhaps for finitely many elliptic curves. This is the content of Andre--Oort in the modular surface .
So, for instance, there are only finitely many CM j-invariants such that is also CM because the function is not a Hecke operator.
Of course this is just another point of view on the same discussion for Diophantine equations but I hope this second point of view seems less artificial.
There are, however, completely unrelated applications. For instance and at a more technical level, it is known that the more general version of the Andre--Oort conjecture has applications in transcendence theory, see for instance chapter 6 in "A Panorama of Number Theory or The View from Baker's Garden''.
I should add that it is not just the applications that make the problem interesting, it is also the techniques developed to attack it (e.g. generalizations of Ax's transcendence theorem) as well as the other problems that were solved in the way of finding the final answer (e.g. the average Colmez conjecture).
If this does not go in the direction you expected, please let me know what kind of examples or applications you would consider as more natural or interesting.
I appreciate this answer, thanks.
Just to add on the model theory being necessary to some extent. Most of the applications use structures (eg real field with restricted analytic functions and the unrestricted exponential) that are quite difficult to show o-minimality for. So, in addition to the Pila-Wilkie theorem being formulated in o-minimal terms, if you have the ambition for whatever reason to “remove the model theory”, you’ll need to reprove these difficult results in real analytic geometry.
This is all just to say, the model theory is all tied up with the analytic geometry in various ways going far beneath the surface.
Just to add on the model theory being necessary to some extent. Most of the applications use structures (eg real field with restricted analytic functions and the unrestricted exponential) that are quite difficult to show o-minimality for. So, in addition to the Pila-Wilkie theorem being formulated in o-minimal terms, if you have the ambition for whatever reason to “remove the model theory”, you’ll need to reprove these difficult results in real analytic geometry.
This is all just to say, the model theory is all tied up with the analytic geometry in various ways going far beneath the surface.
The standard foundational reference on o-minimal structures, at least for applications to number theory and algebraic geometry, seems to be van den Vries's "Tame topology and o-minimal structures". Flipping through that book, I see that he introduces the most basic notions of a "language" and of a "structure", but it's not obvious to me that any theorems of model theory that I once learned and have now forgotten (completeness, Lowenheim-Skolem,...) get used in the book.
What's the "theoretical minimum" of model theory needed for these types of applications? What does seem important is Tarski-Seidenberg, which is a rather concrete statement with a constructive proof. It has consequences in model theory, but on another planet it could very well have been proved without any model-theoretic notions, no?
My impression is that the groidy way to say these things is that an o-minimal structure defines a certain very simple kind of Grothendieck topology, and that geometry in this setting is the study of sheaves in that Grothendieck topology. As far as I understand, this is the more general approach of Kashiwara-Schapira to tame topology, and in the o-minimal setting is the point of view of Baker-Brunebarbe-Tsimerman:
Just to add on the model theory being necessary to some extent. Most of the applications use structures (eg real field with restricted analytic functions and the unrestricted exponential) that are quite difficult to show o-minimality for. So, in addition to the Pila-Wilkie theorem being formulated in o-minimal terms, if you have the ambition for whatever reason to “remove the model theory”, you’ll need to reprove these difficult results in real analytic geometry.
This is all just to say, the model theory is all tied up with the analytic geometry in various ways going far beneath the surface.
The standard foundational reference on o-minimal structures, at least for applications to number theory and algebraic geometry, seems to be van den Vries's "Tame topology and o-minimal structures". Flipping through that book, I see that he introduces the most basic notions of a "language" and of a "structure", but it's not obvious to me that any theorems of model theory that I once learned and have now forgotten (completeness, Lowenheim-Skolem,...) get used in the book.
What's the "theoretical minimum" of model theory needed for these types of applications? What does seem important is Tarski-Seidenberg, which is a rather concrete statement with a constructive proof. It has consequences in model theory, but on another planet it could very well have been proved without any model-theoretic notions, no?
My impression is that the groidy way to say these things is that an o-minimal structure defines a certain very simple kind of Grothendieck topology, and that geometry in this setting is the study of sheaves in that Grothendieck topology. As far as I understand, this is the more general approach of Kashiwara-Schapira to tame topology, and in the o-minimal setting is the point of view of Baker-Brunebarbe-Tsimerman:
One doesn’t need a huge amount of model theory to study o-minimality (almost none), but the model completeness/QE results powering these applications are serious results. As you say, I suppose they could be proved by non model theorists on some other planet or an alternate timeline.
But few things in few subjects are completely unavoidable. Deeper results in sometimes make the model theory very hard to remove from applications (eg uses of Zilber trichotomy). Of course this comes up in applications of model theory, but not in what has been discussed so far in this thread.
Just to add on the model theory being necessary to some extent. Most of the applications use structures (eg real field with restricted analytic functions and the unrestricted exponential) that are quite difficult to show o-minimality for. So, in addition to the Pila-Wilkie theorem being formulated in o-minimal terms, if you have the ambition for whatever reason to “remove the model theory”, you’ll need to reprove these difficult results in real analytic geometry.
This is all just to say, the model theory is all tied up with the analytic geometry in various ways going far beneath the surface.
The standard foundational reference on o-minimal structures, at least for applications to number theory and algebraic geometry, seems to be van den Vries's "Tame topology and o-minimal structures". Flipping through that book, I see that he introduces the most basic notions of a "language" and of a "structure", but it's not obvious to me that any theorems of model theory that I once learned and have now forgotten (completeness, Lowenheim-Skolem,...) get used in the book.
What's the "theoretical minimum" of model theory needed for these types of applications? What does seem important is Tarski-Seidenberg, which is a rather concrete statement with a constructive proof. It has consequences in model theory, but on another planet it could very well have been proved without any model-theoretic notions, no?
My impression is that the groidy way to say these things is that an o-minimal structure defines a certain very simple kind of Grothendieck topology, and that geometry in this setting is the study of sheaves in that Grothendieck topology. As far as I understand, this is the more general approach of Kashiwara-Schapira to tame topology, and in the o-minimal setting is the point of view of Baker-Brunebarbe-Tsimerman:
There are various, sometimes hidden, uses of compactness throughout various foundational theorems of o-minimality (though maybe you'll just say such uses are not fundamentally model theoretic?). Even model theorists have taken up some of the sheaf theoretic point of view you mention (e.g. Edmundo's papers come to mind). Anyhow, I agree with the earlier comment that there are very few times in which some point of view is absolutely essential in a given problem, so I believe one could remove a bunch of the model theory from this stuff, but I don't see the point, since their point of view has sort of driven the subject over the past 40 years.
[...]
The standard foundational reference on o-minimal structures, at least for applications to number theory and algebraic geometry, seems to be van den Vries's "Tame topology and o-minimal structures". Flipping through that book, I see that he introduces the most basic notions of a "language" and of a "structure", but it's not obvious to me that any theorems of model theory that I once learned and have now forgotten (completeness, Lowenheim-Skolem,...) get used in the book.
What's the "theoretical minimum" of model theory needed for these types of applications? What does seem important is Tarski-Seidenberg, which is a rather concrete statement with a constructive proof. It has consequences in model theory, but on another planet it could very well have been proved without any model-theoretic notions, no?
My impression is that the groidy way to say these things is that an o-minimal structure defines a certain very simple kind of Grothendieck topology, and that geometry in this setting is the study of sheaves in that Grothendieck topology. As far as I understand, this is the more general approach of Kashiwara-Schapira to tame topology, and in the o-minimal setting is the point of view of Baker-Brunebarbe-Tsimerman:
There are various, sometimes hidden, uses of compactness throughout various foundational theorems of o-minimality (though maybe you'll just say such uses are not fundamentally model theoretic?). Even model theorists have taken up some of the sheaf theoretic point of view you mention (e.g. Edmundo's papers come to mind). Anyhow, I agree with the earlier comment that there are very few times in which some point of view is absolutely essential in a given problem, so I believe one could remove a bunch of the model theory from this stuff, but I don't see the point, since their point of view has sort of driven the subject over the past 40 years.
Also, just to add - what is necessary are various general version of Tarski-Seidenberg. And most of these results are hard theorems of model theorists (e.g. Wilkie on reals with exponentiation)
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