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Can somebody add a paragraph on what’s interesting about the conjecture, apart form it being hard to prove?
The reference of Barry Mazur linked in the page (pdf) is dedicated much to convey his view to answer your question, but being not excited about number theory myself, I can hardly digest this into a short statement. Mazur says
The beauty of such a Conjecture is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare; the precision of the Conjecture goads one to investigate this rarity quantitatively. Its very statement makes an attractive appeal to perform a range of numerical experiments that would test the empirical waters. On a theoretical level, it is enlightening to understand its relationship to the constellation of standard arithmetic theorems, conjectures, questions, etc., and we shall give some indications of this below.
Among the relation to standard great theorems of proved importance one should note for example the implication in
Thanks! I put that quote into the entry.
That quote still doesn’t make the question interesting to me. But it is interesting to see that this is what is meant to be interesting :-)
I saw a ’discussion’ on Linkedin and there was a quote from Brian Conrad (Stanford):
The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory
This is very much seen as related to the anabelian problem (or so I am told). ;-)
Even implying the Mordell conjecture does not make it interesting ?
You probably meant to say: “the Mordell conjecture”. ;-)
I find all these texts that I have seen about these conjectures are lacking a paragraph that explains why they are interesting beyond their plain statement.
For instance Mazur’s quote above essentially just re-iterates the abc conjecture while saying that it is “beautiful”. I wonder if one left out the explicit mentioning of “beauty”, if the reader would recognize what the beauty is. I don’t think Mazur actually says it. But I guess one could say it.
From the Polymath link you can reach an interview by John Baez of Minhyong Kim which is worth reading.
…his work probes the very core of mathematical language such as what we might really mean by a number or a geometric figure, and how they might be interpreted in a manner quite different from usual conventions. In fact, it relies on deep relations of a geometric nature between such varying interpretations. Such questions have occupied philosophers for millennia, but are usually quite distant from the consciousness of modern mathematicians. But then, these seemingly philosophical questions have to be recast in the robust language of precise mathematics. You have to add to that some of the most sophisticated portions of 21st century arithmetic geometry.
what we might really mean by a number or a geometric figure, and how they might be interpreted in a manner quite different from usual conventions.
I can’t but think: “Namely? What is that new notion of number and of geometric figure?”
I guess I should ask Minhyong Kim or some other expert. But if anyone here knows, I’d be interested in hearing about it.
And what are those “other universes” that the proof allegedly makes use of? I saw a quote on this somewhere, and that looked to me as if this is just about enlarging set-theoretic universes as usual. But possibly I misinterpreted that sentence which I saw. Somewhere…
Minhyong Kim also says in that interview the following:
Alternative foundations for mathematics is an idea that’s somewhat in the air in our times even for working mathematicians not particularly close to philosophy, partly as a result of new notions of space and time demanded by a certain kind of mathematical physics. Vladimir Voevodsky at the Institute for Advanced Study in Princeton comes to mind as well as Jacob Lurie at Harvard.
I sure wonder what that refers to. There is really only one suggestion which I know of that relates mathematical physics to homotopy type theory and higher topos theory. And this is related to the ABC conjecture?! :-)
I should ask Minhyong.
I have looked at Minhyon Kim’s more technical discussion here on MO. From the very last paragraph there I am pretty sure that the “other universes” that the abc proof allegedly involves are nothing but the usual set theoretic universes. I guess there is just one universe enlargement at one point. If that’s the case, then it’s not quite the mystery that it is made to look like in some G+ discussion…
So I have looked around a bit now. The impression solidifies that what Shinichi Mochizuki does is good topos theory/Galois theory. I haven’t found any indication that the “geometric figures” he uses are not those that topos theory is all about. (If I am wrong about this, please don’t hesitate to set me straight.) In particular the point that is being emphasized is that he takes etale homotopy theory seriously. So maybe we can say he does good higher topos theory, even. All in all, it seems to be work that squarely fits into our realm here.
I have read the section 3 of IUTT IV, which deals with the ’interuniversal’ aspect, and really it looks like considering models of sufficiently rich theories in different toposes. For example, the theory of covering spaces of a given space, which Mochizuki takes as being defined by a set-theoretical formula, which can easily be interpreted in the internal language of a (certain sort of) topos. Mochizuki says that he deals with a collection of universes, and not just universe enlargement. But it seems to me what he really wants to get at is the universal topos carrying a model of the theory, but just doesn’t have the terminology.
However, I’ve searched through the rest of the papers, and there is no substantial reference to ’species’ or ’mutations’, the key concepts of section 3, outside of that section or the introduction outlining that section. Even in the parts of the paper that are referred to from section 3 that are said to need the universes, there is no mention of them or the techniques involving them.
Mochizuki says himself
Remark 3.3.2. One somewhat naive point of view that constituted one of the original motivations for the author in the development of theory of the present series of papers is the following. In the classical theory of schemes, when considering local systems on a scheme, there is no reason to restrict oneself to considering local systems valued in, say, modules over a finite ring. If, moreover, there is no reason to make such a restriction, then one is naturally led to consider, for instance, local systems of schemes [cf., e.g., the theory of the “Galois mantle” in [pTeich]], or, indeed, local systems of “entire set-theoretic mathematical theaters”. One may then ask what happens if one tries to consider local systems on the schemes that occur as fibers of a local system of schemes. [More concretely, if $X$ is, for instance, a connected scheme, then one may consider local systems $\mathcal{X}$ over $X$ whose fibers are isomorphic to $X$; then one may repeat this process, by considering such local systems over each fiber of the local system $\mathcal{X}$ on $X$, etc.] In this way, one is eventually led to the consideration of “systems of nested local systems” — i.e., a local system over a local system over a local system, etc. It is precisely this point of view that underlies the notion of “successive iteration of a given mutation- history”, relative to the terminology formulated in the present §3. If, moreover, one thinks of such “successive iterates of a given mutation-history” as being a sort of abstraction of the naive idea of a “system of nested local systems”, then the notion of a core may be thought of as a sort of mathematical object that is invariant with respect to the application of the operations that gave rise to the “system of nested local systems”.
So the desire to avoid potential ill-founded chains $a\in b_1 \in b_2 \in \cdots \in a$ seems to arise from the issue of having these systems which are apparently self-referential. Consider the case of a quotient map $A \to A/\sim =: B$ where the fibres are isomorphic to $B$ and we are working in material set theory with elements of $B$ the fibres of the map. If one of these fibres was $B$ we would have an ill-founded $\in$. In practice I think what happens in Mochizuki’s work is that there is a sequence of constructions like this and at some point he wants to get back to where he started. Unfortunately he doesn’t know about the principle of equivalence, and is working in material set theory where these sorts of questions can pop up.
Really the ’species’ and ’mutations’ are just categories and (isomorphism classes of) functors which are definable in the language of set theory. Mochizuki is trying to get categorical invariants from these. In a post on MO, Vesselin Dimitrov says that what Mochizuki is trying to do is construct a formal quotient that doesn’t exist, by embedding his algebro-geometric objects (hyperbolic curves, number fields etc) into some category whose objects are categories and taking some diagram of these categories (the ’log-$\Theta$ lattice’) as being the quotient in some sense. This is of course a long-established tradition in category theory. In particular, he is dealing with functors between these categories which are not induced by maps of schemes as replacements for maps (like Frobenius) which are ’missing’ in the context in which he is working.
Oh, and the excitement about the link between abc and Mordell is that it gives effective bounds on the number of rational solutions, rather than just assuring us that there is less than $\aleph_0$ of them. And I’ve heard it said that Mochizuki’s work is very explicit in the bounds it gives, which is even better (because he proves something stronger than abc, in fact).
For the quote in 13, about local systems, that starts looking to be some form of infinity category formulation. I had a glance at his idea of Galois mantle, but cannot see what he means as I do not have a good enough alg. geom. background. His p-Teichmuller book has a lot of the same ideas with an attempt to explain them in simpler language.
Hi David,
in remark 3.1.4 of IUTT IV it says explicitly:
Note that because the data involved in a species is given by abstract set-theoretic formulas, the mathematical notion constituted by the species is immune to, i.e., unaffected by, extensions of the universe — i.e., such as the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ … ∈ Vn ∈ … ∈ V that appears in the discussion preceding Definition 3.1 — in which one works. This is the sense in which we apply the term “inter-universal”. That is to say, “inter-universal geometry” allows one to relate the “geometries” that occur in distinct universes.
So explicitly: the universes here are just Grothendieck universes and “inter-universal” just refers to universe enlargement.
Ah, I see. I did read that but thought he might in the end use something more general than mere universe extension
Well, I may be missing something. But that’s my impression.
My general impression is: hopefully somebody with a bit of general abstract background eventually takes the time to go through all this and brush it up.
My understanding is that he looks at equivalence class of formulas where their effect is compared in models in different Grothendieck universes, including infinite chain. The category of definable functors in that sense behaves rather different than if one changes universes one at the time.
Another minor point may be that one has axiom that a larger Grothendieck universe exists but when looking at infinite chains of inclusions one uses some rather strong axiom of choice to handle definitions which utilize such chains.
You see the groups in definable context behave rather different than just usual groups in sets, even when we talk usual model theory. So if one defines scheme not by a functor but by a definable functor in inter-universal context one may have internally rather different geometry than the totality of geometries for various sizes of universes. The latter is handled by a change of universe. The former can not. Every feature, “point” in the theory itself is studied as entity depending on all universes simulateneosly. This is my suspicion.
To come back to #2, I’ll play advocatus diaboli. May the following provoke somebody to rebuke.
What is interesting about the abc-conjecture except that it was easy to state and hard to prove? In which sense is it not just a technical lemma?
So if Mochizuki is right, we learn that the abc conjecture has a place as a phenomenon in topos theory (which subsumes Galois theory and anabelian geometry). And topos theory is beautiful and all. But the abc conjecture? Let it be a powerful technical lemma, as so many technical lemmas are. But isn’t it still a technical lemma?
Isn’t much of the excitement about these “big” number-theory conjectures just that: that they are easy to state, hard to prove and therefore make an excellent story of bravehood, struggle and eventual success that the public perception can appreciate?
All the proofs of these “big” number theory conjectures (Fermat, now abc, etc.) proceed by interpreting the problem by Isbell duality as a question in geometry in the general sense of topos theory and then showing that certain geometric phenomena of general and genuine interest dually involve the given conjecture.
I’d say: great for the conjecture that it finds this way a home in an actually interesting theory and is thereby knighted to carry genuine meaning itself, instead of being a problem as any random crossword-puzzle: easy to state but hard to fill out.
For instance I find the Taniyama-Shimura conjecture already much more beautiful and interesting than the Fermat theorem that it implies. Because it has more conceptual meaning, less the feeling of a random problem. The huge increase of excitement about it after Frey’s remark that it implies FLT has its source not really in mathematics as a subject, has it? Instead this kind of excitement has its source in mathematics regarded as a sport.
Or so I think currently. Am I wrong?
I more or less agree with that. It reminds me of the Weil conjectures that Grothendieck developed all of the topos theory and cohomology stuff to try to prove. The conjecture acted as a goal, but the interesting thing for me is the concepts that were developed to attempt to prove it. (I do not like ’maths as sport’ and after a summer of a surfeit of sport in the UK, now people are asking how to use the Olympics to improve general fitness.) The recent work on the abc conjecture hopefully will improve the general health of mathematics and give new (or nearly new) tools for us to work with.
I just worked on the entry abc conjecture a bit. It seems to me that until a few minutes back the statement of the conjecture there was lacking several qualifiers:
it needs to be positive integers
it needs to be coprime integers
the bound is not $P(a,b,c) \geq \epsilon$ for $\epsilon \gt 0$ but $P(a,b,c) \geq 1+ \epsilon$.
Experts please check. I am just comparing to Wikipedia.
NO!! why positive integers, not at all, that is why we can move $c$ to another site, and that is why in the definition of the power one has absolute value. It is irrelevant as it is stated in terms of prime factors which do not include $-1$ anyway. Finiteness does not depend on ordering taking minus sign etc. It just can get to some double counting. Secvond, I said relatively prime what is the same as coprime. The original form is $a+b+c = 0$ and I said that both shapes are in the literature, why do you want to exlude the other choice which I put ? The bound $1+\epsilon$ vs. $\epsilon$ depends on the variant of the definition of power.
I said relatively prime what is the same as coprime.
Okay, I see. In your statement of the conjecture further below you in fact just referred to “the abc equation”. So you were thinking of “the abc equation” as including the condition of relative primacy.
All right, but it’s better to say it explicitly I think.
I must have mistyped in the version which got preserved (as I was editing many times). At least some version had relatively prime.
In which sense is it not just a technical lemma?
Usually a lemma is meant to help prove a specific result. Abc conjecture, on the other hand implies a plethora of results of seemingly different nature in number theory. I do not understand that you are saying that Mordell conjecture is not interesting. Having a general finiteness statement on all higher genus curves over rational numbers is clearly a very natural and non-obvious statement.
To me
A (strong) technical lemma is a technical statement that provides for instance (strong) size estimates using possibly intricate technical conditions.
A beautiful theorem is a statement that provides deep concetpual insight in a transparent way.
To me, anything that involves intricate inequalities such as
$\frac{log|c|}{log(rad(a\cdot b\cdot c))} \geq 1 + \epsilon$already utterly fails to be beautiful or receive non-lemma status. This is clearly not a transparent conceptual statement but is some hard technical condition. The next guy may come and fiddle more with it, getting better/different bounds under weaker/different conditions etc. There is an element of randomness involved. This is what I’d call a technical lemma.
Notice that I am not saying that there is anything wrong with technical lemmas. Technical lemmas propel the world. But I find reactions like “I am quivering of excitement in view of the beauty of this statement” out of place. It’s not beautiful. It’s intransparent and technical.
By the way: I am not saying that it’s not interesting.
I am saying (in #2): “Can somebody say why it is interesting?”
So far we have
Mazur: it’s beautiful because it, ahm, is what it is.
Kim: it’s cool because it involves us rethinking what “numbers” and “geometric figures” are.
Škoda: because it implies the Mordell conjecture.
The first point I just can’t see how it answers the question.
The second I in princple very much agree with, but it seems to be putting the question from its feet to its head: it’s interesting because we can answer it in a cool theory? That can’t be. Clearly not every statement provable in an cool theory is thereby already interesting.
The third I need to think about. But the evident next question is: can you say what you find interesting about the Mordell conjecture? And if you do, preferably: can you put it into the entry Mordell conjecture?! :-)
Re #28
To me
Obviously those are key words. Or are you trying to offer objective standards as to what is ’beautiful’ in mathematics?
Without attempting to argue why abc might be considered beautiful to some people (because it would probably take me a really long time to come up with a statement I’d be satisfied with), I’d say this one might partially be a matter of acquired taste. Beauty in the eyes of some might not imply ’transparency’ (whatever that word means exactly – the beauty of $(\infty, 1)$-topos theory is only ’transparent’ after a pretty long period of struggle of getting used to the concepts).
Perhaps there is a beauty in trying to and succeeding in crystallizing hazy intuitions into precise (and delicate) statements. Minhyong tried to indicate the hazy intuition in terms of small and large prime factors of numbers and their sums. It’s interesting trying to make that intuition more precise in a way that is both deep and has a chance of being true.
I am also suspicious about the division of results into ’beautiful theorems’ vs. ’technical lemmas’. Recall here Paul Taylor’s mot “Lemmas do the work in mathematics: Theorems, like management, just take the credit. A good lemma also survives a philosophical or technological revolution.” Is the Yoneda lemma a strong technical lemma, or a beautiful theorem? And: while the proof of the Yoneda lemma is easy, is the deep meaning ’transparent’? I’d say it’s probably not so transparent to most people, until they see its wide ramifications. Interesting questions.
Hey, how do you turn on that smart quotes thing? Can you do it with single inverted commas as well?
Edit: “Testing”, ’1, 2, 3’… hm, I tried to set to the default setting. Didn’t work for the single inverted comma.
I would just like to see those entries be better. Currently neither the entry abc conjecture nor Mordell conjecture convey any sense of what it is about these things that is interesting. You should not try to argue with me. If you care and have the energy, you should write something helpful into these entries.
Currently we have the state of affairs that there is excitement and nobody says why. I come and try to provoke a reaction by saying that I don’t see what’s exciting. You should react to that by saying why its’s exciting to you and not by trying to tell me in turn that I am wrong about not being excited.
If you see what I mean.
But if nobody has the energy to work on these entries. Let’s leave it at this. Because: this discussion here is in “Latest Changes” not in “Mathematics, Physics, Philosophy”!
I will reply to you privately.
The statement that Mochizuki claims to prove (namely
$log|\Delta_{min}(E)|\le(6+\varepsilon)log N_E+O_\varepsilon(1),$according to Vesselin Dimitrov on MO - in fact it is something slightly different which is a bit more complicated) deals with conductors ($N_E$) and discriminants ($\Delta_{min}$) of a curve $E$ which is defined, like they Frey curve, from a solution to the abc equation (actually it is something more general, but for application to the abc conjecture it is this). Numbers like these are bread and butter of number theorists. But as you say, Urs, perhaps it has a more structural interpretation we haven’t seen yet. I think what is exciting number theorists more is the toolbox Mochizuki has built, which will be there even if the proof fails.
EDIT: I should say what these are. The conductor of a modular elliptic curve is the minimal $N$ such that the curve is the quotient of $J_0(N)$, which is the Jacobian of the famous modular curve $X_0(N)$. This number pops up in the functional equation of the $L$-function of the curve for any elliptic curve, so it is an important number. The discriminant is a measure of the singular points of the curve under reduction various different primes (I think there is a cohomological interpretation of this). An upper bound for the discriminant helps tell us that the amount of singularity is bounded, and in fact there is an algorithm to compute the conductor.
I added a section on Mason’s theorem and its relation to the abc conjecture to abc conjecture.
28 The inequality there is much more conceptual than many of the homological algebra sequences and “obstructions” used in nLab. It has two ingredients. First is that the things are stratified according to the exponentials. Mazur motivates this step before getting into stating abc conjecture as a natural step in studying the set of solutions. Second, the radical of abc simply removes the repetitions of prime factors, one can hardly imagine simple quantities more akin to the nature of integer numbers. Of course your taste and my taste is not about excitement about numbers and their prime factors. But this is such a subject. Didn’t you claim elsewhere that higher categories will get everywhere. But you see, some subjects have different conceptual ingredients, which do not boil down to “topos” nonsense but can freely combine with it.
Consider for example the Alexander polynomial which typically fits into nLab gadgetry. Any of its definitions is much more intricate in construction than the inequality here.
32 I do not see a need to improve the Mordell conjecture to make the statement interesting. The very sentence of the idea section (which is in fact sufficiently precise) is striking enough. Higher genus curve over rationals has finitely many points. Period. Finiteness of something what is not a priori finite, and in such a generality…
David Roberts wrote
Really the ’species’ and ’mutations’ are just categories and (isomorphism classes of) functors which are definable in the language of set theory. Mochizuki is trying to get categorical invariants from these.
Correct. A $0$-species in Mochizuki’s sense is simply a definable set (with a specific that he works with model theory with ZFC models with the axiom of Grothendieck universes). Then he can combine $0$- and $1$-species to get Mochizuki species as definable categories (in the sense as in Hrushowski 2006 or entry definable groupoid) and so on. He complicates much more than needed. These notions can be defined in any set theory. Then at some point he can simply specialize to work with ZFC with Gorthendieck universes, but the basic machinery of model theory as essentially used there is quite set-theory independent as usually.
In a sense I guess he then proceeds to definable schemes in appropriate sense and so on.
Remark: Of course, traditionally the definable sets are with respect to a language, not a theory, but it works the same way and many modern works now take the equivalence classes just taking the models of a theory and not all structures of the underlying language. Here one has definable sets in the sense of theory ZFC with Grothendieck universes.
Mochizuki also works with multisorted parametrizations.
Mochizuki wrote:
The relationship between the classical “set-theoretic” approach to discussing mathematics — in which specific sets play a central role — and the “species-theoretic” approach considered here — in which the rules, given by set-theoretic formulas for constructing the sets of interest [i.e., not specific sets themselves!], play a central role — may be regarded as analogous to the relationship between classical approaches to algebraic varieties — in which specific sets of solutions of polynomial equations in an algebraically closed field play a central role — and scheme theory — in which the functor determined by a scheme, i.e., the polynomial equations, or “rules”, that determine solutions, as opposed to specific sets of solutions themselves, play a central role.
In fact this can be formalized by noting that the definable sets correspond to certain class of presheaves on the category of models and elementary monomorphisms. So if one works with appropriate notion of a scheme then I can make two remarks. First that the presheaf is typically here not representable. Second if it were representable the morphisms are typically different already in the underlying category: the elementary monomorphisms are not the typical kind of morphisms in mathematical practice. Rather one looks at natural transformations (morphisms of presheaves) and the naturality is here quite a constraint, while at the level of components one has also the usual restrictions of kind of morphisms in usual mathematics one desires for that kind of objects.
some subjects have different conceptual ingredients, which do not boil down to “topos” nonsense but can freely combine with it.
Funny that you say that, given that the conjecture is now apparently proven by way of anabelian geometry.
Mochizuki wrote:
The notion of anabelian geometry dates back to a famous “letter to Faltings” [cf. [Groth]], written by Grothendieck in response to Faltings’ work on the Mordell Conjecture [cf. [Falt]].
So far as Mordell conjecture being not clearly interesting.
Urs said
Funny that you say that, given that the conjecture is now allegedly proven by way of anabelian geometry.
The fact that this is one of the main ingredients used is not in contradiction with what I am saying; some other deep theorems (related to progress partial results on Vojta’s conjecture and so on) are allegedly also used. This did not boil down to SOLELY anabelian geometry (except in the logical trivial sense that any theorem boils down to any single step in its proof while all other steps being done) nor the latter (anabelian geometry) boils down only to topos theoretic generalities (while it can be formulated in a specific instance of a such a topos theoretic setup).
The subject of number theory has its own conceptual ingredients DIFFERENT from topoi. The mathematics is of course intertwined and whenever the things COMBINE they become truly deep. The things which are just one kind of nonsense are more superficial than non-obvious combinations of things. As Manin once said, the deep problems in mathematics in their solution usually involve at least 5 different fields…
some other deep theorems (related to progress partial results on Vojta’s conjecture and so on) are allegedly also used
Can you say more concretely which ones?
Can you say more concretely which ones?
Yes, if I go back to recheck my readings in last couple of days. But tomorrow and over the next week I am not likely to spend more time on this issue (our group collaborator is arriving on Saturday, among other things). I’ll try some evening though if time permits.
I added a link to the ‘letter to Faltings’ at anabelian geometry.
Thanks. I have also copied it to and cross-linked with Mordell conjecture.
@David C.,
re #8: it turns out that this interview is not actually by John Baez. It is by some reporter and John has only been forwarding it. John just pointed this out in the comment section. See at the very end currently.
I see. Actually reading those questions with hindsight it’s fairly obvious it wasn’t John.
To question Minhyong properly, it would probably be as well to understand his MO answer.
“Can somebody say why it is interesting?”
Jürgen Habermas tried to detail the relation of knowledge and human interest (see also the enty in the german wikipedia web). In short he follows the pragmatism of Charles Sanders Peirce, who says that process of acquiring knowledge is bound to the collective of researchers who solve their task by communication. This process is conducted by knowledge-interests which need to be communicable. So Habermas would probably say that a necessary condition for the abc-conjecture to be interesting is to be communicable. Because it is hard to proof it can moreover constitute a long term research program involving a lot of people who can communicate with each other very well since they are doing similar things.
But I admit that some aspects of this have been said above. Also it gives only a necessary and no sufficient condition for hard problems in number theory to be interesting.
a lot of people who can communicate with each other very well since they are doing similar things.
Hopefully you are not deducing that something is the more interesting the more people talk about it! ;-)
From time to time I keep googling for information on why these conjectures are interesting. I keep thinking that there will be some interesting implication of Mordell’s conjecture to something like Gromov-Witten theory or the like. But it seems to be tough.
Googling for
“abc conjecture is interesting”
yields 0 hits.
Googling for
“Mordell conjecture is interesting”
yields a single hit. That says that it is interesting because it says something about Fermat’s last theorem.
“From time to time I keep googling for information on why these conjectures are interesting.” This must mean that you don’t find number theory in and of itself to be interesting, because I’d say most results in number theory are interesting because they relate to these ones not the other way around. They are quite fundamental.
Some examples: The Mordell conjecture completed a classification of the behavior of rational points on curves over Q. For genus 0 you get no points, or something isomorphic to $P^1$ and hence infinitely many. For genus 1, the Mordell-Weil theorem tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the case of finite can occur for higher genus. This is amazing!
This result also implies many non-trivial results. For example, fix a finite set of primes $S$, a dimension $n$, and a polarization degree $d$. There are only finitely many abelian varieties of dimension $n$ and polarization degree $d$ with bad reduction inside $S$. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3’s.
It proves the Isogeny Theorem which says that abelian varieties are isogenous if and only if they have isomorphic Tate modules. This one should fascinate anyone interested in geometry. It tells you exactly how much geometric information you can recover just by knowing a single cohomology group!
Is the point that you want it to be interesting by relating to something that can be tied back to physics? I think there are results in math that can be quite interesting just by relating them to more math. I dare say that this result is one of the foundations of modern number theory because of the use of Galois representations to prove something geometric.
That discussion reminded me of the following:
There is a paper: Moore and Seiberg equations, topological field theories and Galois theory, by Pascal Degiovanni. It attempts to show the links between TFTs and the questions in Grothendieck’s Esquisse. There are conjectured links between Rational Conformal Field Theories, and the Teichmuller tower, but from there to IUTT seems a very long way. (The paper is in the Dessins d’Enfants LMS lect. Notes 200, volume from 1994. There is a much longer (85 pages) version on the web in French. The short version is also on the web.
A Google search starting from there did give some other links, but not directly linked to the more purely number theoretic conjectures, more to Grothendieck’s starting point.
49 I find this link you found counterproductive to my interest, and you will likely agree. If somebody (Mordell, Faltings etc.) tells me that a striking and conceptual result of finiteness of number of points of ALL higher genus curves over rationals should be interesting because some particular series of diophantine equations (Fermat one) has no solution than I am really perplexed. Every statement in math (even the least interesting) can by Matiasevich theorem be reduced to solving a diophantine equation (the reduction is complicated and never illuminating to the very problem at hand). The Fermat equation may look nicer than average equation but one can easily write many diophantine equations which look nice to a person literate in elementary school algebra but which are of no importance in math and whose solution is often easy to diophantine experts. Truly, eventually it occurs that solving Fermat’s problem people found that it is related to some beautiful mathematics of elliptic curves and the problem is more known than Mordell among school boys and girls, and among mushroom hunters, but I would say the Mordell’s conjecture is appealing in its very statement (especially if you know the facts about genus 0 and 1, as 50 has kindly explained), while the Fermat problem becomes conceptually interesting only after a long chain of deductions resulting in deeper facts like modularity.
Hopefully you are not deducing that something is the more interesting the more people talk about it! ;-)
Well, talking is not always communication. If you ”accept” an (incoming) communication (and you actively participate in it), you put the things being said in relation to what you already know and there is some resonance in your feeling and you will formulate an answer and thereby develop the subject in discussion… However the outcome of this communication may be that there is no strong relation between the discussed topic and the concepts you are interested in, but for the time the communication lasts - I think - you have an investigative interest in the topic, since you wish to put yourself into perspective to it.
But if we consider a group of people having no mathematical background and who just understand that a difficult problem has been solved, then they might have a discussion on difficult problems in general, but they won’t have a rich communication on the abc-conjecture.
@hilbertthm90,
The Mordell conjecture completed a classification of the behavior of rational points on curves over Q. For genus 0 you get no points, or something isomorphic to P 1 and hence infinitely many. For genus 1, the Mordell-Weil theorem tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the case of finite can occur for higher genus. This is amazing!
This result also implies many non-trivial results. For example, fix a finite set of primes S, a dimension n, and a polarization degree d. There are only finitely many abelian varieties of dimension n and polarization degree d with bad reduction inside S. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3’s.
Thanks, that’s useful! I have moved that into the entry. Feel invited to further expand there.
Is the point that you want it to be interesting by relating to something that can be tied back to physics?
No. The point of my request since #2 is that I think an $n$Lab entry on something important/interesting should explain why that something is important/interesting. I am not an expert on this stuff, but I guess I know enough math that somebody who is can explain to somebody like me why he is fascinated by these conjectures. You just made a very good start. That helped, thanks.
I haven’t been following the developments. But the discussion you linked to was of a very general sort and had little mathematical content.
Something in this situation reminds me of what Thurston wrote in his On Proofs and Progress in Mathematics (p. 15):
I also gave many presentations to groups of mathematicians about the ideas of studying 3-manifolds from the point of view of geometry, and about the proof of the geometrization conjecture for Haken manifolds. At the beginning, this subject was foreign to almost everyone. It was hard to communicate—the infrastructure was in my head, not in the mathematical community. There were several mathematical theories that fed into the cluster of ideas: three-manifold topology, Kleinian groups, dynamical systems, geometric topology, discrete subgroups of Lie groups, foliations, Teichmu ̈ller spaces, pseudo-Anosov diffeomorphisms, geometric group theory, as well as hyperbolic geometry.
We held an AMS summer workshop at Bowdoin in 1980, where many mathematicians in the subfields of low-dimensional topology, dynamical systems and Kleinian groups came. It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour.
At that time, there was practically no infrastructure and practically no context for this theorem, so the expansion from how an idea was keyed in my head to what I had to say to get it across, not to mention how much energy the audience had to devote to understand it, was very dramatic.
I can only hope that Mochizuki is patiently explaining his ideas in similar sorts of workshops.
I can only hope that Mochizuki is patiently explaining his ideas in similar sorts of workshops.
As somebody in that blog discussion pointed out, on Mochizuki’s Home page – What’s new‘? is announced a workshop lecture that he is going to give next month, and the slides for that lecture he has here. I gather the key idea is that alluded to on slide 4.
I had re-organized the references-section, in order to disentangle general discussion from IUT stuff.
Then I also added this pointer:
Relating the abc conjecture to D=4 N=4 super Yang-Mills theory:
Ah, now I see where my first edit went. It was deleted by Anonymous in rev 21.
If that article has problems, add a comment and let us know.
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