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Unbounded ranks of elliptic curves: another premature faith

I generally don't like arrogant people who claim to be certain about something even though there is no solid basis for that certainty. Many climate fearmongers are textbook examples of these folks. The list of these arrogant people also includes Scott Aaronson – but also many other people in computer science – who claim (not only that the Earth will evaporate soon but also) that their word and influence is enough to be almost certain that e.g. \(P\neq NP\), even in the absence of a proof in either way.

Exactly 3 months ago, I discussed an interesting article by Kevin Hartnett in the Quanta Magazine that described an exciting story of Mr/Ms Ewin Tang, an ex-student of Aaronson's in Austin who is now a grad student at University of Washington. Tang was ordered to prove a proposition, basically a miniversion of \(P\neq NP\), as if it were a fact, except that he was finally led to prove the converse. Needless to say, lots of people had previously wasted their time with efforts to prove something that couldn't have been proven – and the activities done in order to prove X are often substantially different from those needed to prove non(X) which is why most of the mental energy was completely incorrectly allocated.

Now, the same Kevin Hartnett wrote another story with a similar lesson – in the absence of a proof, the mathematicians' belief in a certain conclusion may very well be a prejudice that is gonna be reversed. His text

Without a Proof, Mathematicians Wonder How Much Evidence Is Enough
talks about a 2016 paper by Melanie Wooden-Machete Trump and her 3 pals (OK, fair enough, I wanted to increase the number of views of their preprint page).

First, let me answer the question from that title. If the questions are of a purely qualitative, binary type, e.g. the question "whether the supremum of a set of ranks is finite or infinite", then no amount of "evidence" that is short of a proof is enough! If we can't complete a proof, we should really say that no other comments are truly relevant so the amount of evidence is zero.

OK, what's going on? We may write down curves such as\[

y^2 = x^3 - 4x + 1

\] i.e. elliptic curves and search for the list of all possible rational solutions \((x,y)\in {\mathbb Q}^2\) to this equation. Well, there are infinitely many.

Here, I need to warn you: the elliptic curve doesn't mean that it is an ellipse. An ellipse would only have at most the second powers of the coordinates, my example has the third power as well. Instead, what is "elliptic" about it is that if you extend \(x,y\) to complex numbers \(\CC\), the curve will have a complex dimension 1 and the topology of a torus – which has one extra "handle" on top of the topology of a sphere. The sphere ends up being a rational curve and because the "handle" needs some square-root branch cuts to be created from the sphere, it is called the next-to-rational i.e. elliptic curve. OK, if you want to know why it's really called "elliptic", it comes from the elliptic integrals.

(The torus, a complex elliptic curve, is the most rudimentary part of the complex algebraic geometry you need to learn if you want to study F-theory compactifications in string theory.)

OK, let's return to the list of rational solutions. Imagine that you find one point \((x,y)\in{\mathbb Q}^2\) on that curve. From that single point, you can find other points: draw any line through that point whose slope \(dy/dx\) is rational. That line will intersect the elliptic curve at some other points, and you can prove that their coordinates will be rational, too. The "offspring" may be created to further generations, and you find many solutions. The animation at the top sketches this proliferation of the descendants.

Amusingly, you get a significant percentage of the rational solutions in this way. If you think about the construction of the new rational solutions, you will see that the set of rational solutions actually forms an Abelian group of a certain kind. The only "hard" part of the construction of all solutions is the "starting point".

If you had a circle instead of an elliptic curve, one rational starting point would actually be enough to find all rational solutions – through the process of picking the rational slope and intersections, using the points that you already know. So we say that the rank of the circle is one. This statement can be proven. Circles are simple.

But if you already think about all the possible elliptic curves, the ranks may be higher than one. Hartnett describes another story of "group think" whose general sociological skeleton is something that you have heard of many times. There is some "lore" that everyone is obliged to believe although it's not really proven. The "lore" has some particular influential people who force it on others. In this particular story, it is especially Noam Elkies – at his time, the youngest mathematician who got tenure at Harvard.

The simplest way to see that sociologically, the "lore" is being pushed everywhere, is to look at Wikipedia that states:
A common conjecture is that there is no bound on the largest possible rank for an elliptic curve. In 2006, Noam Elkies discovered an elliptic curve with a rank of at least 28: [an elliptic curve with two terrible googol-like integer coefficients]
"A common conjecture" really means nothing else than "rationally unjustified group think" but "common conjecture" probably sounds more intimidating which is what the writers wanted. The reference for the "common conjecture" is a Croatian pop math page which actually calls it a "folklore conjecture", not a "common conjecture", but certain people never hesitate to "improve" a statement because the propagation of an answer to this not yet settled question is something like a religion with its (biased) missionaries.

OK, this Dr Elkies, and less importantly, a few others (in Hartnett's story, the number of people who actually "guarantee" the group think is incredibly small – it seems that there has been a silent majority that simply allowed the loud minority to determine the group think – like in discussions about political issues), have been saying it's almost certain that the ranks are unbounded – because he could produce some ranks above 20, namely 19 and 28 etc., and it's almost infinity. But is it true? Well, one comment we could make is that Elkies isn't quite impartial – he has a clash of interests because his name has been rather linked to these statements about the unbounded ranks. If you study the atmosphere in that field, you will see that people are pushed to pay lip service to this faith. Those who didn't were considered heretics or idiots by a network of folks who enforce the group think. But there's no proof.

In 2016, Melanie Wooden-Machete Trump et al. have presented the heuristic paper I mentioned. Well, it's called a heuristic paper because it doesn't contain any rigorous proof of the most important propositions. But it's still over 40 pages of heavy professional mathematics that I would be visually unable to distinguish from fourty pages with a very finely constructed, complete, perfectionist proof. When we say that the paper is a heuristic, it simply doesn't mean at all that the paper looks like Leo Vuyk's Strawberry Universes. ;-)

They didn't attack the rank-of-the-elliptic-curve problem itself but a similar problem with "alternating integer matrices" and they discussed some properties of primes etc. But it "looks like" the possible values of the ranks are analogous for their "matrix model" as it is for the original problem involving the elliptic curves.

Some parts of their models are rigorously proven – I don't quite know whether "everything aside from the equivalence to the elliptic curve problem" has been proven rigorously. There's obviously a lot of things that I don't know here.

But the conclusion is: The rank is bounded, thus the religion is incorrect. In fact, the last step before they declare that the rank is bounded proves a much stronger and interesting statement: the number of elliptic curves (well, elliptic curves' "caricatures" in their model) with the rank exceeding 21 is finite. If there are finitely many above 21, the allowed ranks can't be unbounded because you can pick the maximum rank among the finitely many ranks above 21.

Great. So this heuristic paper doesn't rigorously settle the problem, either. But it gives a strong argument in the opposite direction than what the lore used to say. Now, Elkies offers a plausible counter-argument: maybe Melanie has only studied some "generic" elliptic curves that are close enough to the mean of a distribution in some parameter space – while those hypothetical curves that have ever larger ranks are increasingly far from the mean and non-generic in some way.

Maybe. But at this level, it becomes clear that he's just guessing. Even if his counter-argument were relevant, there could be another counter-counter-argument that could weaken his counter-argument and have the last laugh, after all. He's proposing something that may be relevant and reverse their conclusions – or not. The real point is that there is no proof in one way or another but there are always "incomplete arguments to make you think in one way or another". A funny thing about incomplete arguments is that they may always be cherry-picked or artificially fabricated. So if you wish, you may collect a large number of arguments that support your case and claim that all arguments that you could find agree and point in a certain direction.

However, that "uniformity" is purely due to your bias, prejudice, and/or dishonesty. Some other people could have cherry-picked (and helped to artificially construct) the arguments pointing in the opposite direction, too!

Because of Melanie's model and a previous model that's been around for years, it seems rather clear that "something special" is changing when the rank reaches the critical value of 21. But that has moral consequences. If your car – let's call it a Tesla (how do you call a Tesla on a hill? A miracle) – half-breaks after 21 miles, it is rather easy to believe that the car will completely break after a finite number of miles. The maximum survival of such a Tesla car may be as small as 28 miles – 28 is the maximum known rank of an elliptic curve. It may still be infinite but if it could survive infinite usage, why would it half-break after 21 miles?

My feeling is that the argument by Melanie et al. is "finer" because they actually found an important and actual finite value of the rank where things start to change a lot, 21. On the other hand, Elkies says that the supremum of the ranks is \(\infty\) because, apparently, his \(28\approx \infty\): it is the relatively largeness of the number \(28\) that should impress you and make you believe that he can also get to infinity. His rough argument is equivalent to \(1/28\approx 0\) while Melanie et al. are able to see \(1/21\neq 0\) (because something that is possible below 21 isn't possible above 21, i.e. because \(28\approx 21 = O(1)\)) which means – in another approximation \(21\approx 28\) – that they probably have a better resolution! And when we want to see whether there is one star or two stars, \(0\) and \(1/{\rm rank}_{\rm max}\), it is better to have a finer resolution. Do you get my point? I am half-joking but the qualitative lesson I want to convey is meant totally seriously. Melanie et al. have found a characteristic finite scale in the possible values of the rank. They have disproven some kind of a "scaling invariance" for ranks!

(Analogously, it's plausible that sometime in the future, a bright algorithmic complexity theorist will find something special happening for some perhaps incomplete calculations of \(NP\) problems that last \(N^{21}\) steps – e.g. that it seems to be much easier or more likely to find a solution if the exponent is above 21. This would analogously suggest that polynomial-time algorithms exist because we would see that "the exponent around 21 already starts to be enough". Maybe all \(NP\) problems could be solved in \(CN^{28}\) steps.)

There are clearly lots of "spectra" in mathematics that are bounded. Take all simple compact Lie groups and study their ranks and dimensions. The dimension of \(SO(N)\) is \(N(N-1)/2\), isn't it? So it scales like \(N^2\). To make the behavior uniform for large \(N\), let's divide the dimension of a Lie group by the squared rank and call the ratio Lie-Motl Rank. The group \(SO(N)\) has the rank \(N/2\) for an even \(N\) so the dimension over the squared rank goes to \(2\) from below. A similar Lie-Motl rank will be found for symplectic groups and odd \(N\) orthogonal ones. For unitary groups, \(SU(N)\), rank \(N-1\), and \(N^2-1\) dimension, the Lie-Motl Rank will be around \(1\).

Are there simple compact Lie groups with the Lie-Motl Rank above \(2\)? You bet. Just look at the five exceptional groups. \(E_8\) will be found to maximize the Lie-Motl Rank, at \(248/8^2 = 31/8\) – it is almost four.

My model is obviously much less "directly relevant" for the rank of the elliptic curves (unless you find some shocking equivalence) but the general lesson is very similar. Infinitely many simple compact Lie groups have the Lie-Motl Ranks around \(1\) or \(2\) but well above \(2\), or perhaps some \(2+\epsilon\) to eliminate the problem of ranks going to two from above, the number of simple compact Lie groups that may get this high in the rank is limited – it is the exceptional groups. And the Lie-Motl Ranks simply must be at most \(31/8\) of the \(E_8\) group.

These distribution-like issues may be analogous for the ranks of the elliptic curves. You may see that I am not proving anything. I am just inventing excuses. I am rationalizing one possible answer, making it look more plausible. But I don't really have any strong faith that the rank must be bounded. Instead, I want to say that it's important to be open-minded in the absence of the proof – because the other side's arguments are just excuses and rationalizations, too.

And rationalizations aren't proofs. In fact, in pure mathematics dealing with discrete qualitative propositions (those that you need to assume to be 50-50 a priori), I think it is correct to say that a rationalization isn't even evidence. It doesn't mean that I am never using "rationalizations". But I only talk about them as "evidence" if I believe that they are actual sketches of a proof that has a certain chance to be completed. If that interpretation is impossible, we've been just wasting our time because the rationalization doesn't really imply anything.

More typically, I am defending intense research into something – e.g. string theory – even in the absence of certainty that it is correct (or the right theory of the Universe, in this example). Why? Because I am not taking any resources away from any meaningful alternative research. You can't get any interesting results just from the assumption that "string theory isn't the right theory of the Universe". We can prove that string theory leads to vastly more convincing and interesting results than the known alternatives which is a sufficient justification for a frantic research into string theory. String theory may only compete with ideas that are already out there, not with some non-existent, future, or hypothetical competitors. But in the case of the mathematical propositions, there could exist interesting and heavily unequivalent strategies to do research that assume both possible truth values of the conjectures.

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