Number theory: how many clues replace a proof?
Is there a limit to the rank of elliptical curves? This seems to be the assertion of a new statistical model, against the generally accepted idea. But how much should we trust him, when only demonstrations count in mathematics?
Four researchers have recently developed a model that challenges an accepted idea in their field. On the basis of data obtained by a computer calculation, they suggest that the dominant opinion for decades on a major concept is wrong.
They are not biologists, climatologists, or physicists. They do not come from an area where empirical models and simulations are used to determine what is true. They are mathematicians, representatives of a discipline whose current currency — the incontestable logical proofs — normally spares them this kind of debate which agitates other scientific fields. Yet, as a model, they argue today that it may be time to reassess certain beliefs that are firmly rooted in number theory.
This model, developed by Bjorn Poonen, of the Massachusetts Institute of Technology (MIT), Jennifer Park, of the University of Michigan in Ann Arbor, John Voight of Dartmouth University, in New Hampshire, and Mélanie Wood of the University of Wisconsin, Madison, was pre-published online in 2016 (https://arxiv.org/abs/1602.01431) and is to be published in the Journal of the European Mathematical Society. It concerns a mathematical concept known as the “rank” of an algebraic equation. It is a measure that indicates how many solutions to this equation are rational numbers (the numbers that can be written as a fraction). High-ranking equations have larger and more complex sets of rational solutions than the others.
Since the beginning of the 20th century, mathematicians have wondered if there is a limit to the rank of an algebraic equation. At first, almost everyone thought about what should be the case. But in the 1970s, the majority opinion changed. Most mathematicians have come to think that the rank is not bounded, which means that it should be possible to find algebraic curves with an arbitrarily high rank. And it is at this position that opinion stopped even if, in the eyes of some mathematicians, there were no solid arguments in its favor.
“Some mathematicians asserted that there was no limit in a sufficiently authoritarian way. But when you looked at their arguments, they seemed very thin”.
Says Andrew Granville, a mathematician at the University of Montreal, and who was Jennifer Park’s supervisor.
Now the clues point in the opposite direction. In the two years since the publication of the empirical model, many mathematicians have been convinced that the rank of a particular type of algebraic equation is well limited. But not everyone finds this model convincing. The absence of a direct solution to the problem raises questions that are not often addressed in mathematics: what weight to give to empirical evidence in a field where what ultimately counts are demonstrations?
“There is no mathematical justification for why this model is correct,” says Jennifer Park. “Except that experimentally, many things seem to work “.
Point by point
Consider an equation with two variables. We can trace the curve of its solutions. Among other things, mathematicians wonder how many of these solutions are rational points, that is, whose coordinates can be expressed as a ratio of two integers, such as 1/2, -3, or 4483/929.
Rational solutions are difficult to determine systematically, but there are techniques that work in certain circumstances. Take for example the equation x² + y² = 1. The solution curve of this equation is a circle. To find all the rational points of this circle, start by taking a particular rational solution, for example, the point of coordinates (x = 0, y = 1), then draw a straight line passing by this point which cuts the circle in another point. As long as the slope of this line is rational, the second point of intersection will also be a rational solution. Thanks to this procedure, you have built a rational solution into a second.
And there is no reason to stop there. Repeat the process by drawing a different rational slope line passing through your second rational point: this line will cut the circle at a third rational point. And so on … By repeating the operation an infinite number of times, you will end up finding all the rational points on the circle, in infinite number.
In the case of the circle, it is enough to start with a single rational point to find them all. The number of rational solutions that must be known at the outset to discover all the others is known as the “rank” of an algebraic curve. Rank is a way of characterizing an infinite set of rational solutions with a single number. “It’s sort of the most efficient way to describe rational solutions to these curves,” comments Bjorn Poonen.
The circle is a degree 2 equation (the highest power of the variables), also called a quadratic equation. We have known perfectly how to find rational solutions to degree 2 equations for more than a century.
The next type of equation is elliptic curves, which have a variable raised to the power of three (for example, y² = x³ + ax + b). Elliptical curves present a good compromise: they are more complicated than quadratic equations, which makes them interesting to study, but they are not too complicated. A modified form of the process of drawing rational lines also applies to elliptic curves to find rational solutions, but it stops working with equations of degree four and more.
Elliptical curves have varying ranks. With certain elliptical curves, for example, you can start with a rational point, apply the procedure of drawing lines, and not discover all the rational points. A second rational starting point, unrelated to the first, may be necessary. A curve for which two rational points must be known at the start in order to find all the rational points is of rank two.
There is no known limit to the value of the rank of an elliptical curve. The higher the rank of an equation, the larger and more complex the set of rational solutions of the curve. “Rank sort of measures the complexity of all the solutions”, explains Bjorn Poonen.
However, the rank resists the efforts of mathematicians to circumscribe it in a general theory. If we consider any elliptical curve, there is no obvious relationship between its appearance and its rank. “If you take an elliptical curve and just change the coefficients of the variables, its rank changes dramatically”, says Jennifer Park. “Adding the coefficients one at a time can increase the rank of a million. No one knows how the rank behaves”.
This lack of theory has forced mathematicians to rely on the few clues available to try to guess whether the rank is bounded or not. “The current position seems to be to say that there is no limit because we are constantly discovering curves of increasing rank”, said Andrew Granville. The current record is held by a rank 28 elliptical curve, discovered in 2006 by Noam Elkies, a mathematician at Harvard University.
But the model for Jennifer Park and her colleagues has arrived, and he suggests that in fact there is certainly a limit and that it is not very far.
A surprise at 21
Scientists use models to study phenomena that are too complicated or that cannot be observed directly. By creating an “acoustic black hole” in the laboratory, analogous in certain aspects to real black holes, we can, for example, learn a lot about the behavior of these strange objects without being blocked by their event horizon.
Mathematicians do the same. The study of prime numbers is a good example. Mathematicians wonder if there is an infinite number of pairs of consecutive prime numbers (i.e. whose difference is 2, like 3 and 5, or 11 and 13) — this is the conjecture of prime numbers twins. A formal proof is currently out of reach, but mathematicians have developed models that predict how often twin prime numbers should appear; and, in fact, there seem to be infinite numbers.
The new model on the row of elliptical curves does not directly attack the curves themselves. It establishes a relationship between curves and simpler mathematical objects, namely the nuclei of certain matrices defined on finite fields. These objects are somewhat elliptical in shape what mice are to humans in medicine: they are different, but easier to study, and assumed to be similar enough that it is reasonable to draw conclusions about one based on experiences on the other. By examining the distribution of the dimensions of the nuclei, the four mathematicians thus hoped to have an idea of the distribution of the ranks of the elliptic curves. More precisely, they bet that the distribution of the ranks of the nuclei and the elliptical curves are mutually reflective.
“This is where the jump into the void comes in”, admits Jennifer Park. “We hypothesize that this other set of mathematical objects, much easier to understand, has the same distribution as the rows of elliptical curves”.
When the four researchers started their work, most mathematicians still believed that the rank of elliptical curves has no limits. However, their model tells another story: he claims that there is only a finite number of elliptical curves having a rank greater than 21. One of them must therefore necessarily have the highest rank in the lot, which means that the rank is limited. When the four mathematicians discovered this, they understood that they had an important result in their hands.
“This prediction did not match what everyone believed, at least not publicly”, says Melanie Wood. “No one thought that the rank of elliptical curves was limited“.
If it takes an act of faith to believe in a model, it takes an even greater one when that model says that a commonly accepted assumption is false. But there are many reasons to take this result seriously. This model was built from previous models, developed by other mathematicians to study different properties of elliptical curves. These models have held up well over time and some of their predictions have even been proven.
“It wasn’t like we started from scratch and built a model from scratch,” says Melanie Wood. “Rather, we asked ourselves how we can enrich the existing models that people trust.”
Another reason to believe the new model is that rank 21 does not seem to be an arbitrary limit. Ten years earlier, Andrew Granville had created a different model which had also led to the conclusion that there should only be a finite number of elliptic curves of rank greater than 21. Now the first Granville model had nothing to do with this new model, so much so that the fact that the two reports that 21 is a particular rank seemed too precise to be a coincidence in the eyes of many mathematicians. “The fact that two completely different heuristics result in the same number, 21, is rather surprising”, summarizes Jennifer Park.
But perhaps the most convincing reason why the model of Park and his colleagues seems credible is that he makes other predictions that correspond almost exactly to results established on elliptical curves. The main conclusion of the model — that there are only a limited number of elliptic curves with a rank greater than 21 — applies to elliptic curves in general. But elliptical curves can be classified as subfamilies, and mathematicians have established that there is a limit rank for many of them. The model of the four mathematicians predicts what the limit rank should be for many of these subfamilies, and these predictions are consistent, even equal to the limits already established.
“Our model accurately predicts many of the cases already studied,” says Jennifer Park. “When I give a lecture, the audience is generally very skeptical at first, but when I mention it to the others, they say to me: It is really incredible”.
The gray area between the clues and the evidence
The model for Jennifer Park and her colleagues has a lot of advantages, but not everyone believes it, and it may well be wrong. The best-known skeptic is Noam Elkies, a Harvard mathematician, who holds the record for the highest-ranking elliptical curve. In the decades since he became the youngest professor to secure a permanent position at Harvard, he has compiled a number of results which indicate that rank is limitless. “My position has been what it has been for a long time, that I don’t think we understand this question well enough to support guesswork one way or the other,” Elkies wrote in an email.
Noam Elkies thinks that the model could stumble on several points. It takes into account curves chosen at random, or “average” curves in a certain sense. However, there are indications, notably in the works of Elkies, that there may exist whole subfamilies, each containing an infinite number of elliptic curves, which deviate significantly from the behavior of typical elliptic curves. “Heuristics that rely on the expected behavior of randomly selected representative curves could miss the behavior of extreme curves,” writes Elkies.
Even one of the authors of the new model is not entirely convinced: “I would say that I am agnostic about the question of limit on the rank,” admits Melanie Wood. It recognizes that this model could fail for the reasons cited by Elkies. But if it doesn’t hold, it’s because it doesn’t take into account some hidden and unexpected properties of elliptical curves. “The interesting question is: if you don’t think the rank is bounded, where is the error in the model?” Asked the mathematician.
“They are probably right, unless someone finds a valid reason to the contrary. I have no idea whether this reason exists or not ”, said Alexander Smith, a doctoral student who works with Elkies at Harvard on the rank of elliptical curves.
The authors of the model are not dogmatic as to its scope. They know the difference between clues and proof — and they know that in mathematics, no matter how many clues, it doesn’t prove in the end. But they think their work at least provides a reasonable basis for thinking about a major mathematical concept that has long been speculated on.
“Perhaps this will act as a challenge for mathematicians, who will try to find elliptical curves of ever-higher rank”, says Jennifer Park. Or maybe these “should reconsider what we thought was a folk guess”.
