Mathematicians shed light on the minimalist hypothesis

This tablet, originally from Babylon, made around 1800 BC, lists Pythagorean triples — integers a, b, and c, which satisfy the polynomial equation a ² + b ² = c ² . To this day, the search for rational and integer solutions of polynomial equations remains a serious problem for mathematicians.

In the fifth century BC The Greek mathematician made a discovery, shaken the foundations of mathematics, and, according to legend, cost him his life. Historians believe that it was Hippas from Metapont , and he belonged to the Pythagorean school of mathematics, the main dogma of which was that any physical phenomenon can be expressed by integers and their relations (what we call rational numbers). But this assumption collapsed when, according to historians, Hippas considered the lengths of the sides of a right-angled triangle, which should satisfy the Pythagorean theorem — the famous relation a ² + b ² = c ² . It is said that Hippas showed that with the same length of the legs of a triangle expressed by a rational number, its hypotenuse cannot be expressed by a rational number.

According to one of the versions of the story, Gippas made this discovery while at sea, and his colleagues, shocked by this discovery, threw him overboard.

Modern mathematicians are no longer confused, like the ancient Greeks, by irrational numbers (and in general they discovered that there are more irrational numbers than rational ones). But the Pythagorean love for rational solutions of equations continues to feed the mathematicians with information. It underlies the theory of numbers, the traditional theoretical branch of mathematics, which, in our digital era, found unexpectedly many applications.
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Now two young mathematicians have advanced to the forefront of science in their study of rational solutions of cubic equations. Polynomial equations in which variables are in some degree, such as y = 3x ³ + 4 or x ² + y ² = 1, are among the fundamental objects studied by mathematicians and are used in various practical applications, as well as in branches of mathematics .

Polynomial universe

It is easy to see that a polynomial equation in which the degree of variables does not exceed 1, such as y = 3x + 4, has an infinite number of rational solutions. Any rational value of x gives a rational value of y, and vice versa.

How to find rational solutions for polynomials with degree 2, such as x ² + y ² = 1 or y = 3x ³ + 2x - 7, has been known for a thousand years. They may have no solution at all, or have infinitely many solutions. The graphs of such curves are conic sections — circles, parabolas, ellipses, and hyperbolas. If one rational point P is on the graph, then there is a beautiful way to find all the other rational points. You just need to take all the lines passing through P with a rational bias, and calculate the second intersection of this line with a conic section.

In 1983, Gerd Faltings [Gerd Faltings], today occupying the post of Director of the Mathematical Institute. Max Planck in Bonn, dealt with polynomial equations with powers of more than 3. He showed that most of them can only have a finite number of rational solutions. And there remained the cubic equations, the stubborn deviators of the universe of polynomials.

The cubic equations resisted the attempts of mathematicians who classified their solutions. Attempts to classify rational solutions of cubic equations — more precisely, families of cubic equations, known as elliptic curves, since it is they, with the exception of a few others, can have rational solutions — all great number theory specialists have been involved since the 17th century French mathematician Pierre Fermat , says Benedict Gross of Harvard University.

Elliptic cubic equations can have zero, finite, or infinite number of solutions. Mathematicians have so far managed only to guess how often these options arise.

Elliptic curves have an inexplicable ability to occur in unexpected places, both in theoretical and applied mathematics. Their understanding has become a key element in the proof of Fermat's 1995 theorem , although it seems that elliptic curves are not related to its formulation. Operations using elliptic curves have become central components of many cryptographic protocols encoding bank card numbers in online transactions. Rational solutions of elliptic curves are at the very center of various Pythagorean-style geometric problems, for example, the search for right-angled triangles with rational side lengths and at the same time a rational area.

“Intellectual stimulation, great structure, practical applications - all this is on elliptical curves,” says Manjul Bhargava from Princeton University.

Bargawa is 38 years old, his colleague, Arul Shankar - 26, they work at the Institute of Advanced Studies in Princeton and have already taken one of the biggest steps in the past few decades to understanding rational solutions of elliptic curves.

In their work, there is no recipe for finding rational solutions for a specific elliptic curve; instead, it explains what are the most likely scenarios for rational decisions if you choose a curve randomly.

The discoveries of Bargava and Shankar “are beginning to shed light on a large area of ​​our ignorance,” Gross said. “After their work, the whole world looks different.”

Elliptical security

If we take two rational points on an elliptic curve, then the straight line passing through them will almost always cross the curve at another point, also with rational coordinates. It is very simple to use two different rational points to generate a third one, but it’s very difficult to do the opposite — take one rational point and find two other rational points that would generate it. This property makes elliptic curves useful for cryptography: cryptographic security is based on operations that are easy to do in one direction and difficult in another.

“Elliptic curves are involved in many amazing things,” said Peter Sarnak of Princeton University. “They are complex enough to carry a large amount of information, but simple enough for in-depth study.”

Fun ride

Finding rational solutions to an elliptic curve reduces to finding points on its graph on the xy plane, such that their x and y coordinates are rational numbers. And often it is quite difficult to do. But if you have found several rational points, it becomes possible to generate more, using simple procedures, first discovered two thousand years ago by the Alexandrian mathematician Diophantus. For example, if you draw a line through two rational points, it will usually cross the curve at exactly one point, also rational.

This process is “a very complex structure, there is something special about cubic equations that gives them depth,” said Bargava.

In 1922, Louis Mordell proved something astounding. For any elliptic curve, even with infinitely many rational points, you can generate all rational points, starting with a small number of them, and then connecting them together. If the number of rational points on an elliptic curve is infinite, then the minimum number of points necessary to generate them all is called the rank of the curve. When the number of these points is finite, the rank of the curve is 0.



For decades, mathematicians pondered a minimalist hypothesis that assesses the rank of elliptic curves, with mixed evidence. The hypothesis says that statistically, about half of the elliptic curves have a rank equal to 0 (that is, they have either a finite number of rational points, or zero), and the other half have 1 (that is, their infinite number of rational points can be generated from ). According to this hypothesis, the number of all other cases is vanishingly small. This does not mean that there are no exceptions, or even a finite number of them - but if you take ever larger collections of elliptic curves, then the curves that fall into other categories, in percentage terms, will become less and less, and their number will tend to 0% .

This assumption was first formulated in 1979 by Dorian Goldfeld of Columbia University, referring to a certain class of elliptic curves. “It has long been a folklore,” says Barry Mazur of Harvard University.

Partly the minimalist hypothesis is supported by the widespread belief that elliptic curves should not have too many rational points. Indeed, on the numerical line of rational numbers there is a minority.

“The rational points of elliptic curves are random gems of mathematics, and it is very difficult to imagine that there would be too many such precious coincidences,” wrote Mazur and his three co-authors in 2007 for the journal Bulletin of the American Mathematical Society .

At first glance, this suggests that most elliptic curves should have rank 0. But many mathematicians believe in the parity hypothesis, suggesting that elliptic curves with even and odd ranks occur 50 to 50. If you combine the hypothesis of parity with the rarity of rational points, we get the minimalist hypothesis - the division of 50 by 50 between the lowest possible rank, 0 and 1.

In support of the minimalist hypothesis, there are also experimental data, according to which elliptic curves are really difficult to have high ranks. Elliptic curve specialists used computers to search for curves with high ranks. The current record is set at 28 - but there are very few such curves and their coefficients are gigantic.

But the other data is not so inspiring. Mathematicians calculated the ranks of hundreds of thousands of elliptic curves, and so far 20% of all curves have rank 2. A small but not very small percentage of curves has a rank of 3. According to the minimalist hypothesis, their percentage should tend to zero if all elliptic curves are taken into account. "Apparently, the data are opposed to the assumption," said Mazur.

Usually, when the data does not match the hypothesis, it will be correct to discard it. But many mathematicians cling to the minimalist hypothesis. Although computers have reworked many examples, mathematicians indicate that these calculations are only the tip of the iceberg. “It may also happen that until we prove the hypotheses, no data collected by us, even very solid in number, will reassure theorists,” wrote Mazur and his colleagues.

They also added that a rather large part of the calculated elliptic curves with a rank of more than 1 is similar to dark matter in physics. “This large mass of rational points is clearly there. We have no doubt about this. We only doubt how to give a satisfactory explanation for what they are there. ”

Because of the conflict of data and theory, they write, for decades the minimalist hypothesis "was either rejected, then taken for granted."

New methods

Until recently, Manjul Bargava, the rising star of the mathematical world, was in the camp of the doubters. One of Popular Science journals ranked him among the “ten geniuses” in 2002, and the next year at 28 he became one of the youngest people who received the title of professor at Princeton University. His colleagues admire not only his mathematical achievements, but also his kind and creative disposition.


Manjul Bargawa, 38 years old

“Manjul is a very unusual guy,” said Gross. “He looks at things in a way different from most people, and this is where his genius lies.”

Bargava, an expert in number theory, became interested in the apparent contrast between the calculated data and the minimalist hypothesis. “This suggests that something interesting is happening there,” he said. “I went to a colleague, Peter Sarnak, and asked him:“ How can you believe this assumption? ”Recalls Bargava. "For me, it looked ridiculous."

But Sarnak believed that the data as a result would begin to lean in the opposite direction, when it would be possible to shortcut elliptic curves with much larger coefficients. “He was very confident in this hypothesis,” said Bargava.

Bargava decided one way or another to find out something specific about the hypothesis. “It’s time to prove something,” he says. He began to study sets of algorithms that calculate the ranks of elliptic curves that originate in the procedure introduced by Fermat in the 17th century. This family of algorithms, called descent algorithms — exists for each of integers greater than 2 by an algorithm — they worked with knowledge and found elliptic curves with rational points. But despite numerous attempts, no one could prove that these algorithms will always work.

Bargava decided to try a different approach. “I had an idea to try the descent algorithm for all elliptic curves at the same time, and then prove that in most cases it will work,” said Bargava. After all, to study the minimalist hypothesis, it is not necessary to know what each elliptic curve looks like - it is enough to know what type they want.

This approach included work in the field of geometry of numbers, engaged in the calculation of lattice nodes in various figures (the lattice node is a point with integer coordinates). In the simplest forms such as a circle or a square, the number of lattice nodes corresponds approximately to the area of ​​the figure. But the task of Bargava was for more complex figures, and when a figure has complex features, such as tentacles, it may have much more or less lattice nodes than predicted by its area.


Arul Shankar at 26

Before embarking on such forms, Bargawa set a similar but simple task to Arul Shankar, his graduate student. Frequently, graduate students have been battling with problems from dissertations for years, but Shankar brought the solution in just three months. Therefore, says Bargava, “I asked him if he would like to join me.”

Bargawa and Shankar have developed a set of new techniques , whose importance is likely to go far beyond the limits of the original problem they are solving, as Mazur says. "The geometry of numbers has always been a deep and powerful method, and now they have seriously increased its power." He added that the genius of their technology "opens up new possibilities in number theory."

These new techniques “will influence the theory of numbers for many more years,” agrees Gross.

Clear pattern

If the minimalist hypothesis is true, then the average rank of the elliptic curves should be ½, but before the work of Bargawa and Shankar, mathematics could not even prove that the mean would be finite. Using the algorithm of descent 2 orders, Bargava and Shankar were able to show that the average rank for all elliptic curves does not exceed 1.5. Using orders 3, 4 and 5 for some curves that were not processed in the previous step, they were able to lower the upper bar to 0.88.

And although between this value and the average predicted by the minimalist hypothesis, as long as there is a gap, the discovery of Bargawa and Shankar represents a leap forward. “This is only the first step, but already very large,” says Sarnak. “It's great to see how two such young people are actively moving forward.”

Moreover, having shown that the average rank is less than 1, Bargawa and Shankar proved that a rather large piece of elliptic curves - at least 12% - have a rank of 0 (otherwise the average would be higher). They used this to show that the same part of the curves satisfies the famous Birch – Swinnerton-Dyer hypothesis , the old question about elliptic curves, for which Clay’s mathematical institute was awarded a million dollars reward .

At the Bargava lecture at the Clay Institute, one of the listeners jokingly asked if Bargawa and Shankar now rely on 12% of the million prize. “Representatives of the institute were at the lecture, and they immediately said that no, they were not supposed to,” said Bargava sadly.

The discoveries of Bargava and Shankar overwhelmed number theory specialists, many of whom did not expect progress in the field of middle rank. “You ask me a month before Manjul told me about his work,” says Gross, “I would answer you that this is hopeless.” Now, according to him, the minimalist hypothesis looks more and more promising. "I would put money on it."

One of the possible ways - which will probably require the infusion of new ideas, as mathematicians say - try to use algorithms that descend orders of magnitude higher than 5 in order to clarify more and more the limits of the average rank. “With the use of descents of 2, 3, 4 and 5 orders, a clear pattern emerged, and most likely it will continue,” said Bargava.

Bargava does not consider himself the sole owner of the rights to this idea, and hopes that their work will inspire young mathematicians to further research in the field of rational points of elliptic curves. “The minimalist hypothesis is not an end in itself,” he says. - Each time, opening the door, it turns out that you need to open many more doors. The more people do this, the more doors we can open. ”