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A “stunning” mathematical bridge that extends beyond Fermat’s Great Theorem

Mathematicians figured out how to lengthen the mysterious bridge connecting the two distant continents of the mathematical world




When Andrew John Wiles proved Fermat's Great Theorem in the early 1990s , it was a monumental step not only for mathematicians, but for all of humanity. The statement of the theorem is very simple - it claims that the equation x n + y n = z nthere are no whole positive solutions for n> 2. However, this simple statement attracted a huge number of people wishing to prove it for more than 350 years, since the French mathematician Pierre de Fermat casually sketched the statement of the theorem in 1637 on the margins of Diophantus's “Arithmetic”. Fermat’s formulation is also famous: he “found truly wonderful evidence for this, but the margins of the book are too narrow for him.” For centuries, professional mathematicians and amateur enthusiasts have been looking for Fermat's proof - or whatever else.

The proof ultimately obtained by Wiles (with the help of Richard Taylor ) would never have occurred to Fermat. It did not directly affect the theorem, but built a huge bridge, which, according to mathematicians, was supposed to exist - a bridge between two distant mathematical "continents." Wiles's proof came down to defining this bridge connecting two small patches of land between two continents. The proof was full of new and deep ideas, and spawned cascades of new results on both sides of this bridge.

From this point of view, the awesome evidence of Wiles solved a tiny part of a much larger mystery. His proof was “one of the best math events of the 20th century,” said Toby Guyfrom Imperial College London. And yet it belonged to the "tiny stretch" of the bridge, known as the geometric correspondence of Langlands .

The entire bridge would allow mathematicians to shed light on the vast expanses of mathematics, conveying concepts from one part to another. Many tasks, including Fermat’s Great Theorem, seem to be difficult on one side of the bridge, but quickly turn into easier tasks, moving to the other side.

After Wiles came up with his proof, other mathematicians began enthusiastically expanding this bridge to larger sections of the two continents. And then they ran into an obstacle. There are two natural directions for expanding this bridge, but in both of them the Taylor-Wiles method seemed to encounter an insurmountable barrier.


Mathematician Andrew Wiles, who proved Fermat’s Great Theorem and received the Abel Prize in 2016

“People have long wanted to do this,” said Anna Karayani of Imperial College London. But “we, generally speaking, did not think that this is possible in principle.”

Now two works - representing the culmination of the works of more than ten mathematicians - have overcome this barrier, essentially solving both problems. Someday, these discoveries can help mathematicians prove Fermat’s Great Theorem for a numerical system that goes beyond positive integers.

These are “top results,” said Matthew Emerton of the University of Chicago. “They reveal a few fundamental phenomena from number theory, and we are just beginning to understand what they are.”

Needle in vacuum


One of the sides of the Langlands bridge is concentrated on almost the simplest equations that can be written down: these are Diophantine equations, or combinations of variables with exponentials and integer coefficients, for example, y = x 2 + 6x + 8, or x 3 + y 3 = z 3 . For millennia, mathematicians have tried to figure out which combinations of integers satisfy a particular Diophantine equation. Basically, their motivation is based on the simplicity and naturalness of this issue, but recently, part of their work has received an unexpected continuation in areas such as cryptography.

Since ancient Greece, mathematicians have known a way to find integer solutions of Diophantine equations with only two variables and no degrees greater than 2. However, in the case of higher degrees, finding integer solutions is by no means a simple matter - starting with elliptic curves. These are equations with y 2 to the left of the equal sign and a combination of terms with a maximum degree of 3 to the right, for example x 3 + 4x + 7. Guy said that compared to equations with lower degrees, these are „ a radically more complex problem. "

On the other side of the bridge live objects called automorphic forms, which are similar to coloring tiles with a very high degree of symmetry. In the cases studied by Wiles, the tiles may resemble something similar to Escher's mosaic , where the fish or angels with demons shown on the disks decrease as they approach the border. In the more general Langlands universe, tiles can pave a three-dimensional ball or other figure in higher dimensions.

These two types of mathematical objects are completely different from each other. Nevertheless, in the middle of the 20th century, mathematicians began to reveal deep relationships between them, and by the beginning of the 1970s, Robert Langlands of the Institute for Advanced Studies expressed the hypothesis that Diophantine equations and automorphic forms can be correlated in a certain way with each other.


Robert Langlands, who put forward the hypothesis of compliance 50 years ago, gives a lecture at the Institute for Advanced Studies in Princeton, New Jersey, in 2016.

Namely: both in Diophantine equations and in automorphic forms there is a natural way to generate infinite sequences of numbers. With Diophantine equations, the number of solutions in modular arithmetic can be calculated (it can be represented as numbers located on the clock face; for example, in the case of the 12-hour dial, 10 + 4 = 2). And for such automorphic forms that appear in accordance with the Langlands, you can get an endless list of numbers similar to the levels of quantum energy.

If we use modular arithmetic based only on primes, then, according to Langlands, these two kinds of sequences will coincide in a stunningly wide range of different conditions. In other words, for any automorphic form, its energy levels control the modular sequence of a Diophantine equation, and vice versa.

This connection is “weirder than even telepathy,” Emerton said. “The way these two sides communicate with each other seems amazing and unbelievable to me, although I have been studying this phenomenon for more than 20 years.”

In the 1950s and 1960s, mathematicians found the first signs of the existence of this bridge in one of the directions: how to move from certain automorphic forms to elliptic curves with coefficients that are rational numbers (fractions consisting of integers). Then, in the 1990s, Wiles, with Taylor, found another direction for the bridge for a particular family of elliptic curves. Their result automatically produced a proof of Fermat’s Great Theorem, since mathematicians had already shown that if it were incorrect, then at least one of these elliptic curves would not have a corresponding automorphic form.

Fermat’s great theorem was far from the only discovery that followed from the construction of this bridge. For example, mathematicians used it to proveThe Sato-Tate hypothesis , a problem of several tens of years of age related to the statistical distribution of the number of modular solutions of an elliptic curve, as well as to prove the hypothesis regarding the energy levels of automorphic forms, which was expressed by the legendary mathematician of the beginning of the 20th century Srinivasa Ramanujan Iyengor .

After Wiles and Taylor published their findings, it became clear that their method was still full of possibilities. Soon, mathematicians realized how to extend it to elliptic curves with rational coefficients. Later, mathematicians figured out how to cover coefficients with simple irrational numbers, such as 3 + √2.

But what they did not succeed in was to extend the Taylor-Wiles method to elliptic curves with complex coefficients, such as i (√-1) or 3 + i, or √2i. Also, they could not cope with diophantine equations with powers much more than elliptic curves. Equations with degrees 4 on the right side of the equal sign instead of 3 were easily solved using the Taylor-Wiles method, but as soon as the degree increased to 5, the method already stopped working.

Mathematicians gradually began to realize that the problem with these two natural extensions of the Langlands bridge was not just to come up with a slight improvement to the Taylor-Wiles method. Apparently, the obstacle was fundamental.

These were “just the following examples that occurred to me,” Guy said. “But they told you: No, these things are hopelessly out of reach.”

The problem was that the Taylor-Wiles method finds an automorphic form corresponding to the Diophantine equation by successively approximating it using other automorphic forms. However, when complex numbers or a power higher than the 4th occur in the coefficients of equations, there are very few automorphic forms - so few that almost any automorphic form will most likely not have the nearest automorphic forms that could be used to approximate it.

Under Wiles, the automorphic shape we need is similar to a “needle in a haystack, but this stack does exist,” said Emerton. “And this can be compared with a stack of metal filings, to which you bring a magnet - the filings are aligned and point to the needle you need.”

However, in the case of complex coefficients or degrees of a higher order, according to him, it more "resembles a needle in a vacuum."

Flight to the moon


Many of today's number theory experts were growing up at a time when Wiles came up with his proof. “This was the only example of mathematics that I saw on the front pages of newspapers,” recalls Guy, who was 13 years old at that time. “It inspired many people, they wanted to figure it out, and as a result, it was for this reason that they started working in this area.”

Therefore, when in 2012 two mathematicians - Frank Kalegari of the University of Chicago and David Gerati (now a Facebook researcher) - proposed a way to overcome the obstacle that did not allow expanding the Taylor-Wiles method, this idea provoked rave reviews from a new generation of number theory experts.

Their work demonstrated that “this fundamental obstacle that impeded our progress was not an obstacle at all,” Guy said. He explained that, in fact, the apparent limitations of the Taylor-Wiles method suggest that “you only felt the shadow of the real, more general method presented to us by Calegari and Gerati.”


David Geraty at Boston University in 2015

In cases where an obstacle suddenly arises, automorphic forms live on tiles of higher dimensions than the two-dimensional Esher-style tiles that Wiles studied. In these higher-dimensional worlds, it is uncomfortable that automorphic forms are very rare. But tiles of higher dimensions often give a richer structure than two-dimensional ones can offer. Kalegari and Gerati came up with the idea of ​​using this rich structure to compensate for the lack of automorphic forms.

More precisely, for each specific automorphic form, you can use the “coloring” of its tiles as a measuring tool that can calculate the average color of any portion of your chosen tile. In a two-dimensional situation, automorphic forms, in fact, are the only such measuring tool available. But the higher dimensions tiles have new tools, the so-called torsion classes, and with their help, each tile section can be assigned not the average color, but the number from modular arithmetic. And such classes of torsion are a dime a dozen.

Kalegari and Gerati suggested that for some Diophantine equations it can turn out to find the corresponding automorphic form through approximation not by other automorphic forms, but by twisting classes. “This idea of ​​them turned out to be fantastic,” said Karajani.

Kalegari and Gerati presented a scheme for constructing a much wider bridge from Diophantine equations to automorphic forms in comparison with what Wiles and Taylor built. However, their idea could not be considered a full-fledged bridge. To make it work, first it was necessary to prove three large theorems. According to Kalegari, this can be compared with the fact that their work with Gerati describes the flight to the moon, if only there is a spaceship, rocket fuel and spacesuits. And these three theorems were “perfect out of our reach,” said Kalegari.

In particular, the method of Calegari and Gerati demanded the presence of a ready-made bridge going in the other direction, from automorphic forms to diophantine equations. And this bridge was supposed to combine not only automorphic forms, but also twisting classes. “I think many people considered this a hopeless task when Calegari and Gerati first described their program,” said Taylor, now at Stanford University.

Less than a year after the publication of the work of Kalegari and Gerati, Peter Scholze is a young genius from the University of Bonn who received the Field Prize, the highest award for mathematics, amazed specialists in number theory, figuring out how to switch from twisting classes to the side of Diophantine equations in the case of elliptic curves, whose coefficients are simple complex numbers like 3 + 2i or 4 - √5i. “He did a lot of amazing things, but this is probably his most amazing achievement,” Taylor said.


The mathematician Peter Scholze

Scholze proved the first of the three theorems of Calegari and Gerati. And the couple of subsequent joint work by Scholze and Karayani came very close to proving the second theorem, demonstrating the presence of the correct properties at the bridge found by Scholze.

There was a feeling that this program could be easily mastered, so in the fall of 2016, Karajani and Taylor organized, according to Kalegari, the “secret workshop” at the Institute for Advanced Studies, aimed at achieving further progress. “We occupied one audience there and didn’t let anyone in,” Kalegari said.

After a couple of days of preparatory conversations, the workshop participants began to understand how to simultaneously deal with the second theorem and get around the third. “And maybe within a day after formulating all the tasks, we all solved them,” said Guy, one of the project participants.

The remainder of the week, the participants devoted to a detailed study of various aspects of the evidence, and over the next two years formalized their discoveries in the work.authorship of ten people - such an amount is unheard of for works on number theory. In fact, their work establishes the existence of a Langlands bridge for elliptic curves with coefficients from any number system composed of rational numbers and simple irrational and complex numbers.


Anna Karayani and Richard Taylor

"The workshop was organized mainly in order to understand how close you can get to the solution," said Guy. “I don't think any of us expected us to prove everything.”

Continuation of the bridge


Meanwhile, another story was unfolding related to the continuation of the bridge beyond the elliptical curves. Calegari and Guy worked with George Boxer (now working at the Higher Normal School in Lyon, France) on cases where the highest degree of Diophantine equations is 5 or 6 (rather than 3 and 4, as already known cases). However, three mathematicians were stuck at a key point in their proof.

And then, the very next weekend, after holding the “secret workshop,” Vincent Pilloni of the Higher Normal School published a paper showing how to get around this very obstacle. “Now we need to slow down our work and begin cooperation with Pilloni!” - So, according to Kalegari, three researchers immediately told each other.

Within a few weeks, four mathematicians solved this problem, although it took a couple of years of work and almost 300 pages of a detailed description of ideas. Their work , as well as the work of authorship of 10 people, was published on the Internet in December 2018, with a difference of four days.


Frank Calegari, Toby Guy and Vincent Pilloni

“This is a very serious achievement,” commented Emerton on these two works. He called them and the building blocks that preceded them a “work of art."

Although these two works, in fact, prove that the mysterious telepathic connection between Diophantine equations and automorphic forms is transferred to the new conditions, there is one catch: they do not build an ideal bridge between two mathematical shores. The works only state the “potential presence of automorphism”. This means that each Diophantine equation has a corresponding automorphic form, but we do not know for sure whether this automorphic form lives on that part of the continent where, according to scientists, it should be located. However, potential automorphism is enough for many applications - for example, for the Sato-Tate hypothesis about statistics of modular solutions of Diophantine equations, whose operability on a much wider landscape than before, could be proved by ten authors.

Mathematicians are already beginning to understand how to improve these results with potential automorphism. In October, three mathematicians — Patrick Allen from the University of Illinois at Urbana-Campaign, Chandrasekar Hare from the University of California at Los Angeles and Jack Thorne from the University of Cambridge — proved that a significant part of the elliptic curves considered in the work with 10 authors have bridges coming just to the right places.

Bridges with such higher accuracy in the future may allow mathematicians to prove a whole bunch of new theorems, including a generalization of Fermat’s Great Theorem a century ago. The latter claims that the equation of this theorem will still have no solutions, even when instead of x, y and z we will substitute not only integer values, but combinations of integers and an imaginary unit .

Two work under the Calegari-Gerati program provides important evidence of the concept being operational, said Michael Harris of Columbia University. They, he said, "demonstrate that the method is applicable in a wide range."

And although new works connect bridges to much wider sections of the Langlands continents than before, they still leave vast territories uncharted. From the Diophantine equations side, these equations include all equations with degrees greater than 6, as well as equations with more than two variables. On the other hand, uncharted territories belong to automorphic forms living on more complex symmetrical spaces than those studied to this day.

“Today, this work represents the pinnacle of success,” Emerton said. “But at some point they will be regarded as one of the steps towards achieving the goal.”

Langlands himself never thought about twisting, studying automorphic forms, so one of the difficult tasks for mathematicians will be to find a unified view of these two different approaches. “We are expanding our range,” Taylor said. “We went off the road in some way with the Langlands and we don’t know where we were going.”

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