The top student must be diligent.

Chapter 278 Ating Conjecture
Time passed quietly, and what probably surprised everyone was that the influence of this major event in the mathematics world has continued to expand.

It has even gone beyond the academic circle and affected the international political world.

Even politicians from various countries have expressed great concern about this matter.

For example, in the United States, their president personally came to the Department of Mathematics at Princeton University and met with many scholars in the Prime Number Pioneer Project. He also expressed his condolences to them and proposed that the US government was willing to provide them with considerable convenience to help them conduct mathematical research more without worries in the process of trying to challenge the Riemann hypothesis.

Of course, the president has made no secret of why he is doing this.

In a speech he gave at Princeton, he said directly: "I have always held the view that the United States is the country with the most geniuses in the world and the country with the greatest scientific research potential in the world. No country and no person can compare with our scientists!"

"Especially when our scientists come together, no one in the world has more potential than us!"

"It reminds me of the great scientists who were secretly working on the atomic bomb at Los Alamos, Oppenheimer, Fermi, etc., and the scientists who are now trying to build our own nuclear fusion reactor in Utah. They are undoubtedly so great that I feel like crying every time I think of these things."

When he said this, the president acted as if he had tears in his eyes. He wiped his eyes and looked very moved.

However, he then cheered up again and said passionately: "So now, I also believe that the mathematicians of the Prime Number Pioneer Project can successfully achieve their ultimate goal and completely solve the Riemann hypothesis."

"As for that Xiao Yi, he is just one person after all. He will definitely lose!"

"We not only have the smartest people, but also the most talented team. Any mathematical problem is just a simple problem."

"So, let's fight! Fight! Fight!"

After saying this, he waved his fist like the world-famous painting that was taken after the failed assassination attempt, expressing his belief to the people present.

The President's remarks were indeed inspiring, and when related news broke, they also attracted quite a lot of praise from "Wolf Warriors" in the United States.

After all, due to the issues concerning nuclear fusion over this period of time, there has been a growing consensus in their country that their scientific potential is no longer as good as before, and they are even facing the possibility of being surpassed by China.

So the president probably wants to do this to boost the scientific research atmosphere in their country and hope to strengthen people's confidence in this regard.

However, given the president's identity as a comedian, there is still a lot of fun. For example, there is a video that captured the president's speech. The expressions of many mathematicians below were almost as if their ears were filled with magic sound and they could not hear a word. Probably in their eyes, this was a group of top mathematicians listening to a "monkey" who knew nothing about mathematics encouraging them to solve the Riemann hypothesis.

Among them, Terence Tao's expression was particularly eye-catching, which made many netizens burst into laughter.

Terence Tao already has a bad relationship with the president, and being forced to listen to his speech is already a very uncomfortable thing, let alone listening to this kind of thing.

Netizens directly gave him an emoticon: Stop talking, stop talking.

Tao turned around and started complaining online.

[I really don't understand why this guy insists on paying attention to such things. This is our own business in the mathematics community, but he wants to elevate this matter to the political level. This is simply an insult to our mathematics!
Now I even kind of hope that Xiao Yi can succeed ahead of time and then slap this guy hard in the face!

Of course, considering that this is the Riemann hypothesis, I will not give up, and neither will any of us. Come on! Xiao Yi, now we know that your paper is your response, so we will do our best to make this challenge even more epic! 】

……

"Well... this guy still has a young mentality."

Xiao Yi naturally also saw what Terence Tao said, and couldn't help but sigh.

This year, Terence Tao is 52 years old and already an old man, but his mentality is always very young.

On the other hand, perhaps because his mentality became more rational as his ruthless learning BUFF level became higher and higher, he felt more and more like an older man.

“Maybe sometimes, I should also find some things for young people to do?”

Xiao Yi laughed at himself.

At this time, Wang Hao, who was right beside him, said, "Maybe you can try to have a relationship? Isn't dating something that young people do?"

Xiao Yi's eyebrows suddenly raised, he turned his head to look at Wang Hao, and asked: "If it's not a young person's love, then it's not called love?"

Wang Hao smiled and said, "If it's not a love affair between young people... I think it's probably a love affair for the purpose of passing on the family line."

Xiao Yi smiled and said, "If I were in a relationship, I would probably do it for the sake of continuing the family line. Otherwise, given my current situation, do you think I really have time to enjoy the experience of young people in love?"

Wang Hao spread his hands and said, "If you want to take a leave to date, I think the leaders will agree."

Xiao Yi immediately narrowed his eyes and looked at Wang Hao, saying, "Don't tell me that you have also received tasks from some leaders to persuade me to get married or something?"

Wang Hao immediately waved his hands and said, "No, absolutely not! I am really just thinking about you."

"What is your relationship with Wang Li?"

Xiao Yi asked again.

"Uh..." Wang Hao didn't know whether to laugh or cry: "Brother Li and I really have nothing to do with each other. As you know, Brother Li's hometown is Qinxi, and I am from Anhui Province."

"Brother Li has already called him." Xiao Yi waved his hand and said, "Then don't say anything. Even if you two had no relationship before, you are related now."

Wang Hao shrugged helplessly.

Are you kidding? One is Xiao Yi’s guard and the other is his assistant. It’s hard not to know each other!

Xiao Yi didn't say anything more. He lowered his head and looked at the pile of draft papers on his desk again, and continued his research on the Riemann hypothesis.

However, for now, his main research focus is on the development of analytical methods for elliptic inverse curves.

The more he studied, the more he discovered that this method he had created by accident had many possibilities.

Starting from the elliptical angle and radiating out, it can gradually cover a considerable area.

Whether it is number theory, algebraic geometry, representation theory, modular forms, or further refinement to various automorphic forms, Dirichlet L-functions, etc., corresponding places can be found.

To the extent that, the direction of his real exploration now is no longer the Riemann hypothesis, but the Langlands program.

Langlands program, not the geometric Langlands program.

Moreover, what he is now dealing with is not the superficial level of the Langlands Program, but is directly related to the most important conjecture of the Langlands Program, the functor conjecture. The functor conjecture is an important prerequisite for the realization of the Langlands Program, mainly because it has shown a very huge role in the field of representation theory, number theory and geometry.

In representation theory, it provides a unified framework to understand the relationship between representations of different groups; in number theory, it connects automorphic representations with many important number theory objects, such as L-functions, Galois representations, etc.; in geometry, it has inspired many profound thoughts and ideas, such as the development of the geometric Langlands program, which was inspired by the functor conjecture.

Once the functor conjecture can be proved, it will be of great help to the realization of the Langlands Program.

However, judging from the current research status, it is still a long way to prove the functor conjecture. Judging from Xiao Yi's current achievements, the thing he is most likely to accomplish is the Artin conjecture.

The Artin conjecture is a classic example of a functor conjecture.

If Artin's conjecture can be proved, it will be of great help to the proof of the functor conjecture.

However, what Xiao Yi is more concerned about now is the role of proving the Artin conjecture in proving the Riemann hypothesis.

Xiao Yi made a simple deduction on the draft paper and was able to easily come up with a relationship equation.

"Well... to put it simply, before, because the classical Riemann hypothesis does not correspond to any Galois representation, even if the Artin conjecture is proved, it will not be of much help in proving the classical Riemann hypothesis. Instead, it is very helpful in proving the generalized Riemann hypothesis of Artin L-function."

"But just by hearing the name you can tell it's very helpful."

Xiao Yi smiled.

The generalized Riemann hypothesis refers to various generalizations of the Riemann hypothesis. There are many types, and the Riemann hypothesis of Artin L-function is just one of them.

However, for the most classic Riemann hypothesis, the result of Artin's conjecture is completely unhelpful.

But now, with the help of elliptic anticurve analysis, even though the classical Riemann hypothesis does not have a corresponding Galois representation, Xiao Yi can still make a connection between the two from another elliptic form.

And so...

If the Artin conjecture can be proved, it will be of great help in proving the Riemann hypothesis!
In fact, it is equivalent to coming directly to a place extremely close to the final proof of the Riemann hypothesis.

It's like a shortcut.

Of course, this shortcut is not so easy to take. After all, its prerequisite is to prove the Artin conjecture first.

After all, the difficulty of Artin's conjecture is there.

Although Artin's conjecture was not listed as one of the seven Millennium Problems, the difficulty of proving it is no less than that of the seven Millennium Problems.

However, he has solved the Millennium Problem before. Since he dared to come up with such an idea, it means that he already has the idea of ​​proving Artin's conjecture.

still the same.

Analysis of elliptical recurve!

The possibilities of elliptical recurve analysis are endless.

Even in Artin's conjecture, it can play an extremely huge role!

Xiao Yi raised his eyebrows slightly.

Now, a lot of ideas have emerged in his mind, and each of them can become a way to prove Artin's conjecture.

Therefore, with regard to Liang Qiushi's reply to Bihu in which he was praising him, one point he did not agree with was that the analysis of elliptic anticurve was not really an ordinary paper among his many papers, but a very important one.

In other words, there is still not much research on the analysis of elliptic anticurves in the mathematical community. If it were not for the paper he published, it would probably take some time for people to truly realize that there are more ingenious applications of the analysis of elliptic anticurves.

Without further ado, he began his in-depth research.

"First, we give the Galois representation of the elliptic curve."

"Given an elliptic curve E over the field Q of rational numbers, consider its Tate module T(E), which is a Z-module generated by all-bisected points of E. The Galois group Gal(Q/Q) naturally acts on T(E), which gives a Galois representation."

【ρ:Gal(Q/Q)→GL(2,Z)】

"Then we need to use the L function."

Associated with the Galois representation ρ above is the L-function L(s, E) of the elliptic curve E, which can be defined by the Euler product.

【L(s, E)=∏(p) 1/(1-a_p p^(-s)+p^(1-2s)), where p is all prime numbers (E can be easily restored), and a_p is the trace of E in the restoration modulo p】

There are more and more derivations on the draft paper, and the elliptic curve itself can play a very important role in proving Artin's conjecture.

For example, Taniyama-Shimura's theorem can itself be seen as Artin's conjecture in an elliptic context, and the Artin conjecture that Xiao Yi is going to prove now can be regarded as a more general form of the Artin conjecture.

Therefore, the proof process of Taniyama-Shimura's theorem can also serve as a reference in the process of proving Artin's conjecture.

"Then, using the Langlands correspondence method to study it is the best angle."

Xiao Yi raised his eyebrows and chose this angle from the various ideas that emerged in his mind.

Since it involves the problem of the Langlands Program, it would be very appropriate to solve it using the methods of the Langlands Program.

……

Just like that, time passed quietly.

Whether you want to solve the Riemann hypothesis or the Artin conjecture, it is destined to be a task that requires a long time and energy.

This is the Long March of mathematics, and there are only a handful of people, or even a dozen people, who can take part in it.

There may even be some people among them who are just there to make up the numbers.

Just like in the past, there will only be one person who can ultimately solve a problem.

……

(End of this chapter)