The top student must be diligent.

Chapter 290: Riemann Hypothesis Report (II)
The report will start at 400 a.m., and according to Xiao Yi's request, this report may last at least four hours. After all, after including the analysis of elliptic anticurves, the total number of pages of the paper will exceed pages. By then, it will be a question whether it can be fully completed in four hours.

Therefore, about two hours will be reserved at noon for those present to rest. At the same time, the hotel next door has also made preparations to entertain the mathematicians present. These are also specially prepared by the Feicheng Municipal Government for this report. Just like when Xiao Yi held a report on the proof of the NS equation, the Feicheng Municipal Government also specially held a mathematics exchange meeting and prepared several days of banquets for the mathematicians who attended Xiao Yi's report at that time.

We even discovered some talents and kept them in the country, several of whom have already settled in China.

This time, maybe we will have a chance to retain a few talented people.

……

The time had already come to half past nine and the conference hall was already full of people.

These seats all need to be reserved, and before the reservation, invitations will be sent to those well-known mathematicians. As long as the mathematicians accept the invitation, they will be arranged to sit in the front so that they can listen to Xiao Yi's talk at a closer distance and it will be more convenient for them to ask questions afterwards.

After these well-known mathematicians have confirmed the invitation, the remaining positions will be opened to other mathematicians.

“I’m really here again.”

Sitting in the front seat, Terence Tao sighed.

Next to him, Qiu Chengtong also said with a smile: "It seems that the place where we sat last time was also here."

"It is indeed here." Fefferman nodded, "I remember it."

"That's great, I'm back to visit an old place," said Tao. "I think if these conference halls had their own consciousness, this one would be very happy to be able to hold two heavyweight seminars in mathematics in succession."

"Yes." Qiu Chengtong also nodded.

"It's like a Steinway grand piano. If a piano master plays it, it can fully display its timbre. But if an ordinary person plays it, no matter how excellent its timbre is, it will only be covered in dust."

In the past, Qiu Chengtong would never have thought that he would be able to come to Fei City, China to attend an international mathematics academic conference.

I never thought that I would come here twice.

Ok……

Considering Xiao Yi's current age, there might be a third, fourth, or even more times in the future.

After all, although Xiao Yi has solved almost three of the seven problems of the millennium, excluding the Poincaré conjecture, there are still three that have not been solved.

Maybe Xiao Yi will solve the remaining three problems by then.

However, in his heart he was secretly looking forward to the day when Xiao Yi would solve the P=NP problem, because he was also very curious about what method should be used to solve this problem.

"But then again, did Xiao Yi choose to hold this report meeting today? Was it really not intentional?"

Beside him, Fefferman couldn't help but speak again.

Tao Zhexuan smiled and said, "Maybe it was intentional?"

Why do you say that?
That’s because the presidential election is just a few days away.

As a result, he now holds lectures on such days.

"I heard that our respected President has been cursing Xiao Yi in his office these days." Tao Terence said with a smile.

Anyway, he is happy when the current president is angry.

Feferman also laughed, full of joy.

……

Apart from them, mathematicians sitting in other places were also talking about their own things.

As for the mathematicians who were not invited, they were also discussing whether Xiao Yi's lecture would be a success, or marveling at the bigwigs sitting in front.

And now these big guys have come from far away just to listen to a Chinese person's report for a few hours, which also makes many Chinese people present feel honored.

At this moment, they were also quite grateful in their hearts because Xiao Yi was their compatriot.

……

Time passed quietly, and finally, it was approaching ten o'clock.

"Professor Xiao, there are still three minutes before you go on stage."

The staff came to the lounge and spoke respectfully to Xiao Yi inside.

"Okay, thank you. I understand."

Xiao Yi nodded, then stood up, walked to the mirror and looked at himself.

Ok……

He is still young, but he has changed a lot compared to his past.

Although he has a very high IQ, his body is still a human body. As cells continue to divide, it will eventually age.

What will I become at that time?

Suddenly he sighed.

Then he shook his head, returned to his seat, and waited for ten o'clock.

Soon, it was ten o'clock.

He stood up, walked to the door, and stepped out.

There was no host for this conference, he was both the host and the speaker.

At this time, the conference hall had become quiet and no one spoke. As footsteps sounded from the stage, everyone present immediately turned their eyes to the backstage exit.

Soon, a figure familiar to many people present appeared.

Applause suddenly broke out, and many mathematicians in the front row stood up to welcome this God of Mathematics.

Although the correctness of his proof remains to be determined, it does not prevent them from recognizing him.

"He's still so young."

Deligne in the audience exclaimed.

"Yes." Bombieri also nodded.

Finally, Xiao Yi walked to the center of the podium. Facing the many familiar faces present, although he hadn't seen them for a long time, he smiled and waved.

"Friends, long time no see, I miss you so much."

Everyone present smiled, it had indeed been a long time since they last saw each other.

"Then please take a seat."

Xiao Yi pressed his hands together, and then the audience sat down one after another.

Then, Xiao Yi said: "First of all, thank you all for coming to attend my report meeting."

"Today's lecture will be very long, and I will try my best to explain to you in a more comprehensive way how I proved the Riemann hypothesis."

"Well, let's not waste any more time talking and get started."

Xiao Yi turned around, walked to the extra-long blackboard behind him, and wrote the Riemann hypothesis on it.

"I will not elaborate on the origin and statement of the Riemann hypothesis." "All non-trivial zeros of the Riemann zeta function fall on the straight line Re(s)=1/2, which is the goal we want to prove."

"And it represents one of the goals that we number theorists pursue, which is to make more accurate predictions about the distribution of prime numbers."

"Of course, many years from now, will we still be able to find a general formula that can directly generate prime numbers?"

Xiao Yi smiled, then changed the subject and said, "Okay, then let's start with the first step to prove the Riemann hypothesis."

"Analysis of the elliptic recurve."

Xiao Yi wrote these words on the blackboard again.

"Elliptic inflection analysis is one of the most core methods in my entire proof. He provided a lot of help, the most important of which was helping us prove the Artin conjecture and helping us give the Riemann hypothesis the properties of Galois representation itself."

“I believe many of my friends have already noticed this when reading my paper.”

"Then, let's first give a more comprehensive explanation of the analytical method of elliptical inverse curves."

Then, Xiao Yi began to write the deduction process of the analytical method of elliptic inverse curve on the blackboard.

The mathematicians present also watched quietly.

Although they have also reached a relatively in-depth understanding of the analysis of elliptic anticurves, they are also very happy to listen to Xiao Yi's more in-depth explanation of this, which may bring them a lot of inspiration.

And sure enough, as the creator of the analysis of elliptic recurve, Xiao Yi's simple demonstration revealed many detailed thoughts that could not be described in the paper.

"…The most important role of elliptic inverse analysis is that it successfully helps us connect the Riemann hypothesis with the Galois representation, and the most important step is in the fourth paper, "CM Elliptic Curves and Hecke Characteristic"."

"CM elliptic curves are a special type of elliptic curve. We can easily associate them with the fact that their complex multiplication structure brings special properties to their L-functions. At this point, we can try to construct them through this elliptic curve, thereby constructing an ellipse that can be analyzed using elliptic inverse analysis..."

Xiao Yi slowly explained and deduced, and finally, he revealed the core ideas of the paper "CM Elliptic Curve and Hecke Characteristic".

It also made the mathematicians present sigh with emotion.

It was Xiao Yi's way of thinking that made them always feel deeply impressed by him.

“…Specifically, for a CM elliptic curve E, its L-function L(s, E) can be decomposed into the product of two Riemann Zeta functions.”

[L(s, E)=ζ(s) L(s, χ)]

"where χ is a Dirichlet characteristic."

"And this decomposition connects the Riemann Zeta function to the L-function of the elliptic curve."

"Then we can naturally make a connection at this time. If we can associate L(s, E) with a Galois representation, then through the above decomposition, we can also associate ζ(s) with a Galois representation."

"In this way, we have achieved a key step."

"Therefore, the Hecke characteristic theory came into our sight."

"The Hecke character is a fundamental and powerful tool in the theory of modular forms. The basic idea is that given a modular form f and a positive integer n modulo N, we define a new modular form Tnf, called the nth-order Hecke character of f."

“For a CM elliptic curve E, we can construct a special Hecke characteristic λ_E that encodes the complex multiplicative structure of E into a Galois representation.”

"Specifically, λ_E is a representation from the Galois group Gal(K/K) of a number field K to GL_2(C), which satisfies the following properties..."

As Xiao Yi told the story, time passed quietly.

Although what he is talking about now is from the fourth paper, in fact, this problem should have been solved at the beginning, because its most important purpose is to give the Riemann hypothesis the properties of Galois representation, so that the Artin conjecture proved later can be applied to the proof of the Riemann hypothesis.

"As expected, I still have to listen to Xiao Yi's own story to understand the details."

While listening, Terence Tao sighed.

He has now received a lot of inspiration, and some of the problems that had existed in his mind before in this regard have been solved at this moment.

This is the significance of listening to reports, which allows them to understand the author’s own ideas.

In this way, the first paper and the fourth paper were combined and finally, Xiao Yi basically finished explaining the analysis of elliptic recurve.

And time, an hour has passed.

Generally speaking, an academic conference should last about an hour, and at least some time should be provided for the scholars present to rest.

But at this moment, no scholar wanted to take a break and just wanted to continue listening.

"Now, the key points of elliptic recurve analysis have been explained, but it should be noted that the influence and role of elliptic recurve analysis will also be present in many of the following steps."

“Then let’s move on to the second paper, which is the discussion of high-dimensional modular curves, or generalized modular curves.”

Hearing this, all the mathematicians present suddenly became serious.

Generalized modular curve!
This is the most important thing in the entire paper!

Before reading the paper, when they saw it, they were only slightly interested in how Xiao Yi performed high-dimensional processing on modular curves.

But after reading it, they all clearly realized that this generalized modular curve can be regarded as a new kind of mathematics!
It is the most critical new theory in the entire proof!
The reason why people are looking forward to the proof of the Riemann hypothesis is not only because of the chain reaction brought about by the proof of the Riemann hypothesis, which can make many propositions based on the Riemann hypothesis become real theorems, but also because of the expectation that new theories will be born in the process of proving the Riemann hypothesis.

And this generalized modular curve is the new theory they have been expecting, and its significance has lived up to their expectations!

"Generalized modular curves come from modular curves."

"The reason I thought of it was mainly because I wanted to connect the extended L-function and the modular L-function in the process of proving Artin's conjecture."

"If we can establish this connection, then we can study the extended L-function through the properties of modular forms, and then study the arithmetic properties of elliptic curves."

"So I thought of the content of Weil's conjecture, which has similar properties to the extended L-function."

"More precisely, for an elliptic curve E defined over the field of rational numbers, its extended L-function L(s, E,) seems to satisfy such a functional equation."

【Λ(s,E,)=ε(E,)Λ(1-s,E,)】

"Where Λ(s, E,) is the complete L-function obtained by multiplying L(s, E,) by some Gamma factor, and ε(E,) is a constant called sign."

"So it's easy to imagine that the extended L-function might be related to the Zeta function of some geometric object."

"Then I started experimenting with all kinds of geometric objects I could think of."

“But to be honest, in the first two months, I didn’t find the geometric objects I wanted, including modular curves. I tried them, but when I first tried them, they didn’t make me think more deeply about them.”

Xiao Yi spread his hands and said.

When the mathematicians heard him say this, they all felt that the God of Mathematics before them was a little more real.

It turns out that God also encounters situations where the answer is right in front of him but he doesn't notice it.

"But fortunately, one day when my partner and I were at an amusement park, I saw a roller coaster and the structure of a Ferris wheel, which gave me inspiration."

After hearing the story he told, everyone present immediately showed expressions of interest and surprise.

He was interested in gossip, but was surprised that the amusement park could also help Xiao Yi make mathematical associations?

Really not cheating?
But then, Xiao Yi began to tell them how he gradually associated the structure of the amusement park with the modular curve, and began to try to discuss this from a higher dimension.

The audience present gradually immersed themselves in his narration.

……

(End of this chapter)