The top student must be diligent.

Chapter 291 Riemann Hypothesis Seminar (Part )
"…In short, it was the geometric trajectories I saw in the amusement park that gave me some inspiration."

"Geometry exists naturally in our world, and mathematics is known as the language of the universe. So it might be a good idea to use these things to inspire us."

Xiao Yi smiled slightly: "This is just a small suggestion I brought to you."

Many mathematicians present immediately nodded in approval.

Although this method still sounds a bit too magical to them, it does not prevent them from learning from Xiao Yi.

Now, as long as it is a method recommended by Xiao Yi, they are willing to try it, maybe it will be suitable for them.

"Well, now that I have explained to you how I came up with the idea of ​​developing modular curves from higher-dimensional situations, let's move on to discussing how I ultimately derived generalized modular curves."

"At the beginning, I tried modular curves, but it was easy to find that although modular curves provide a geometric framework for studying extended L-functions, they cannot fully explain all the characteristics of extended L-functions. In particular, for some types of extended L-functions, their special values ​​do not seem to be consistent with the geometry of modular curves."

"Then, I have to thank my research in theoretical physics for giving me some inspiration."

"We all know that in physics, some high-dimensional geometric spaces are used to study some physical phenomena, such as Calabi-Yau manifolds, so this gave me some inspiration."

"So, for this high-dimensional modular curve, it should contain the usual modular curve as an independent case, but it should also contain more information to characterize the extended L-functions beyond the usual ones."

"So now we can simply give the definition."

"For an n-dimensional generalized modular curve, we denote it as X_f^(n), which is an n-dimensional complex manifold, it parameterizes a special class of n-dimensional Abelian varieties that have some modular properties similar to those of ordinary elliptic curves."

"Then we need to use some special tools to process it."

"So I thought of Shimura clusters and Siegel modular forms."

"For an n-dimensional Siegel modular form f, we define a Shimura variety Sh_f that parameterizes all n-dimensional Abelian varieties with modular properties described by f."

"In this way, we can prove that there is a natural isomorphism."

【X_f^(n) Sh_f】

……

Xiao Yi began to demonstrate on the blackboard how he proved this natural isomorphism.

However, for most mathematicians in the audience, they really couldn’t understand it.

How could Xiao Yi think of these things?
How did he come up with the idea of ​​using Shimura clusters and Siegel mold forms?

How do we give such precise structures and definitions?
Is this something humans can do?
They were all in a state of confusion.

For mathematics, finding the tools that can be used to solve problems is only the first step; how to use these tools is the second step.

Sometimes, even if they find the tool, they may not be able to successfully solve the problem with this tool. The main reason is that in the process of using it, they still have not found the "keyhole" that can well embed the tool into the problem. Therefore, the problem is still a problem and the tool is still there.

This kind of situation occurs quite frequently in the mathematics community.

Just like Andrew Wiles, when he first proved Fermat's Last Theorem, other mathematicians discovered a key error in his proof, so much so that he almost admitted failure.

But it was not until the end that he found a way to solve the problem from other existing mathematical tools and finally successfully completed the proof of the paper.

For example, Perelman, who proved the Poincare conjecture, mainly used a mathematical tool called the Ricci flow in his proof. Since the birth of this mathematical tool, the mathematical community has seen the great role this tool may play in proving the Poincare conjecture, but for a long time, mathematicians have not been able to successfully complete the proof.

It was not until later that Perelman found a way to embed the Ricci flow into the "keyhole" of the Poincare conjecture and finally completed the proof.

Therefore, finding the tools is only the first step. How to apply the tools is also a very critical step.

And now, Xiao Yi has shown an ability as if he had opened his clairvoyance. He can not only discover new tools such as generalized modular curves, but also find auxiliary tools to embed generalized modular curves into the "keyhole", namely Shimura clusters and Siegel modular forms.

Is this really not open?

"Oh my god...oh my god...oh my god..."

Below, many mathematicians looked at the steps demonstrated by Xiao Yi in amazement, and their hearts were filled with shock.

"Yeah, who says he's not God?"

Deligne said, shaking his head with emotion.

"I never thought that the content extended from the Weil conjecture could be expanded in this way."

Bombieri, who was standing next to him, spread his hands and said, "You just proved the Weil conjecture. What do you know about the Weil conjecture?"

Deligne shrugged his shoulders and said, "Yes, I completely agree with you."

On the other hand, Terence Tao also expressed his amazement.

"Did he really come to this conclusion without even trying anything? I...I just can't believe it..."

Fefferman on the side shook his head and said, "Even if he did make some wrong attempts, how long do you think he stayed on these mistakes? I think it might not even be a month."

"That's true..." Tao Zhexuan sighed.

If they were to take this step, God knows how long it would take them to try and make mistakes.

If they had not seen this paper, and only understood the definition of the generalized modular curve, and knew that the two methods of Shimura clusters and Siegel modular forms were also needed, if they were lucky, they might be able to complete this step in a month, but if they were unlucky, it might take several months or even a year.

Although they didn't think that with their abilities, they wouldn't be able to find the final answer in more than a year, but the possibility was definitely not zero. On the contrary, the possibility was even a little high...

In the past, they might have attributed this more to luck, with a dash of mathematician’s intuition thrown in for good measure.

But now, after listening to Xiao Yi's story, they began to doubt this.

Is it really?

Could it be that this trial and error process can actually be solved directly through mathematical ability alone?
But they obviously could not get answers to such questions. After all, they were not Xiao Yi, so they naturally could not understand what Xiao Yi's experience was like when he encountered such questions.

……

Xiao Yi had no idea what the mathematicians below the stage were thinking at this moment.

If they asked Xiao Yi, Xiao Yi would probably be a little confused about their asking this question.

After all, he really relies entirely on mathematical intuition, so he wouldn't understand why they would have such problems.

This situation is generally referred to as the "curse of knowledge."

Simply put, people who regard certain knowledge as common sense will think that other people who do not understand this knowledge are very stupid. A common situation is like when adults help children with homework, they will feel very confused because the children can never understand a problem that seems very simple to adults, and they may even think that the children are very stupid.

Now, these top mathematicians are like children, and Xiao Yi is an adult. The latter can rely on their mathematical intuition to save a lot of trial and error time, while the former find it difficult to understand what kind of mathematical intuition can help Xiao Yi find the answer so accurately.

Of course, Xiao Yi was not clear about their problem at the moment and continued to explain his proof.

"...So we can get a theorem like this -"

"Let E be an n-dimensional Abelian variety and f be an n-dimensional Siegel modular form. If the modular properties of E are described by f, then the extended L-function L(s, E,) of E is equal to the Zeta function ζ(X_f^(n), s) of the generalized modular curve X_f^(n)."

"This theorem significantly extends previous results on elliptic curves and modular curves by showing that generalized modular curves provide a natural geometric framework for uniformly treating Abelian varieties of various dimensions and their extended L-functions."

"Now we can fully study all types of extended L-functions."

"By associating each extended L-function with a generalized modular curve, we can use the geometric properties of the generalized modular curve, such as dimension, Betti number, Hodge structure, etc., to characterize the characteristics of the extended L-function..."

"Naturally, we can then move towards the final step of Artin's conjecture."

At this point, Xiao Yi paused, then looked at the time, and then said with a smile: "Well, now it's twelve o'clock, so according to the previous arrangement, it's lunch time. In the hotel next door, we have prepared a sumptuous meal for you. Everyone is welcome to come and taste it."

Everyone who was expecting Xiao Yi to explain how to prove Ating's conjecture suddenly wailed.

Although Artin's conjecture has become less important after the Riemann conjecture was proved, this is only relative. People are still extremely shocked that Xiao Yi was able to prove Artin's conjecture at the same time. So now the situation is like a cessation.

The only people who would be grateful to Xiao Yi for being able to "get off work" so punctually are probably the audience members who are challenging the limits of their bladders but are reluctant to leave.

Seeing Xiao Yi entering the backstage without even looking back, many audiences could only give up trying to save the relationship.

"Well, it looks like we'll have to wait two hours."

Qiu Chengtong said helplessly.

Fefferman stood up with a smile: "It's okay, it's Huaguo's tea break, I have been looking forward to it for a long time, and I still haven't forgotten what I ate here last time."

He patted Qiu Chengtong on the shoulder and said, "Qiu, I'm going to trouble you to introduce some delicious food to me again."

Qiu Chengtong smiled and said:
"No problem at all."

……

Afterwards, the audience left the place at the invitation of the staff and went to the hotel next door to enjoy lunch.

Of course, they did not forget at all that they would have to come back in two hours to listen to Xiao Yi talk about the next proof process.

As for the lunch, they were not idle at all. They discussed what Xiao Yi had just said, and exchanged their inspirations and gains. Even Fefferman, who had originally planned to enjoy the Chinese specialties, gave up these delicacies and joined their discussion.

Anyway, after the lecture in the afternoon, I can come back here and eat as long as I want.

In this way, during various exchanges, they were more and more impressed by Xiao Yi's brilliant ideas and wonderful methods of proof, and at the same time, they were looking forward to the second half of the report meeting.

Soon, before 14 p.m., all the audience returned to the scene again.

Even when queuing to enter, everyone complied with the order and no unexpected situations occurred, because no one wanted the report that started at 14 p.m. to be delayed because of such things.

At 14 p.m., the second half of the Riemann hypothesis lecture started on time.

Xiao Yi once again stepped into the venue and stood on the podium. Looking at the audience, he smiled and said, "Well, let's continue with our report."

"Now, let's officially begin the proof of Artin's conjecture."

What follows is a series of complex proof processes.

The key tools have been mastered, and the next thing to do is to actually apply all these tools to the proof process.

In between are several dozen pages of thesis content.

Of course, Xiao Yi omitted all these processes, which are probably equivalent to the words commonly used by mathematicians such as "obvious", "noted", "easy to get", etc.

However, the mathematicians present could completely understand it. After all, it was impossible for Xiao Yi to write out the entire proof process for them here. That was a total of more than 400 pages of paper content.

Therefore, what can be omitted is omitted directly, and only the key steps are retained.

This continued until the end of the proof of Artin's conjecture.

“We will be able to make a final judgment here.”

"Let f be an n-dimensional Siegel modular form and X_f^(n) be the corresponding generalized modular curve, then there exists a natural Galois representation -"

【ρ_f: Gal(Q/Q)→ GL_n(Z_)】

"This Galois representation makes it so that for any prime number p, the characteristic polynomial of the Frobenius element Frob_p under ρ_f is equal to the Zeta function ζ(X_f^(n)，T) of X_f^(n) at p."

"In this way, we have successfully established a connection between the geometric properties of generalized modular curves and the arithmetic properties of Galois representations."

"With this result, we were able to successfully transform the Artin conjecture into a question about Galois representations."

"Specifically, we have achieved such a result."

"Let E be an elliptic curve and L(s, E) be its Hasse-Weil L-function, then the following two conditions are equivalent."

"First, L(s, E) is a holomorphic function on the entire complex plane and satisfies a functional equation; second, there exists a modular form f such that the Galois representation ρ_E of E is isomorphic to ρ_f."

“…eventually, we can start trying to embed every elliptic curve into a generalized modular curve.”

"Now, we know that ρ_X comes from a Siegel modular form f, namely ρ_Xρ_f. Combining these two results, we have——"

【ρ_Eρ_X i_*ρ_f i_*. 】

"This suggests that ρ_E also comes from a modular form, namely the 'pullback' of f."

"This means that L(s, E) is integral and satisfies the functional equation. In summary, we have successfully proved the Artin conjecture."

Xiao Yi turned around, faced the audience, and said with a smile.

Everyone present exclaimed in amazement.

Artin guessed!

This problem, which originally seemed extremely difficult to them, was solved in this way, and even became the "preface" to the proof of the Riemann hypothesis.

At this moment, they didn't know how many times they had been shocked by Xiao Yi's proof process.

Exquisite, perfect, almost no loopholes can be found...

"And this generalized modular curve..."

Schultz murmured.

He studied arithmetic geometry.

The quasi-complete space he created back then can be regarded as an important breakthrough in arithmetic geometry, and can be applied to the study of various problems, especially in the fields of algebraic geometry and the Langlands program.

And now, the generalized modular curve created by Xiao Yi has made an even more powerful expansion of arithmetic geometry from another level. It is a true and extremely close combination of the methods of algebraic geometry and number theory. For the entire mathematical community, this can be regarded as a great innovation.

Not to mention the future, how much help this generalized modular curve may bring to solving other problems in mathematics, let’s just talk about now. Just the process of Xiao Yi thinking out this generalized modular curve, the logic, analysis, etc., may be able to bring some inspiration to mathematicians like them, and let them think about other existing theories, whether they can also be expanded in this way like the modular curve, etc.

This is the other more important meaning of the generalized modulus curve, and it is also the important reason why they are so looking forward to the generalized modulus curve.

Now, after seeing the whole process of how Xiao Yi thought out the generalized modular curve and applied it to solve the problem, they can now be said to have gained some understanding.

"Now, I have figured out what to write for my next paper."

"I've thought about it, too. I hope your idea is different from mine."

"I'm thinking of the theory of moduli spaces. What about you?"

"Damn!...Haha, I'm just kidding. I'm different from you. I'm thinking about the Shimura cluster that Xiao Yi just mentioned. I think that maybe the Shimura cluster can be further developed."

"Very good. I just had this thought, but in the end I still want to choose the moduli space theory. I think there will be more things worth digging into here."

"Good luck to you then."

"Good luck to you too."

"..."

The mathematicians below seemed somewhat excited.

Xiao Yi on the stage had already started the last part of his report.

This is the final proof of the Riemann hypothesis.

(End of this chapter)