Section 361

To put it simply, 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 an adult helps a child do homework, because the child has a certain attitude towards someone in the world. When adults still cannot understand a problem that seems to be very simple, they feel very confused and may even think that the child is very stupid.

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

Of course, Xiao Yi didn't know what their problem was at the moment, and he continued to tell his proof.

"...So we can get such a theorem——"

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

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

"At this point, we can completely study all types of extended L-functions."

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

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

Speaking of this, Xiao Yi paused, then looked at the time, and then said with a smile: "Okay, now the time has reached twelve o'clock, so according to the previous arrangement, it is lunch time now, in the hotel next door , we have prepared sumptuous meals for you, everyone is welcome to come and taste them. ”

Everyone who was looking forward to Xiao Yi's explanation on how to prove Ating's conjecture suddenly burst into mourning.

Although after the Riemann Hypothesis was proved, Atting's Hypothesis no longer seemed so important, this was only relative. People were still extremely shocked that Xiao Yi was able to prove Ating's Hypothesis at the same time.

So the current situation is almost as if it has stopped.

The only ones who were grateful to Xiao Yi for "getting off work" on time were probably the only viewers who were challenging the limits of their bladders but couldn't bear to leave.

Seeing Xiao Yi entering the backstage without looking back, many viewers could only give up on trying to save him.

"Well, it seems we can only wait until two hours later."

Qiu Chengtong said helplessly.

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

He patted Qiu Chengtong and said, "Qiu, this time I have to trouble you to introduce some delicious food to me."

Qiu Chengtong smiled and said:

"Of course no problem."

…

Afterwards, the audience present were greeted by the staff and left, then went to the hotel next door to enjoy lunch.

Of course, they have not forgotten at all that they will come back in two hours to listen to Xiao Yi explain the next proof process.

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

Anyway, after I finish speaking in the afternoon, I can come back here and eat as much as I want.

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

Soon, before 14:00, all the audience returned to the scene again.

Even during the process of queuing up to enter the venue, everyone completely obeyed the order and there were no unexpected incidents, because everyone did not want to delay the report starting at 14:00 because of this kind of thing.

When 14:00 arrived, the second half of the Riemann Hypothesis report meeting started on time.

Xiao Yi once again entered the stadium and stood on the podium. Looking at the many spectators, he smiled slightly and said, "Then, let's continue with the report."

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

What comes next 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 the middle, there are almost dozens of 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", "noticed", "easy to obtain", etc.

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

Therefore, what can be omitted will be omitted directly, and the key steps will be retained.

In this way, we came to the end of the proof of Artin's conjecture.

"At this point we will be able to make a final judgment."

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

【ρ_f: Gal(Q/Q)→ GL_n(Z_)】

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

"In this way, we have successfully created a connection between the geometric properties of the generalized modular curve and the arithmetic properties of the Galois representation."

"With this result, we can successfully transform Artin's conjecture into a question about Galois representation."

"Concretely speaking, we achieved such a result."

"Suppose E is an elliptic curve and L(s, E) is 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 function equation; second, there is a modular form f such that the Galois representation of E is isomorphic to ρ_E and ρ_f."

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

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

【ρ_Eρ_X i_*ρ_f i_*. 】

"This shows that ρ_E also comes from a modular form, the "pull back" of f."

"And this means that L(s, E) is integral and satisfies the function equation. To sum up, we have successfully proved Artin's conjecture."

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

Everyone present immediately exclaimed in amazement.

Artin guessed!

This problem, which originally seemed to them to be extremely difficult, 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 were 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 seemingly complete space he created back then can be regarded as an important breakthrough in arithmetic geometry and can be applied to the study of a variety of problems, especially in the fields of algebraic geometry and the Langlands Program.

Now, the generalized modular curve created by Xiao Yi is a more powerful expansion of arithmetic geometry from another level. It is a true fusion of the methods between algebraic geometry and number theory. The extremely tight combination is a great innovation for the entire mathematical community.

Not to mention the future, how much help this generalized modular curve may bring to the solution of other problems in mathematics. Now, it is just the process of Xiao Yi thinking about this generalized modular curve, in which logic, analysis, etc. may all be involved. It can bring some inspiration to mathematicians like them and let them think about whether other existing theories can also be extended through this method, such as modular curves, etc.

This is the more important meaning of the generalized modular curve on the other hand, and it is also an important reason why they look forward to the generalized modular curve so much.

Now, after watching the whole process of how Xiao Yi thought about the generalized modular curve and applied it to solve the problem, now they can be regarded as somewhat enlightened.

“Now, I’ve already figured out what to write about for my next paper.”

"I've already thought about it, and I hope you don't think the same thing as me."

"I'm thinking about moduli space theory, what about you?"

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

"It's very good. I had this idea just now, but in the end I still want to choose the modulus space theory. I think there will be something more worth exploring in it."

"Good luck to you then."

"Good luck to you too."

"..."

The mathematicians below seemed a little happy.

Xiao Yi on the stage has also begun the last part of his report.

This is the ultimate proof of the Riemann Hypothesis.

Chapter 292 Riemann Hypothesis Report Meeting (4)

The following process of proving the Riemann Hypothesis was much faster. After all, the more important processes have been basically explained before, including at the beginning, Xiao Yi also assigned Gallo to the Riemann Hypothesis in the fourth paper. Wa said that the familiar process has been explained together.

Therefore, the remaining main content basically focuses on the technical issues in the process until the step of successfully proving the Riemann Hypothesis.

So, another hour or so passed in the blink of an eye.

The time has reached 16:00, and the second half of the report meeting has passed for two hours.

The vast majority of the audience present were more or less tired.

Especially those who were sitting in the back.

Although there were mathematics professors among them, there were also students, enthusiasts, etc.

For them, this lecture was not like the top mathematicians in front, who regarded this lecture as an art appreciation meeting.

They regarded this lecture as a gathering to witness miracles.

Unfortunately, before they could really witness the miracle, the creator of the miracle standing at the top had to chant the miracle summoning spell with them for several hours.