The top student must be diligent

They could understand each word of these spells, but once they were connected, they really didn't understand them at all.

So much so that they all began to become a little confused.

But at this moment, Xiao Yi suddenly said:

"Now, we have the fifth inference. If λ_E is an automorphic representation for any CM elliptic curve E, then the Riemann hypothesis holds."

[Riemann hypothesis holds].

This sentence immediately triggered the key words, causing many confused people present to sit up straight and look at Xiao Yi.

It's finally here?

The miracle they want to witness is finally coming?

"This inference finally successfully transformed the Riemann hypothesis into a question about Hecke characteristics."

"So, at this point, in order to prove the Riemann hypothesis, we only need to prove that the Hecke characteristics of all CM elliptic curves are self-conserving."

"At this point, we can continue to borrow the idea of ​​proving the Artin conjecture before, consider embedding each CM elliptic curve into a generalized modular curve, and then use the modularity of the generalized modular curve to prove the self-conserving property of λ_E."

"In this way, we have: for any CM elliptic curve E, there exists a generalized modular curve X and an embedding i: E→ X, so that i induces isomorphisms between Hecke characteristics."

[λ_Eλ_X i_*]

"Where λ_X is the Hecke characteristic of X, and i_* is the homomorphism between Galois groups induced by i."

"And now, we can get a result that for any CM elliptic curve E, we have a generalized modular curve X and an embedding i: E→ X, so that λ_Eλ_X i_*. ”

“According to the proof of Artin's conjecture, we know that λ_X is an automorphic representation, so λ_Eλ_X i_* is also an automorphic representation. ”

“So far, recalling a theorem we gave earlier, if and only if λ_E is an automorphic representation, all zeros of L(s, E) are located on the straight line Re(s)=1/2. ”

“Therefore, it is also equivalent to that all non-trivial zeros of the Riemann zeta function all fall on the complex plane Re(s)=1/2, this straight line. ”

Speaking of this, Xiao Yi paused, and the pen in his hand that was deducing on the blackboard also stopped here.

Then, he turned around, opened his arms to the audience, and said: "That is to say, at this point, the Riemann hypothesis has been proved."

"Riemann in the 19th century probably would not have thought that the short eight-page paper he wrote by chance would eventually leave the mathematical community with such a problem that has been haunting the mathematical community for more than a hundred years."

"But until now, I think I can officially announce to you that this problem has become a thing of the past."

"I believe this is a memorable achievement for mathematics, but of course, we must look at everything from a developmental perspective. The proof of the Riemann hypothesis is only a phased victory for us. In the future, there are still many problems waiting for us to discover and explore."

"Okay, so that's it, the main content of my report is all here."

"Thank you for your patience, then..."

Just as Xiao Yi was about to say the next words, the whole audience burst into applause.

The audience who had been ready to applaud for a long time applauded after hearing Xiao Yi's thanks.

However, it was not until they saw Xiao Yi's helpless expression that they realized that it seemed that it was not time yet.

So the applause gradually stopped again.

Xiao Yi spread his hands helplessly, and then said to everyone present: "Then, next is the question session."

"Everyone can start asking questions now if they have any doubts about my proof process."

As Xiao Yi's voice fell, the whole audience fell into silence. Those who had no questions, or who could not ask questions, looked around, wondering if anyone could ask questions.

Although many mathematicians had some doubts in their hearts when they just finished reading the paper.

But in Xiao Yi's narration just now, these problems have basically been solved.

And now, if anyone can still ask questions, it must be a relatively tricky question.

Until a moment later, someone still raised his hand.

Peter Schultz.

Seeing him, everyone was not surprised. As one of the most outstanding mathematicians in arithmetic geometry today, it was not surprising that he could find the problem.

People began to wonder what questions this mathematician, who was once a genius and is now over 40 years old, could ask?

Xiao Yi on the stage smiled and said, "Peter, please ask your questions."

Schultz also smiled and nodded at him. Looking at Xiao Yi, he seemed to recall the afternoon when he sent an email to Xiao Yi, inviting him to participate in the academic conference questioning Mochizuki Shinichi's proof of the abc conjecture.

At the time, he felt that Xiao Yi would definitely become a rising star in the mathematics world.

But at that time, he did not expect that this process would be so fast, even beyond his imagination.

Taking the microphone handed over by the staff, he said: "Okay, Xiao, although we all look forward to seeing the Riemann Hypothesis proven, we will not do it easily. Let it be proven.”

"So, my question is - in your proof, there is a key step, which is to connect the Riemann Zeta function with the L-function of the elliptic curve."

"Here, you consider CM elliptic curves and use their special properties to prove that the zero points of their L-functions are all located on critical lines. However, there is a problem: not all elliptic curves are CM curves."

"And your method is only applicable to CM elliptic curves. For general elliptic curves, it cannot decompose its L-function into the product of the ζ function and the Dirichlet L-function like the CM curve."

"So, can you give me an explanation for this?"

Then, he put down the phone and looked at Xiao Yi quietly.

Although he has always maintained a very high opinion of Xiao Yi's proof, it did not prevent him from still finding the problem.

As soon as this question came up, quite a few people present were stunned.

This question...

They suddenly gasped.

This is a problem that goes straight to the core. Once it fails, it will be like a building built with various complex structures, but if one of the load-bearing structures breaks, the entire building will collapse.

So, can Xiao Yi give an answer?

Everyone turned their attention to Xiao Yi.

But when they saw Xiao Yi just smiled slightly, he then said: "Not a bad question."

"This is indeed a point that I did not elaborate on in my proof."

"But maybe it's also because I think... this question is easy to understand?"

Everyone present suddenly had question marks on their faces.

What?

They found this problem quite difficult when they heard it, but Xiao Yi actually said it was easy to understand?

Then, Xiao Yi started to answer.

He began by acknowledging the description in Schulz's question.

"Your observation is correct. Most CM elliptic curves are indeed defined on extensions of the number field. In this case, what we get is some analogue of the ζ function and the Dirichlet L-function."

"However," he then changed the topic: "What I want to emphasize is that although these analogs may not satisfy the classical functional equation, they still satisfy some generalized functional equation."

"Although such generalized function equations may be more complex in form, their essential properties are consistent with the classical case, especially they still contain key information about the zero-point distribution of the L-function."

"In my proof, when I refer to the L-function of the CM elliptic curve, I am actually discussing these generalized L-functions. The key is that these generalized L-functions can still be decomposed into two parts. , and these two parts correspond to some analogues of the Zeta function and the Dirichlet L-function respectively.”

"Then as I further introduced generalized modular curves and discussed their Hecke characteristics, I actually did so under more general conditions. Under these more general conditions, my argument is still valid."

"This is my answer, I wonder if you can understand it."

Many people present were confused. Even the top mathematicians had many expressions of confusion.

It's really because Xiao Yi's answer is a bit too abstract, and I even feel that he is answering randomly.

However, based on their trust in Xiao Yi, those top mathematicians began to think about the truth in Xiao Yi's words.

In the proof, this part has actually been described?

They began to review the paper and what Xiao Yi had just said.

Xiao Yi also gave them time to understand.

It wasn't until a moment later that Schultz suddenly understood and said, "I understand."

Then he sighed: "Indeed, the key proofs all occur in the general process, and this general process is integrated into the entire paper."

"This time, I truly recognize your identity as the God of Mathematics."

"Thank you for your answer."

Then he sat down.

When most of the people present heard what Schultz said, they felt confused again.

No, buddy, do you understand again?

What do you understand?

But probably because of those few words of Schultz, those mathematicians who were also thinking were inspired, and they all showed expressions of sudden enlightenment.

However, the total number of people added up is not more than 5.

Xiao Yi on the stage saw the expressions of the people in the audience, smiled, and said: "I need to explain that this is indeed a bit difficult to understand. This requires a more comprehensive understanding of my thesis. Especially the relevant content I just mentioned, so that this problem can be explained. "

After that, he stopped talking. If no one understood, then he might explain more, but now, since someone understood, there was no need for him to say so much more.

"So, any questions?" he continued.

Everyone present could only recover from the questions just now and continue to wait to see if anyone else had questions.