The top student must be diligent.

Chapter 292 Riemann Hypothesis Lecture (Part )
The following proof process of the Riemann hypothesis was much faster. After all, the more important processes had been basically explained before. Including at the beginning, Xiao Yi also explained the process of giving Galois familiarity to the Riemann hypothesis in the fourth paper.

Therefore, the remaining main content is basically focused on the technical problems in the process until the step of successfully proving the Riemann hypothesis.

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

The time has come to 16: and the second half of the report meeting has already passed for two hours.

The vast majority of the audience present were already somewhat tired.

Especially those who sit in the back seats.

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

For them, this lecture was not like the top mathematicians before them, who viewed it as an art appreciation meeting.

They regarded this lecture as a gathering to witness a miracle.

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

They could understand each word of these spells, but once they were connected together, they really didn't understand anything.

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

But at this moment, Xiao Yi suddenly said:

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

【Riemann hypothesis is established】.

These words immediately triggered a key word, causing many confused people present to sit up straight and look at Xiao Yi.

finally come?

The miracle they wanted to witness was finally coming?

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

"So, at this point, to prove the Riemann hypothesis, we just need to show that the Hecke characteristics of all CM elliptic curves are self-conservative."

"At this point, we can continue to borrow the idea from the previous proof of Artin's conjecture, consider embedding each CM elliptic curve into a generalized modular curve, and then use the modularity of the generalized modular curve to prove the automorphism of λ_E."

"So, we have: for any CM elliptic curve E, there exists a generalized modular curve X and an embedding i: E→ X such that i induces an isomorphism between Hecke characters."

【λ_Eλ_X i_*】

"where λ_X is the Hecke character of X and i_* is the homomorphism between the Galois groups induced by i."

"And now, we can go on to get a result that for any CM elliptic curve E, we have a generalized modular curve X and an embedding i: E→ X such 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."

"At this point, recalling a theorem we gave earlier, all zeros of L(s, E) lie on the line Re(s) = 1/2 if and only if λ_E is an automorphic representation."

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

Having said that, Xiao Yi paused, and the pen in his hand that was writing on the blackboard also stopped here.

Then, he turned around, opened his arms to the audience, and said, "That is to say, the Riemann hypothesis has been proven."

"In the 19th century, Riemann 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 troubled 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 should 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, that's all I have to say. I have finished the main content of my report."

"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 prepared to applaud, 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 not to be the right time yet.

Then the applause gradually died down again.

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

"If you have any questions about my proof process, you can start asking questions now."

As Xiao Yi finished speaking, the whole place fell silent. Those who had no questions, or who could not ask any questions, looked around, wondering if anyone could ask any questions.

Although when many mathematicians had just read the paper, many of them had some doubts in their minds.

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

And now, if anyone is still able to ask questions, they will inevitably be relatively tricky questions.

After a while, someone raised his hand.

Peter Schultz.

Everyone was not surprised to see him. As one of the most outstanding mathematicians in arithmetic geometry today, it was not surprising that he was able to find the problem.

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

Xiao Yi on the stage smiled slightly 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 to question Mochizuki Shinichi's proof of the abc conjecture.

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

But at that time he didn't expect that the process would be so fast, even beyond his imagination.

After taking the microphone handed over by the staff, he said, "Well, Xiao, although I want to say that everyone of us is looking forward to seeing the Riemann hypothesis proved, at the same time, we will not let it be proved easily."

“So, my question is this: There is a key step in your proof that relates the Riemann Zeta function to the L-function of the elliptic curve.”

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

"However, your method is only applicable to CM elliptic curves. For general elliptic curves, it is impossible to decompose their L-functions into the product of the zeta function and the Dirichlet L-function like the CM curve."

"So, can you explain this?"

Then he put down the microphone 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 finding the problem in it.

As soon as this question was asked, many people present were stunned.

this problem……

They immediately gasped.

This goes straight to the core of the problem. If it fails, it will be like a building constructed 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 eyes to Xiao Yi.

But they saw Xiao Yi just smiled slightly, and then said: "Good question."

"That was indeed a point I did not elaborate on in my proof."

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

Everyone present immediately had a look of question marks on their faces.

What?
When they heard this question, they felt it was quite tricky, but Xiao Yi actually said it was easy to understand?

Then, Xiao Yi began to answer.

He first acknowledged the description in Schultz's question.

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

“But,” he then changed the subject: “I want to emphasize that although these analogs may not satisfy the classical functional equation, they still satisfy a certain generalized functional equation.”

"Although such generalized functional equations may be more complicated in form, their essential properties are consistent with those of the classical case. In particular, they still contain key information about the distribution of the zeros of the L-function."

“In my proof, when I mention the L-functions 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 the product of 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 was actually doing so under more general conditions, and my arguments were still valid under these more general conditions."

"This is my answer. I don't know if you can understand it."

Many people present were stunned. Even the top mathematicians had confused expressions on their faces.

It was because Xiao Yi's answer was a bit too abstract, and it even felt like he was 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 realized 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 whole of the paper."

"This time, I truly acknowledge your status as the God of Mathematics."

"Thank you for your answer."

Then he sat down.

When most people present heard Schultz say this, they were confused again.

No, buddy, do you understand?
What do you understand?

But perhaps it was because of Schultz's words that those mathematicians who were also thinking were inspired, and they all showed expressions of sudden enlightenment.

However, the total number of people does not exceed 5.

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

Then, he stopped talking. If no one understood, he might have explained more. But now, since some people understood, there was no need for him to say so much.

"So, any more questions?" he continued.

The people present could only recover from the previous question and continue to wait to see if anyone had any more questions.

After about half an hour, someone finally raised his hand.

Everyone present was stunned. Is there anyone who has questions?
They all turned to look at the person who raised his hand.

Unsurprisingly, this is another well-known mathematician.

Andrew Wiles.

The prover of Fermat's Last Theorem.

Similarly, he was also a mathematician who studied geometry and algebra in great depth.

The Taniyama-Shimura conjecture he proved is itself an important theorem in elliptic curves.

Xiao Yi raised his eyebrows slightly. Although he was well aware of the reputation of this old mathematician, he had not had many opportunities to meet and communicate with him before.

So, what kind of questions can this old mathematician raise?

Wiles took the microphone from the staff and clapped it. After hearing the echo, he smiled and said, "Peter is right. We won't let you prove the Riemann hypothesis so easily."

"To be honest, I didn't understand Peter's question or your answer just now. At first I thought your explanation was a failure."

"But I didn't expect that Peter said he understood, and then a bunch of other guys also understood."

"So, I originally wanted to save this question for when you have truly solved Peter's problem. I will tell you later."

Wiles smiled sheepishly: "I wanted to give you this question as a surprise."

There was a burst of laughter among everyone present.

You call this a surprise, are you sure it's not a shock?

However, Wiles then said: "But now, it seems that I have to tell you my problem in advance."

"My question is also related to the elliptic curve part, and also involves the CM ellipse content."

"In your proof, you exploit the properties of generalized modular curves, in particular the automorphism of their Hecke characteristics, to infer the distribution of the zeros of the L-functions of all elliptic curves. However, there is a subtle logical question here: are you implicitly assuming that all elliptic curves can be embedded in a generalized modular curve?"

Hearing this question, everyone present was stunned.

this problem……

It is more hidden than the previous question and more advanced.

At the same time, it is also more difficult and deadly.

Many people began to circle in their minds, thinking about how to answer this question.

But no matter how they thought about it, they could not find a starting point to solve this problem.

Of course, while other viewers were thinking about how to solve this problem, at the same time, those top mathematicians all looked at Wiles.

The reason is simple, it's just that they all feel particularly familiar with this problem.

Isn’t this the same mistake that Andrew Wiles made in his original proof?
And now, he found the same problem in Xiao Yi's paper?

Does this count as the beginning of a new cycle?

(End of this chapter)