The next proof process for Riemann Hypothesis was much faster. After all, the more important processes have been basically explained before, including the beginning. Xiao Yi also explained the familiar process of assigning Riemann Hypothesis to Galois representation in the fourth paper.
Therefore, the rest of the main content basically focuses on the technical issues in the process until the step of successfully proving 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 sitting in the back.
Although there are also mathematics professors, there are also students, enthusiasts, etc.
For them, this lecture was not treated as an art appreciation meeting like those top mathematicians before.
They regarded this report meeting 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 recite the miracle summoning spell with them for several hours.
They can understand every word of these spells, but once they are connected, they really don't understand them at all.
So much so that they all started to become a little confused.
But at this moment, Xiao Yi suddenly said:
"So now, we have the fifth corollary. If λ_E is an automorphic representation for any CM elliptic curve E, then Riemann Hypothesis is established."
[Riemann Hypothesis established].
This sentence immediately triggered the keyword, causing many confused people present to sit up straight and look at Xiao Yi.
Finally here?
Is the miracle they want to witness finally coming?
"This inference finally succeeded in turning Riemann Hypothesis into a question about Hecke feature."
"So, at this point, in order to prove Riemann Hypothesis, we just need to prove that Hecke feature of all CM elliptic curves is automorphic."
"At this time, we can continue to borrow the idea from the previous proof of Artin Conjecture, consider embedding each CM elliptic curve into a Generalized Modular Curve, and then use the modularity of Generalized Modular Curve to prove the automorphism of λ_E."
"In this way, we have: for any CM elliptic curve E, there is a Generalized Modular Curve X and an embedding i: E→X, such that i induces the isomorphism between Hecke feature."
[λ_Eλ_X i_*]
"Where λ_X is the Hecke feature of X and i_* is the homomorphism between Galois group induced by i."
"Now, we can get a result. 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 Conjecture, we know that λ_X is an automorphic representation, therefore, λ_Eλ_X i_* is also an automorphic representation."
"At this point, recalling a theorem we gave earlier, if and only if λ_E is an automorphic representation, all zero points of L(s, E) are located on the straight line Re(s)=1/2."
"Therefore, it is also equivalent to that all non-trivial zero points of Riemann zeta function all fall on the straight line Re(s)=1/2 in the complex plane."
Having said this, Xiao Yi paused, and the pen on the blackboard in his hand also stopped here.
Then, he turned around, opened his hands to the audience, and said, "In other words, up to this point, Riemann Hypothesis has been proven."
"Riemann in the 19th century probably would not have imagined that a short eight-page paper he accidentally wrote would eventually leave the mathematical community with such a question that has haunted 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 Riemann Hypothesis is just a staged victory for us. In the future, there are still many problems waiting for us to discover and explore."
"Okay, now that we have said that, the main content of my report has been finished."
"Thanks for everyone's patience, then..."
Just when 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, burst into applause after hearing Xiao Yi's thanks.
However, it wasn't until they saw Xiao Yi's helpless expression that they realized it wasn't yet time.
Then the applause gradually stopped again.
Xiao Yi spread his hands helplessly, and then faced everyone present and said, "Then, next is the question and answer session."
"If you have any doubts about my certification process, you can start asking questions now."
As Xiao Yi's words fell, the whole place fell into silence. Those who had no questions, or those who could not ask questions, looked around, wondering if anyone could ask questions.
Although many mathematicians had some doubts in their minds when they first read the paper.
But in what Xiao Yi just said, these problems have basically been solved.
Now, if there are still people who can ask questions, they must be relatively tricky questions.
It wasn't until a moment later that someone still raised their hands.
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 could find the problem.
People have also begun to wonder, what questions can this mathematician who was once known as a genius and is now over 40 years old ask?
Xiao Yi on the stage smiled slightly and said, "Peter, please ask a question."
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 that questioned Shinichi Mochizuki's proof of abc conjecture.
At the beginning, he felt that Xiao Yi would definitely become a new star in the world of mathematics.
But at the time, he didn't expect that the process would be so fast, even beyond his imagination.
Taking the microphone handed over by the staff, he said, "Okay, Xiao, although everyone of us is looking forward to seeing Riemann Hypothesis proven, at the same time, we will not let it be proven easily."
"So, my question is - in your proof, there is a key step, which is to connect 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 the critical line. 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 microphone and looked at Xiao Yi quietly.
Although he has always maintained a very high evaluation of the proof of Xiao Yi, 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 the answer?
Everyone turned their attention to Xiao Yi.
But when they saw Xiao Yi just smiled slightly, they then said, "That's a good 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?
This question seemed quite difficult to them as soon as they heard it, but Xiao Yi actually said it was easy to understand?
Then, Xiao Yi started to answer.
He first acknowledged the description in the Schultz 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 referring to the L-function of the CM elliptic curve, we are 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 Zeta function and Dirichlet L-functions respectively."
"Then as I further introduce Generalized Modular Curve and discuss their Hecke feature, I actually do 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 just said.
Xiao Yi also gives them time to understand.
It wasn't until a moment later that Schultz suddenly realized it 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 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 paper, 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 anymore.
"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.
After about a while, someone finally raised their hand again.
Everyone present was stunned. Does anyone still have questions?
They all turned to look at the person who raised his hand.
Unsurprisingly, this is another well-known mathematician.
ANDREW Wiles.
Proven by Fermat's Last Theorem.
Likewise, this is also a mathematician who has studied geometry and algebra very deeply.
The Taniyama-Shimura conjecture he proved is itself an important theorem in elliptic curves.
Xiao Yi raised his eyebrows slightly. Although he was very 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 ask?
Taking the microphone from the staff, Wiles patted it. After hearing the echo, he smiled and said, "Peter is right, we won't let you prove Riemann Hypothesis so easily."
"To be honest, I didn't understand the question Peter asked and your answer just now. At first, I thought your explanation was a failure."
"But I didn't expect that Peter said he understood it, and then a bunch of other guys also understood it."
"So, the question I wanted to raise was originally intended to be left until you have truly solved Peter's problem before I tell you."
Wiles smiled sheepishly, "I also want to give you this question as a surprise."
There was a sudden burst of laughter.
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 content of the CM ellipse."
"In your proof, you exploit the properties of Generalized Modular Curve, specifically their automorphism of Hecke feature, to deduce the distribution of zeros of L-functions for all elliptic curves. However, there is a subtle logical question here: do you implicitly assume that all elliptic curves can be embedded into a Generalized Modular Curve?"
Hearing this question, everyone present was stunned.
This question...
It is more hidden than the previous question, and at the same time more advanced.
And similarly, 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 what they think, they can't 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, the top mathematicians all looked at Wiles.
There was no other reason than that they all felt extremely familiar with this issue.
Isn't this kind of mistake the same mistake that Andrew Wiles made in his original proof?
And now, he has discovered the same problem in Xiao Yi's paper?
Does this count as ushering in a reincarnation?