"It's getting interesting."
Deligne said with a smile.
"Indeed, this is what is called, does history always repeat itself?"
Bombieri also smiled and nodded.
"That's not necessarily the case. After all, Xiao Yi hasn't expressed it yet, so he can't answer this question."
Deligne shook his head and said.
Then, he moved his gaze away from Wiles, looked at Xiao Yi, and said, "To be honest, with the answer to the previous question, now I don't think Xiao Yi will be able to answer this question."
"Oh? You mean, you understand Xiao Yi's answer to the previous question?"
Bombieri raised his eyebrows, glanced at Deligne, and said in surprise.
"Uh... that's not true." Deligne waved his hand and said helplessly, "Now that I am so old, I can no longer react so quickly. Schultz can understand it, but I can't."
"That's good." Bombieri nodded with a smile and said, "Otherwise, I would have thought how far my thinking ability has dropped."
Then, he turned his head again, looked at Wiles, and said, "But to be honest, I'm still more curious about what Andrew is thinking about now."
"Did he think about how he felt when facing this problem in the past, or did he want to see the young man in front of him who could easily solve the problem that had trapped him for a year?"
Deligne smiled and said, "Maybe both."
On the stage, Xiao Yi raised his eyebrows slightly after listening to this question.
Looking at Wiles with an expectant expression in front of him, he already remembered the problems Wiles had encountered in the past.
Although he had not experienced the original incident, this did not prevent him from having a deeper understanding of the past story in the future.
After all, when he first started to delve into mathematical research, he had carefully studied the proof process of Wiles against Fermat's Last Theorem, and had a very in-depth understanding of it.
And at this moment, this question...
He smiled and said, "Such a question must be a very memorable answer for Professor Wiles, right?"
Wiles's eyes lit up, and then he smiled and said, "So you also know what happened back then. I thought that the young people who came later have never heard of the original story."
When the people present heard the conversation between the two, they suddenly became confused. What on earth were they talking about?
Why do they all sound so confused?
However, people who knew other details soon began to tell others around them what happened.
Soon, one passed to two, and the other to four... In this way, many people present understood what happened when Wiles proved Fermat's Last Theorem.
When Wiles proved Fermat's Last Theorem, he used a technique called the Kolyvagin-Flach method to try to establish the relationship between Selmer group of elliptic curves and the Shafarevich-Tate group.
However, a key step in his proof relied on the existence of certain Euler systems, and upon subsequent review, other mathematicians discovered that Wiles did not fully prove the existence of these Euler systems.
In other words, he also implicitly assumed the existence of these Euler system.
Therefore, this directly constituted the most fatal point in his proof, so that he failed to prove Fermat's Last Theorem for the first time.
Now, the question he raised is also the same: Xiao Yi implicitly assumes that all elliptic curves can be embedded in a Generalized Modular Curve.
But in fact, in the Xiao Yi paper, for those non-CM elliptic curves, there is no clear way to define their Generalized Modular Curve, so this implicit assumption is obviously wrong.
And this is fatal.
When the audience present learned this, they were immediately amazed.
And similarly, they have become more interested in today's historic report meeting.
This time, their trip here was worth it!
At this moment, Xiao Yi continued, "In my spare time, I also have some understanding of the history of mathematics. Of course, I also understand what happened back then."
Hearing this sentence, Wiles sighed, "Is it the history of mathematics? It turns out that what happened more than 20 years ago is already history."
"To be precise, thirty-three years have passed." Xiao Yi said.
Wiles was startled for a moment, and then laughed.
For a top mathematician, it seems that such a mistake should not be made. However, he just sighed, "It was indeed thirty-three years ago. I am very happy that our God of Mathematics can remember this matter so clearly. I am probably older, and my impression of what happened back then is not that deep."
"You know, when we are young, we can clearly remember how many years have passed, but as you spend more and more time, your sensitivity to time gradually becomes less and less."
"Fortunately, at least I didn't get Alzheimer's disease. I don't know if you have seen that movie called "Father Trapped in Time". It was made by us United Kingdom people. It was so scary that we even forgot the time."
Xiao Yi smiled and said, "I know that movie very well. It actually told me a truth. Hannibal's end was miserable in his later years."
Wiles laughed again, because the male protagonist in this movie who got Alzheimer's disease was the actor who played Hannibal in "Silence of the Lambs".
But soon, he stopped smiling, "Okay, our chat is over, but don't forget the question I just asked you."
"Don't even think about using this method to make me forget that I just asked you a question."
"After all, I wasn't Hannibal when I was young."
He stretched out a hand and shook his index finger.
Xiao Yi smiled and said, "Of course I haven't forgotten it."
But after saying this, he fell into silence again.
But obviously, everyone could see that he was thinking.
Maybe, his chat with Wiles just now was also to give himself more time to think?
The voices of the people who were still discussing gradually became quiet, and for a while, there was almost no sound in the entire lecture hall.
No one wants to disturb Xiao Yi's thinking because of the noise they make.
Even Wiles carefully controlled his microphone to prevent it from making noise.
Obviously, this question does directly hit the key weakness of the entire paper.
And this time, it was probably the first time that Xiao Yi showed his thinking appearance to so many people.
They couldn't help but observe Xiao Yi carefully, wanting to see what God of Mathematics looked like when he was thinking about problems.
But now it looks no different from how they usually think.
That is to say, when they are thinking, they may frown, while Xiao Yi's eyebrows relax.
This meditation can probably be regarded as the most critical meditation in the history of mathematics, and it is also a meditation that everyone is looking forward to the result.
No matter what the result is, it can bring different interpretations.
In this way, a long time passed.
It took about two or three minutes.
As a report meeting, two or three minutes is undoubtedly a long time.
Especially now, there are thousands of people in the audience waiting for the results.
Until finally, Xiao Yi raised his head and finally spoke.
"First of all, I have to say that this is a sharp enough and hidden enough problem that I thought about it for a while before I remembered how I considered this problem when I originally proved it."
When everyone heard what he said, they all sat up straight.
Remember how you thought about this issue in the first place?
Could it mean that Xiao Yi was already aware of this problem at the beginning?
So now, he thought about how to answer this question?
But then, Xiao Yi's words made them all stunned.
"Well, what I have to admit is that I did implicitly assume what you said, that all elliptic curves can be embedded in a Generalized Modular Curve."
Huh?
This statement...
Is Xiao Yi going to admit that he failed?
Everyone present seemed to be about to see a myth shattered.
After all, they have never seen Xiao Yi fail.
However, Xiao Yi's next words followed.
"Actually, when I was researching this, my mathematical intuition told me that this hypothesis was completely correct, so I did not discuss this situation during the proof process. Therefore, the mistake I probably made was..."
Xiao Yi paused, then spread his hands and said, "This question is 'obvious' to me."
Everyone was stunned as they listened to his story.
What does this mean?
So, did Xiao Yi admit that he was wrong, or did he admit that this issue was actually an obvious theorem in his eyes, so much so that he felt no need to prove it in the paper?
Wiles was also stunned, not understanding the meaning of Xiao Yi's words.
After all, when he made this implicit assumption in the first place, it was basically because he subconsciously acquiesced to this in his mind, so it became a key flaw in the entire proof, so that he was confused in it for almost a year before he succeeded.
Now, what Xiao Yi said seems to be the same as the reason why he made the mistake in the first place.
But...
Xiao Yi The information revealed in other words is slightly different.
For a moment, he was a little confused.
But then, Xiao Yi's words completely stunned them all.
Xiao Yi turned his head, walked back to the blackboard, and said, "Well, in that case, I will prove it on the spot, proving that all elliptic curves can be embedded into a Generalized Modular Curve."
Then he began to wipe the blackboard that had been wiped countless times in the previous report meeting, and said at the same time, "Of course, the process of this proof may be a little long, so please continue to listen patiently for a while."
Everyone present was stunned for a moment.
Wha... what?
Live proof?
Where should they even start to solve this problem? Xiao Yi is going to prove it to them on the spot?
How long has it been since this question was raised?
For a moment, people couldn't help but start to wonder, what on earth was Xiao Yi thinking about during those few minutes of thinking just now?
Or, as he just said, he was just recalling how he thought about this matter when he first proved it?
They were all a little stunned at this point.
Including the top mathematicians sitting in the front row, they all had the same reaction.
They all looked at Xiao Yi who was wiping the blackboard in disbelief, their faces filled with disbelief.
"He wants to prove it on the spot? Can this kind of problem... such a complicated problem... can really be proved on the spot?"
Schultz said with some disbelief.
"Maybe ordinary people can't do it, but he is Xiao Yi. He might really be able to do it."
Faltings next to him smiled and acted very indifferently to this.
"I..." Schultz didn't know what to say.
On the other side, Deligne smiled and said, "It's starting to get interesting."
Bombieri said, "Isn't it getting more and more interesting?"
Deligne was startled, then laughed and said, "Okay, if you say it more accurately, it is indeed getting more and more interesting."
Terence Tao seemed a little excited at this time.
"So, it is basically impossible to stump him with this kind of problem. In his opinion, such problems are all obvious. He is the real god of mathematics! Buy it!"
As for Feierman and Shing-Tung Yau next to them, they were slightly silent, because in fact, they were all shocked by the behavior of Xiao Yi that they had to prove on the spot.
Does this really still belong to the category of human beings?
They want to ask such questions.
But don't wait for them to think too much.
Xiao Yi on the stage has finished wiping the blackboard, then picked up the blackboard pen and walked to the blank space on the left.
"Then, my proof begins now."
"First of all, let me explain that when I first proved it, I thought it was obvious, or I had such an intuition, mainly because of my understanding of the L-function of elliptic curves."
"For any elliptic curve, its L-function should have some form of automorphism, which reflects some intrinsic symmetry of the elliptic curve. This automorphism, in my proof, is achieved by Hecke feature of Generalized Modular Curve."
"But now, if this process needs to be rigorously proven, then a series of logical inferences are needed."
"Then, we first give the proposition: for any elliptic curve E, there is always a Generalized Modular Curve M, so that E can be equivariantly embedded into M."
"The first step is to consider the L-function L(E, s) of the elliptic curve E. According to the modular properties of the elliptic curve, L(E, s) should satisfy a certain form of functional equation and automorphism. Specifically, there should be a prime number p and an automorphism σ: E→E, such that——"
[L(E, s)=ε(E)* p^(-s/2)* L(σ(E), 1-s)]
"Where ε(E) is a constant that depends on E."
"Next, we consider a formal Generalized Modular Curve M(E) generated by E. As an intuition, we can understand M(E) as a larger object generated by E through some product and quotient operation, similar to a group ring generated by a group."
"In this structure, the automorphism σ on E can be naturally extended to M(E), recorded as σ: M(E)→M(E)."
"Then the third step..."
Xiao Yi wrote down his proof process on the blackboard while telling the story.
Everyone in the audience was completely stunned.
No, buddy, can you really prove it?