The top student must be diligent.

Chapter 293 Riemann Hypothesis Seminar (V)
“It’s getting interesting.”

Deligne said with a smile.

"Indeed, is this what they call history repeating itself?"

Bombieri also smiled and nodded.

"That's not necessarily the case. After all, Xiao Yi hasn't said yet that he can't answer this question."

Deligne said, shaking his head.

Then, he shifted his gaze from Wiles to Xiao Yi and said, "To be honest, with the answer to the previous question, I don't think Xiao Yi will be unable to answer this question."

"Oh? You mean, you understood Xiao Yi's answer to the last question?"

Bombieri raised his eyebrows, glanced at Deligne and said in surprise.

"Uh... that's not the case." Deligne waved his hand and said helplessly, "Now that I'm so old, I can't react that quickly. Schultz can understand it, but I can't."

"That's good." Bombieri nodded with a smile and said, "Otherwise, I would have wondered to what extent my thinking ability has declined."

Then, he turned his head and looked at Wiles, and said, "But, to be honest, I'm more curious about what Andrew is thinking right now."

"Is he thinking about how his past self felt when faced with this problem, or does he want to see the young man in front of him who can easily solve the problem that had troubled him for a year?"

Deligne smiled and said, "Maybe both."

……

On the stage, after hearing this question, Xiao Yi raised his eyebrows slightly.

Looking at Wiles with an expectant expression on his face, he already remembered the problem that Wiles had encountered at the beginning.

Although he did not experience what happened back then, it did not prevent him from having a deeper understanding of this past story later on.

After all, when he first started to delve into mathematical research, he had carefully studied Wiles' proof of Fermat's Last Theorem and had a very deep understanding of it.

At this moment, this question...

He smiled and said, "Such a question must be a very memorable answer for Professor Wiles, right?"

Wiles' eyes lit up, and then he smiled and said, "So you also know what happened back then. I thought young people later on had never heard of the story."

When the people present heard the conversation between the two, they all became confused. What on earth were they talking about?
Why do they all sound so confused?

However, soon someone who knew other details began to tell other people around them what had happened.

Soon, the story spread from one to two, from two to four... and just like that, 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 a relationship between the Selmer group and the Shafarevich-Tate group of elliptic curves.

However, a key step in his proof relied on the existence of certain Euler systems, and in subsequent scrutiny, other mathematicians discovered that Wiles had not fully proved the existence of these Euler systems.

In other words, he also implicitly assumed the existence of these Euler systems.

Therefore, this directly constituted the most fatal point in his proof, so that he failed in his first proof of Fermat's Last Theorem.

And now, the question he raised is the same: Xiao Yi implicitly assumes that all elliptic curves can be embedded in a generalized modular curve.

But in fact, in Xiao Yi's paper, for those non-CM elliptic curves, there is no clear way to define their generalized modular curves, so this implicit assumption is obviously wrong.

And this is fatal.

When the audience learned this, they were all amazed.

Similarly, they also showed a greater interest in today's historic lecture.

This time, their trip here was worthwhile!
……

At this moment, Xiao Yi continued, "In my spare time, I have also learned a little about the history of mathematics, and of course I have learned about what happened back then."

Upon hearing this, Wiles sighed and said, "History of mathematics? It turns out that things that happened more than 20 years ago are already history."

"To be precise, thirty-three years have passed." Xiao Yi said.

Wiles was stunned for a moment, then burst into laughter.

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 glad that our math god can remember this incident so clearly. I am probably old now, and the impression of the incident is not so deep."

"You know, when we are young, we can remember clearly how many years have passed, but as you spend more and more time, your sensitivity to time will gradually decrease."

"Luckily, at least I don't have Alzheimer's. I don't know if you've seen the movie called 'The Father', made by us British. It's really scary. I even forgot the time."

Xiao Yi smiled and said, "I know that movie very well. It taught me a truth. Hannibal had a miserable end in his later years."

Wiles laughed again because the actor with Alzheimer's disease in this movie was the actor who played Hannibal in "The Silence of the Lambs".

But soon, he stopped smiling: "Okay, let's end our small talk. Don't forget the question I asked you just now."

"Don't think you can make me forget that I just asked you a question in this way."

"After all, I wasn't Hannibal when I was young."

He held out a hand and shook his index finger.

Xiao Yi smiled and said, "Of course I haven't forgotten."

But after saying this, he fell silent again.

But obviously, everyone could see that he was thinking.

Perhaps, his casual chat with Wiles just now was also to give himself more time to think?

The voices of those who were still discussing gradually became quieter. For a moment, the entire lecture hall was almost silent.

No one wants to disturb Xiao Yi's thinking because of the noise they make.

Even Wiles was careful to control his microphone so that it wouldn't make any noise.

Obviously, this problem really hits 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 this math god looked like when he was thinking about problems.

But now it seems that there is no difference from the way they usually think.

That is to say, when they are thinking, they might frown, but Xiao Yi's brows were relaxed.

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 results of.

No matter what the result is, it can lead to different interpretations.

And so, a long time passed.

It takes about two or three minutes.

As a report meeting lasts for two or three minutes, it 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 and obscure enough question that it took me a while to think about it before I remembered how I considered this question when I was proving it."

When everyone heard him say this, they all sat up straight.

Do you remember how you thought about this problem at the beginning?
Could it be that Xiao Yi had already noticed this problem from the beginning?

So now, he figured out how to answer this question?

But then, Xiao Yi's words made them all stunned again.

“Well, I have to admit that I did implicitly assume what you said, that all elliptic curves can be embedded in a generalized modular curve.”

what?
This statement...

Is Xiao Yi going to admit that he has failed?

Everyone present seemed to be witnessing the collapse of a myth.

After all, they had never seen Xiao Yi fail.

However, Xiao Yi's next sentence followed.

"In fact, when I first came to this point in my research, my mathematical intuition told me that this assumption was completely correct, so I did not discuss this situation during the proof. So, the mistake I probably made was..."

Xiao Yi paused, then spread his hands and said, "This question has been made 'obvious' by me."

Everyone was stunned listening to his story.

What does this mean?

So, did Xiao Yi admit that he was wrong, or did he admit that this problem was actually an obvious theorem in his eyes, so obvious that he felt there was no need to prove it in the paper?

Wiles was also stunned, not quite understanding what Xiao Yi meant.

After all, he made this implicit assumption basically because he subconsciously accepted it in his mind, which made it a key flaw in the entire proof, so that he was confused for almost a year before he succeeded.

Now, what Xiao Yi said seemed to be the same reason why he made the mistake in the first place.

But...

The information revealed in Xiao Yi's other words was slightly different.

For a moment, he was a little confused.

But then, Xiao Yi's words completely stunned all of them.

Xiao Yi turned around, walked back to the blackboard, and said, "Well, in that case, I will prove it on the spot. I will prove that all elliptic curves can be embedded in a generalized modular curve."

Then he began to wipe the blackboard, which had been wiped countless times in the previous report, and said, "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?
Field proof?

They didn't even know where to start solving this problem, and Xiao Yi was 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 wondering, what was Xiao Yi thinking about during those few minutes?
Or, as he just said, he was actually just recalling how he thought about this matter when he was proving it?
They were all a little stunned at this point.

The top mathematicians sitting in the front row all had the same reaction.

At this moment, they all looked at Xiao Yi who was wiping the blackboard with a little disbelief, their faces full of disbelief.

"He wants to prove it on the spot? This kind of question... this kind of complicated question... can it 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 calm about it.

"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 stunned for a moment, then laughed and said, "Well, if you can be more precise, it's really getting more and more interesting."

At this time, Terence Tao seemed a little excited.

"So, this kind of question is basically impossible to stump him. In his opinion, such questions are actually very obvious. He is the real god of mathematics! Holy crap!"

As for Fefferman and Qiu Chengtong next to them, they were a little silent, because in fact, they were also shocked by Xiao Yi's behavior of proving it on the spot.

Does this really still fall within the realm of humanity?

They really want to ask such questions.

But they didn't have time to think about it.

Xiao Yi on the stage had already wiped the blackboard, then picked up the blackboard pen and walked to the blank space on the left.

"Well, my proof begins now."

"First of all, the reason why I thought this was obvious when I proved it, or why I had this intuition, was mainly due to 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. In my proof, this automorphism is realized through the Hecke characteristic of the generalized modular curve."

"But now, if we need to rigorously prove this process, then we need a series of logical inferences."

"Then, we first give the proposition: for any elliptic curve E, there always exists a generalized modular curve M such 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 property 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 operations, similar to a group ring generated by a group."

"In this construction, the automorphism σ on E can be naturally extended to M(E), denoted by σ:M(E)→M(E)."

"Then the third step..."

As Xiao Yi spoke, he wrote down his proof process on the blackboard.

The people in the audience were completely stunned.

He is not my friend, can you really prove it?

(End of this chapter)