Section 363

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.

Prover of 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 this old mathematician's reputation, he had not had many opportunities to meet and communicate with him before.

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

After 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 the 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 wanted 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 curves, in particular the automorphism of their Hecke characteristic, to deduce the zero-point distribution of L-functions for all elliptic curves. However, there is a subtle logical problem 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 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 one that Andrew Wiles made in his original proof?

And now, he found the same problem in Xiao Yi's paper?

Does this count as ushering in a reincarnation?

Chapter 293 Riemann Hypothesis Report Meeting (5)

"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 true. After all, Xiao Yi hasn't said yet that 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, I don't really think Xiao Yi can 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 true." Deligne waved his hand and said helplessly: "Now that I am so old, I can no longer react so quickly. Schultz can understand that what I am doing Not here.”

"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 brows slightly after hearing 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 later on.

After all, when he first started to study mathematics in depth, he had carefully studied Wiles' proof process of 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' eyes lit up, and then he smiled and said, "So you also know what happened back then. I thought that young people later on had never heard of the story back then."

When the people present heard the conversation between the two, they immediately became confused. What exactly were they talking about?

Why did they all sound confused?

However, soon someone who knew other details began to popularize what happened to others around them.

Soon, one person told two people, two people told four people... In this way, many people present also learned 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 create a relationship between the Selmer group and the Shafarevich-Tate group of elliptic curves.

However, in his proof, there is a key step that depends on the existence of certain Euler systems, and in subsequent reviews, other mathematicians found that Wiles did not fully prove the existence of these Euler systems.

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

Therefore, this directly constitutes the most fatal point in his proof, so that he failed to prove Fermat's Last Theorem for the first time.

And 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 Xiao Yi's paper, there is no clear way to define the generalized modular curves of those non-CM elliptic curves, so this implicit assumption is obviously wrong.

And this is fatal.

After the audience learned this, they were all amazed.

And similarly, they also had a stronger interest in today's historic report.

This time, they did not come here in vain!

...

At this moment, Xiao Yi continued: "In my spare time, I also have some understanding of the history of mathematics, and of course I have learned about the things at the beginning."

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

"To be precise, it has been 33 years." Xiao Yi said.

Wiles was startled, 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 glad that our mathematical god can remember this matter so clearly. I am probably old now, and I don't have such a deep impression of the original incident."

"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 will gradually decrease."

"Fortunately, at least I don't have Alzheimer's disease. I don't know if you have seen the movie called "The Father Trapped in Time". It was made by us British people. It was really terrible. I even forgot the time."

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

Wiles laughed again, because the male protagonist 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 chat, but don't forget the question I asked you just now."

"Don't think of using this method to make me forget that I asked you a question just now."

"After all, I was not Hannibal when I was young."

He stretched out a hand and shook his index finger.

Xiao Yi smiled and said, "Of course I didn't forget."

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 quieted down, and for a while, there was almost no sound in the entire lecture hall.

No one wanted to disturb Xiao Yi's thinking because of the noise they made.

Even Wiles carefully controlled his microphone to prevent it from making noise.