The top student must be diligent

Upon hearing this, the mathematicians present immediately became serious.

Generalized modular curve!

This is the most important thing in the entire paper!

Before reading the paper, when they saw the paper, they were only slightly interested in how Xiao Yi performed high-dimensional processing of modular curves.

But after reading it, they all clearly realized that this generalized modular curve can be called a new kind of mathematics!

It is the most critical new theory in the entire proof!

The reason why people expect the Riemann Hypothesis to be proven is not only because the chain reaction caused by the proof of the Riemann Hypothesis can make many propositions based on the Riemann Hypothesis become real theorems, but it is also an expectation to prove the Riemann Hypothesis. In the process, new theories were born.

And this generalized modular curve was the new theory they were looking forward to, and its significance did not disappoint their expectations at all!

"Generalized modular curves, derived from modular curves."

"The main reason why I came up with it is that in the process of proving Artin's conjecture, I wanted to connect the extended L-function with the modular form of the L-function."

"If this connection can be made, then the extended L-function can be studied through the properties of the modular form, and then the arithmetic properties of elliptic curves can be studied."

"So I thought of the contents of Weil's conjecture, which have similar properties to extended L-functions."

"More precisely, for an elliptic curve E defined on the field of rational numbers, its extended L-function L(s, E,) seems to satisfy such a function equation."

【Λ(s,E,)=ε(E,)Λ(1-s,E,)】

"Where Λ(s,E,) is the complete L-function obtained by multiplying L(s,E,) by some Gamma factor, and ε(E,) is a constant called sign."

"So we can easily think that the extended L-function may be related to the Zeta function of certain geometric objects."

"Then I started experimenting with all kinds of geometric objects I could think of."

"But to be honest, in the first two months, I didn't find the geometric objects I wanted, including the mold curve. I also tried, but when I first tried, it didn't cause me to be interested in it. Deeper thinking.”

Xiao Yi spread his hands and expressed.

And hearing him say this, the mathematicians present suddenly felt that the mathematical God in front of them was a little more real.

It turns out that God also encounters situations where the answer is right in front of his eyes but he doesn’t find it.

"But fortunately, it wasn't until one day when my partner and I were playing in an amusement park that we saw the roller coaster and the structure of the Ferris wheel, which inspired me."

After hearing the story he told, everyone present showed expressions of interest and surprise.

What interested me was the gossip, but I was surprised that the amusement park could also help Xiao Yi make mathematical associations?

Is there really no cheating?

But then, Xiao Yi began to tell them how he gradually associated the modular curve from the structure of the amusement park, and began to try to discuss it from a high dimension.

The audience present gradually became absorbed in his narration.

…

Chapter 291 Riemann Hypothesis Report Meeting (3)

"...All in all, it was the geometric trajectories I observed in the amusement park that inspired me."

"Geometry naturally exists in our world, and since mathematics itself is known as the language of the universe, it may be a good way to inspire us through these things."

Xiao Yi smiled slightly: "This is just a little suggestion I brought to everyone."

Many mathematicians present immediately nodded and expressed their approval.

Although, this method still sounds a bit too magical for them, but it does not prevent them from learning from Xiao Yi.

Now as long as it is the method recommended by Xiao Yi, then they are willing to try it, maybe it will suit them.

"So, I have explained to you the process of how I came up with the idea that modular curves can be developed from high-dimensional situations. Next, we will continue to discuss how I finally derived the generalized modular curves."

"At the beginning, I tried modular curves, but it was easily discovered that although modular curves provide a geometric framework for studying extended L-functions, they cannot fully explain all the properties of extended L-functions. , especially for certain types of extended L-functions, whose special values ​​do not seem to fit well with the geometry of the modular curve."

"Then, this is thanks to my research on theoretical physics, which brought me some inspiration."

"We all know that in physics, some high-dimensional geometric spaces are used to study some physical phenomena, such as Calabi-Yau manifolds, so this gave me some inspiration."

"So, for this modular curve in high dimensions, it should contain the usual modular curve as an independent case, but it should also contain more information to characterize those extended L-functions beyond the usual."

"So now we can simply give a definition."

"For an n-dimensional generalized modular curve, we record it as Modular properties, similar to ordinary elliptic curves.”

"Then, then we need to use some special tools to deal with it."

"So I thought of Shimura clusters and Siegel modular forms."

"For an n-dimensional Siegel modular form f, we define a Shimura variety Sh_f, which parameterizes all n-dimensional Abelian varieties with the modular properties described by f."

"In this way, we can prove that there is a natural isomorphism."

【X_f^(n) Sh_f】

…

Xiao Yi began to demonstrate on the blackboard how he proved this natural isomorphism.

However, for most mathematicians in the field, they really can't understand it.

How could Xiao Yi think of these things?

How did he come up with the idea of ​​using Shimura clusters and Siegel modular forms?

How do you provide relevant structures and definitions so accurately?

Is this something humans can do?

They were all in a state of confusion.

For mathematics, finding the tools that can be used to solve problems is only the first step. How to use these tools is the second step.

Sometimes, even if they find a tool, they may not be able to use it to successfully solve the problem. This is mainly because during the process of using it, they still have not found the "key" to embed the tool into the problem well. "hole", so the problem is still a problem and the tools are still there.

This kind of situation occurs quite a lot in the mathematical world.

Just like Andrew Wiles, when he first proved Fermat's Last Theorem, other mathematicians discovered key errors in his proof, so that he almost admitted failure.

But it was not until the end that he found a way to solve the problem from other existing mathematical tools, and finally successfully completed the proof of the paper.

Another example is Perelman, who proved the Poincaré conjecture. In his proof, he mainly used a mathematical tool called ricci flow. Since the birth of this mathematical tool, the mathematical community has seen the use of this tool. Proving the Poincaré conjecture may have a huge impact, but for a long time, mathematicians failed to successfully complete the proof.

It was not until later that Perelman found a way to embed the ricci flow into the "keyhole" of the Poincaré conjecture, and finally completed the proof.

Therefore, finding the tools is only the first step. How to apply the tools is also a very critical step.

Now, Xiao Yi has shown his ability as if he has opened his eyes. Not only can he discover new tools such as generalized modular curves, but he can also find auxiliary tools for embedding generalized modular curves into the "keyhole", such as Shimura clusters and Siegel. Modular form these two.

Is this really not open?

"I bought it... I bought it... I bought it..."

Below, many mathematicians looked at the steps demonstrated by Xiao Yi in amazement. In addition to being shocked, they were also shocked.

"Yeah, who says he's not God?"

Deligne shook his head with emotion and said.

"I never thought that the content extended from Weil's conjecture could be expanded in this way."

Bombieri next to him spread his hands and said: "You just proved Weil's conjecture. What do you know about Weil's conjecture?"

Deligne shrugged and said, "Yes, I completely agree with you."

On the other side, Terence Tao also expressed his surprise.

"Did he really come up with such a result without any trial and error? I really can't believe it..."

Fefferman on the side shook his head and said: "Even if he really made mistakes in this, but...how long do you think he stayed on these mistakes? I feel like it's not possible. Not even for a month.”

"That's right..." Tao Zhexuan sighed.

If it were them who took this step, God knows how long they would have to try and make mistakes.

If they had not seen this paper and only understood the definition of generalized modular curves, and knew that they also needed to use Shimura clusters and Siegel modular forms, with luck, they might be able to complete this step in a month. But if you're not lucky, it may take several months or even a year.

Although they do not think that with their abilities, they will not be able to find the final answer for more than a year, the possibility is definitely not zero. On the contrary, the possibility is even a bit high...

In the past, they might have attributed this more to luck, and then added a little bit of mathematician intuition.

But now listening to Xiao Yi's story, they began to doubt this.

Is this really the case?

Could it be that this process of trial and error can actually be cracked directly with mathematical ability alone?

But they obviously couldn't get an answer to such a question. After all, they were not Xiao Yi, so naturally they couldn't understand Xiao Yi's experience when encountering such problems.

…

Xiao Yi had no idea what the mathematicians were thinking at the moment.

If they asked Xiao Yi, he would probably be a little confused about their question.

After all, he really relied on mathematical intuition, so he would not understand why they had such questions.

Such a situation is generally called the "curse of knowledge."