The top student must be diligent.

Chapter 282 Generalized Modular Curve!
After getting off the roller coaster, Luo Mingya patted his chest and breathed a sigh of relief.

"Wow, I haven't ridden a roller coaster in a long time. It's really thrilling."

Then she turned to look at Xiao Yi and asked, "Why were you so calm just now? You didn't even shout."

However, at this moment, Xiao Yi had a thoughtful look on his face.

Although he originally planned to have a good time, the inspiration that emerged inadvertently did make some new ideas emerge in his mind.

He had already determined that modular curves might not be the geometric object he wanted when he was studying modular curves before.

But as it stands now, although the model curve cannot meet his requirements, it must be very close to what he wants.

This is not just a result of mathematical thinking, but more importantly, it is what his intuition tells him.

He has always trusted his intuition.

However, what he wanted to think about now was what exactly that geometric object that was close to the modular curve was. In his impression, although there were a few concepts that were close to the modular curve, those concepts had already been verified by him. After thinking about it for a few months, he had naturally tried everything he could think of. As for the ones he couldn't think of...

He felt that there was basically nothing he couldn't think of.

Of course, now that we have thought about it to this point, we have to consider the last possibility.

"So now, if we want to proceed, we need a new geometric concept similar to modular curves?"

If a completely new concept of geometry needs to be developed...

Xiao Yi raised his eyebrows, it seems that it is not impossible.

"Hmm? You're not thinking about the elliptic curve you just mentioned, are you?"

At this time, Luo Mingya, who was standing next to him, saw him thinking and couldn't help but ask.

Xiao Yi came back to his senses and smiled: "Indeed, it even gave me some inspiration."

"Ah?" Luo Mingya was surprised: "Just riding a roller coaster can actually bring you inspiration?"

"Yes." Xiao Yi nodded: "Mathematics is the language of the universe, so everything that obeys the rules of the universe can probably bring some inspiration to mathematical problems."

“I really can’t imagine what the world looks like to you mathematicians.”

Luo Mingya sighed again.

Just like she once said to Xiao Yi, she envied those people who were good at math.

Then she turned around and looked at the many amusement facilities in the amusement park, then pointed at the very obvious Ferris wheel over there and said, "How about we go ride the Ferris wheel? Maybe it can bring you more inspiration?"

Xiao Yi turned his head and looked over, then smiled and said, "Okay."

They went to the Ferris wheel and quickly got in line. Coincidentally, the people in front of them were Qian Wanli's family.

However, Xiao Yi and the others did not greet Qian Wanli, and Qian Wanli did not notice them. He was just enjoying the rare feeling of being with his family.

Luo Mingya looked at them, while Xiao Yi looked up at the Ferris wheel.

The geometric shape of the Ferris wheel looks more complicated than the track of a roller coaster, but it is actually more regular.

After all, it looks like a standard prototype.

Xiao Yi inadvertently activated his perfect engineering ability, and then constructed the entire Ferris wheel model in his mind.

Then he was able to see the entire data about the Ferris wheel in detail.

Then he placed all the data in a coordinate system, and placed this coordinate system in the "Model Curve Amusement Park" he had just thought of.

The thoughts in my mind have inadvertently reached a new level.

"The main problem with modular curves is that, although they provide a lot of useful information, they do not fully explain all the properties of extended L-functions, such as some special values ​​and how these special values ​​relate to the arithmetic properties of elliptic curves."

"So we need an object that is richer than the modular curve..."

A brainstorm started in my mind.

But it soon stopped again.

Because it's their turn to go up now.

Luo Mingya and I got into a cabin and sat opposite each other, and then the staff closed the cabin door.

As the facility's robotic arms slowly turned, they began to rise gradually.

Luo Mingya looked at Xiao Yi who was sitting opposite her.

Xiao Yi was looking at the structure outside.

At this moment, the scene in his mind was still the image of perfect engineering, a circular structure, and they were the points on this orbit, constantly rotating on the orbit and forming a continuous curve.

Although from a purely mathematical point of view, circles and ellipses cannot be considered the same type because their definitions are different. A circle is the set of all points on a plane whose distance to a fixed point (center) is equal to a fixed value (radius); while an ellipse is the set of all points on a plane whose sum of distances to two fixed points (foci) is equal to a fixed value.

However, this is from a strict definition. If we simply talk about the parametric equation of the ellipse, the circle can still be called a special ellipse, because its two foci coincide and the focal length is zero.

So expressed this way, the circle is like a zero-point case of the ellipse.

What if we combine it with the module curve?
At this moment, Xiao Yi's thoughts seemed to be traveling in an endless fog, but until a certain moment, when their cabin rose to a higher altitude, the sunlight passing through the clouds happened to pass through the porthole and shone on his cheek.

Then, his thoughts emerged from the layers of fog.

This led to a completely new idea.

“If I could construct a higher-dimensional analogue of a modular curve, what would it look like?”

He raised his head and looked up at the sun.

The eyes are exceptionally bright.

"This space should contain the usual modular curves, making them a 'slice', but it should also contain more information to characterize those special extended L-functions."

"so……"

He simply pulled out a pen and a small notebook from his pocket and began writing.

He had already thought of a completely new analogue of the modular curve.

A higher dimensional modular curve.

He named it the generalized modular curve.

Now let's denote it as X_f^(n), which is itself an n-dimensional complex manifold that parameterizes a special class of n-dimensional Abelian clusters that have some modular properties similar to the usual elliptic curves.

Ok……

Then we need to add some other things to make this thing more universal.

So his mind took another leap and he recalled the mathematical knowledge he had once mastered.

Shimura cluster, Siegel module form...

The Ferris wheel gradually rose higher, and the sunlight illuminated the entire cabin.

Seeing Xiao Yi taking out a pen and paper to study even at this time, Luo Mingya didn't think he was unromantic, but only felt a faint tranquility, just like at this moment, the noise outside the amusement park was completely isolated. Finally, until they reached the highest point of the Ferris wheel, all they could see was the person in front of them, except for this small space and the sky outside the window.

Young girls like the kind of love relationship with ups and downs like in romance novels, but she prefers this feeling more.

Moreover, although it looks very plain, there is a different mood hidden inside.

She didn't know when she had developed such emotions. She used to particularly like her cousin's life and felt that she was probably a single person in the same way.

But things did change after all.

Seeing the man's eyes gradually clearing up due to the blockage problem, she smiled and felt happy for him from the bottom of her heart.

Today’s team building was not in vain.

In fact, it was she who first proposed the team building. At that time, she saw Xiao Yi frowning slightly because he had not solved a problem, so she suggested taking some time to relax. Maybe relaxing would help him come up with some ideas?

Now, has my wish been fulfilled?

The Ferris wheel gradually descends and the scenery at the top disappears.

Luo Mingya turned her head and saw the cabin where Qian Wanli and his family were.

Qian Wanli sat with his wife, while Qian Huiyin looked out the window with a happy face.

A breeze slipped into the cabin through the window sill, blowing Luo Mingya's hair up with it.

Although she does not pursue the kind of romance with great ups and downs, she still pursues romance after all.

The romance at this moment is just right.

……

Finally, the Ferris wheel turned a circle and gradually returned to the bottom.

Just as he was about to go out, Xiao Yi finally put down the pen in his hand and raised his head.

Noticing the way Luo Mingya was looking at him, he smiled slightly.

"Is the problem solved?" Luo Mingya asked.

"It's not completely solved." Xiao Yi said: "But now we are not far from the final answer."

Luo Mingya raised her eyebrows: "So, the Riemann hypothesis is about to be solved?"

"That's not the case." Xiao Yi smiled and waved his hand: "What we have solved now is just a temporary problem. We are still a little far from truly solving the Riemann hypothesis."

"However, now that we have achieved this point..." Xiao Yi's eyes showed confidence, "I think we have almost done it."

At this moment, the hatch was opened from the outside.

The staff signaled that they could go down.

Luo Mingya and Xiao Yi went down one after another.

Xiao Yi also put the paper and pen back into his pocket at this time, and then said to her with a smile: "Okay, let's go, I won't continue to study the problem for the rest of the day, let's have some fun."

"Okay." Luo Mingya chuckled and nodded, and then they walked side by side towards other places in the amusement park.

……

Sometimes a day seems to pass quickly, and sometimes it seems to pass slowly.

When the past is passing quickly, people will hope it can pass slowly, but when the past is passing slowly, people will hope it can pass quickly.

But for everyone in the Science Island Laboratory, today, they are all the former.

However, time waits for no one. With the closing of the amusement park and the departure of all tourists, the day came to an end. Everyone reported that they had returned to the dormitory or home and were safe.

The night passed.

The next day, the sun rose as usual.

Xiao Yi also slept in the lounge of his office last night, so after getting up and washing up, he could start his research in the office.

He took out a notebook from the pocket of his clothes yesterday and read the notes left on it, a slight smile appeared on the corner of his mouth.

"OK, we can officially start today!"

He contacted Wang Hao and asked him to bring him breakfast from the cafeteria. Then he sat down at his desk, took out draft paper and pen, and started this most critical deduction.

Generalized modular curve, so first we have to review the definition of modular curve.

[For a positive integer N, define the modular curve X(N) as the moduli space of the complex upper half plane H, modulo the equivalence relation produced by the action of Γ(N). Here, Γ(N) is the principal congruence subgroup of the modular group SL(2, Z), defined as...]

"Next, we define the generalized modular curve..."

[Let n be a positive integer, f be an n-dimensional Siegel modular form, i.e., a holomorphic function f:H_n→ C, where H_n is the upper half space of the n×n complex symmetric matrix τ=(τ_ij), which satisfies: for all γ∈Sp(2n, Z), f((Aτ+B)(Cτ+D)^(-1))=det(Cτ+D)^kf(τ), where (AB; CD) are elements in Sp(2n, Z), and k is the weight of f; at each cusp of H_n, f satisfies certain growth conditions. ]

"So, for such f, we can define the generalized modular curve X_f^(n) as the modular space of the Siegel upper half space H_n, modulo the equivalence relation generated by the action of Γ^(n)(f)."

[Here, Γ^(n)(f) is a subgroup of the Siegel module group Sp(2n, Z), which depends on f and is defined as: Γ^(n)(f)={γ∈Sp(2n, Z)|f(γ(τ))=f(τ), for all τ∈H_n}]

"At this point, X_f^(n) has successfully parameterized all n-dimensional Abelian varieties with modular properties described by f."

Having written this, Xiao Yi smiled slightly.

At this point, he has solved the most critical problem.

Although this new expanded geometric concept is named generalized modular curve, it has become something completely new.

It further reflects an important idea in modern mathematics, which is to understand the essence of things at a higher level and discover hidden connections by introducing new mathematical structures.

"Then, next, it's time to return to the extended L-function itself."

With just a simple observation, Xiao Yi was able to easily notice that for every n-dimensional generalized modular curve X_f^(n), there exists a special class of n-dimensional Abelian clusters, whose extended L-functions are closely related to the Zeta function of X_f^(n).

Of course, observation alone is not enough; proof is also required.

But now that we've come this far, there isn't much difficulty there.

After spending several sheets of draft paper, he finally gave a brand new theorem: Let E be an n-dimensional Abelian variety and f be an n-dimensional Siegel modular form; if the modular properties of E are described by f, then the extended L-function L(s, E,) of E is equal to the Zeta function ζ(X_f^(n), s) of the generalized modular curve X_f^(n).

"In this way, the most troublesome step has been successfully completed."

So, what we need to do next is to move towards the final proof!

Atin guessed that it was impossible to stop him now.

By associating each extended L-function with a generalized modular curve, he could use the geometric properties of the generalized modular curve, such as dimension, Betti number, Hodge structure, etc., to characterize the characteristics of the extended L-function.

Finally, the answer was finally placed before his eyes.

half a month later.

……

(End of this chapter)