After getting off the roller coaster, Luo Mingya patted his chest and exhaled.
"Wow, I haven't been on a roller coaster for a long time. It's so exciting."
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 time, Xiao Yi looked thoughtful.
Although he originally planned to have a good time, some new ideas did come to his mind as inspiration emerged inadvertently.
The modular curve may indeed not be the geometric object he wanted. Regarding this, he had already made a determination when he studied the modular curve before.
But looking at it now, although the mold curve cannot meet his requirements, it must be very close to what he wants.
This is not just from mathematical thinking, but more importantly, this is what his intuition told him.
He has always believed in his intuition.
However, what he wants to think about now is what the geometric object that is close to the modular curve is. In his impression, although there are several concepts that are close to the modular curve, those concepts have been verified by him. After thinking about it for the past few months, he has naturally tried everything he can think of. As for the unexpected...
He felt that basically there was nothing he could not think of.
Of course, now that we have thought about it to this point, we have to consider the last possibility.
"So now that means that if you want to continue, you need a new geometric concept that is similar to the modular curve?"
If a completely new geometric concept needs to be developed...
Xiao Yi raised his eyebrows, it seemed that it was not impossible.
"Huh? Are you thinking about the elliptic curve you just mentioned?"
At this time, Luo Mingya next to him couldn't help but ask when he saw how he was thinking.
Xiao Yi came back to his senses and said with a smile, "Indeed, it even brought me a little inspiration."
"Ah?" Luo Mingya was surprised. "Can even riding a roller coaster bring you inspiration?"
"Yes." Xiao Yi nodded. "Mathematics is the language of the universe, so anything that obeys the rules of the universe can probably bring some inspiration to mathematical problems."
"I really can't imagine what the world is like in the eyes of you mathematicians."
Luo Mingya sighed again.
Just like she once said to Xiao Yi, she envied them for being good at mathematics.
Then she turned her head and looked at the many rides in the amusement park, then pointed to 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."
We went to the Ferris wheel and quickly queued up. By coincidence, the few people in front of them were the Qian Wanli family.
However, Xiao Yi and the others did not greet Qian Wanli, and Qian Wanli did not notice them either, just enjoying the rare feeling of getting along with their family.
Luo Mingya looked at them, while Xiao Yi looked up at the Ferris wheel.
Although the geometric shape of the Ferris wheel looks more complicated than the track of the roller coaster, it actually appears to be more regular.
After all, it looks like it is a standard prototype.
Xiao Yi inadvertently activated his Perfect Engineering ability, and then constructed the entire Ferris wheel model in his mind.
He was then able to see the entire data about the Ferris wheel in detail.
Then he placed the entire data in a coordinate system, and placed this coordinate system in the "modular curve amusement park" he just thought of.
The thoughts in my mind reached a new level inadvertently.
"The main problem with the modular curve is that although it provides a lot of useful information, it cannot fully explain all the properties of the extended L-function, such as some special values and the relationship between these special values and the arithmetic properties of the elliptic curve."
"So we need an object that is 'richer' than the modular curve..."
There was a brainstorm in my mind.
But it soon stopped.
Because it's their turn now.
Luo Mingya and got into a cabin, sat opposite each other, and then the staff closed the cabin door.
They began to gradually rise as the facility's robotic arm slowly turned.
Luo Mingya looked at Xiao Yi sitting opposite.
Xiao Yi is looking at the structure outside.
At this time, the scene in his mind was still the image under Perfect Engineering, a circular structure, and they were points on this orbit, constantly rotating on the orbit, and forming a continuous curve.
Although in a purely mathematical sense, circles and ellipses cannot be regarded as the same kind, because the definitions of the two are different. A circle is the set of all points on the plane whose distance from a certain fixed point (center of the circle) is equal to a certain fixed value (radius); and the ellipse is the set of all points on the plane whose distance from two fixed points (focus) is equal to a certain fixed value.
However, this is from a strict definition. If we simply look at the ellipse parametric equation, a circle can still be called a special ellipse, special in that its two foci coincide and the focal length is equal to zero.
So expressed in this way, the circle is like a zero point situation of the ellipse.
What if we combine it with the mold curve?
Xiao Yi's thoughts seemed to be traveling through endless fog at this time, 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 shine on his cheek.
As a result, his thinking emerged from the layers of fog.
This gave rise 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 opened his eyes to the sun.
The eyes are exceptionally bright.
"This space should contain the usual modular curves as a 'slice', but should also contain more information to characterize those special extended L-functions."
"So..."
He took out a pen and a small notepad directly from his pocket and started writing on it.
He had come up with a completely new, analogue of the modular curve.
A higher dimensional modular curve.
He named it, Generalized Modular Curve.
Now record it as
Hmm...
Then some things need to be added to make this thing more universal.
So his mind crossed again and he recalled the mathematical knowledge he had mastered.
Shimura variety, Siegel modular form...
The Ferris wheel gradually rose, and the sun already illuminated the entire cabin.
Seeing that Xiao Yi was able to take out a pen and paper to study even at this time, Luo Mingya didn't feel that he didn't understand the style, but only had a faint sense of tranquility, just like at this moment, the noise outside the amusement park was completely isolated.
Finally, when they reached the highest point of the Ferris wheel, except for the small space here and the sky in the distance outside the window, there was only the person in front of them and nothing else.
Young girls like the kind of love that has ups and downs like in romance novels, but she is more satisfied with this feeling now.
Moreover, although it looks very plain, there is also a different sentiment hidden inside.
She didn't know when she had such an emotion. She clearly liked her cousin's life in the past, and she felt that she was probably single.
But eventually something changed.
Seeing that the blockage in the eyes of the man in front of her was gradually being resolved, she also smiled, feeling happy for him from the bottom of her heart.
Today's team building was not in vain.
In fact, she was the one who suggested team building in the first place. At that time, she saw that Xiao Yi often frowned slightly because he had not solved the problem, so she suggested taking some time to relax. Maybe relaxing would help us gain ideas?
Now, is it considered that you have achieved your wish?
The Ferris wheel gradually descends, and the scenery at the top disappears.
Luo Mingya turned his head and saw the cabin where Qian Wanli's family was.
Qian Wanli and his wife were sitting together, while Qian Huiyin was looking out the window happily.
The breeze slipped into the cabin through the window sill, causing Luo Mingya's hair to rise.
Although she doesn't pursue the ups and downs of romance, she still pursues romance after all.
The romance at this time is just right.
Finally, the Ferris wheel turned around and gradually returned to the bottom.
When he was about to go out, Xiao Yi finally stopped the pen in his hand and raised his head.
Noticing the way Luo Mingya looked at him, he smiled slightly.
"Is the problem solved?" Luo Mingya asked.
"It's not completely solved." Xiao Yi said, "However, now we are not far away from the final answer."
Luo Mingya raised his eyebrows. "You mean, Riemann Hypothesis, it's about to be solved?"
"That's not the case." Xiao Yi smiled and waved his hand, "What is being solved now is just a phased problem. It is still far away from truly solving Riemann Hypothesis."
"However, being able to achieve this step now..." Xiao Yi's eyes showed confidence, "It's probably almost done."
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 fun."
"Okay." Luo Mingya chuckled and nodded, and then they walked side by side to other places in the amusement park.
A day sometimes feels like it goes by very quickly, and sometimes it feels like it goes by very slowly.
When the past is fast, people will want it to be too slow, but when the past is slow, people will want it to be too fast.
For everyone at Science Island Laboratory, today, they are all the former.
However, time will not wait for anyone after all. As the amusement park closed and all the tourists left, the day came to an end. Everyone reported that they had returned to their dormitories or homes to report that they were safe.
The night passed.
The next day, the sun rose as usual.
Xiao Yi also slept in the lounge in his office last night, so after getting up and washing up, he could start research in the office.
He took out the notepad from the pocket of yesterday's clothes and read the notes on it, with a slight smile on his face.
"OK, we can officially start today!"
I contacted Wang Hao and asked him to bring me breakfast from the cafeteria. Then I sat at my desk, took out my scratch paper and pen, and started this most critical derivation.
Generalized Modular Curve, then we must first review the definition of the mold curve.
[For a positive integer N, define the modulo curve Here, Γ(N) is the main congruential subgroup of the modular group SL(2, Z), defined as...]
"Next define Generalized Modular Curve..."
[Suppose n is a positive integer, f is an n-dimensional Siegel modular form, that is, the 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), there is f((Aτ+B)(Cτ+D)^(-1))=det(Cτ+D)^kf(τ), where (A B; C D) is an element in Sp(2n, Z), k is the weight of f; at each cusp of H_n, f satisfies certain growth conditions.]
"So, for such f, we can define Generalized Modular Curve X_f^(n) as the module space of Siegel upper half-space H_n, modulating the equivalence relationship generated by the action of Γ^(n)(f)."
[Here, Γ^(n)(f) is a subgroup of 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 the modular properties described by f."
Having written this, Xiao Yi smiled slightly.
At this point, he has solved the most critical problem.
Although this expanded new geometric concept is named Generalized Modular Curve, it has become a completely new thing.
It further embodies an important idea in modern mathematics, which is to understand the nature 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."
Xiao Yi With just a simple observation, it is easy to notice that for each n-dimensional Generalized Modular Curve X_f^(n), there is a special type of n-dimensional Abelian varieties, and their extended L-functions are closely related to the Zeta function of X_f^(n).
Of course, just observing is not enough, you also need to prove it.
But now that we have arrived here, it is not too difficult.
After spending several scraps of paper, he finally came up with a brand new theorem: Suppose E is an n-dimensional Abelian variety, and f is an n-dimensional Siegel modular form; if the modular property of E is described by f, then the extended L-function L(s, E,) of E is equal to Zeta function ζ(X_f^(n), s) of Generalized Modular Curve X_f^(n).
"In this way, the most troublesome step has been successfully completed."
Then, the next thing to do is to move towards the final proof!
Artin Conjecture, now I can't stop him.
By associating each extended L-function with a Generalized Modular Curve, he can use the geometric properties of Generalized Modular Curve, such as dimensionality, Betti number, Hodge structure, etc., to characterize the characteristics of the extended L-function.
Finally, the answer finally appeared in front of him.
Half a month later.