The top student must be diligent

Xiao Yi shrugged, and then without saying anything, he looked back at the tracks in front of him.

The elliptic curve is also a complex curve. Its shape determines many of its arithmetic properties. If the elliptic curve can be "deformed" into a shape like a roller coaster track in some way, what would it look like?

Perhaps he has been obsessed with the study of this thing recently. He began to build a modular curve amusement park in his mind with the roller coaster track in front of him. In this amusement park, each amusement facility corresponds to a modular curve.

These modular curves parameterize elliptic curves with certain special properties, just like the roller coaster track parameterizes the movement of the roller coaster.

"Modular curve..."

While thinking in his mind, Xiao Yi naturally began to think about this geometric object.

He had thought about connecting the modular curve with the problem he was facing before, and although there were some connections between the two, there was still a little difference.

But now...

He suddenly had some ideas in his mind?

However, before he could continue thinking, the roller coaster started.

As the car slowly reached the highest point and was about to fall, Luo Mingya, who was beside him, grabbed his hand again.

Then——

"Ah!"

The screams of a car spread throughout the amusement park.

...

Chapter 282 Generalized Modulus Curve!

After getting off the roller coaster, Luo Mingya patted her chest and exhaled.

"Wow, it's been a long time since I've ridden a roller coaster. It's so exciting."

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

However, Xiao Yi looked thoughtful at this time.

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

The modular curve may not be the geometric object he wants. He has already made a certainty about this when he studied the modular curve before.

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

This is not only from a mathematical perspective, but more importantly, it is what his intuition tells him.

He has always believed in his intuition.

However, what he wants to think about now is what exactly is the geometric object that is close to the modular curve. In his impression, although there are a few concepts close to the modular curve, those concepts have been verified by him. After thinking about it for a few months, he has naturally tried everything he can think of. As for the unexpected...

He thinks that there is basically nothing he can't think of.

Of course, since he has thought about it to this point, he has to consider the last possibility.

"So now, if you want to continue, you need a new geometric concept that is similar to the modular curve?"

If you need to develop a completely new geometric concept...

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

"Huh? You are not thinking about the elliptic curve you just mentioned."

At this time, Luo Mingya next to him saw him thinking and asked.

Xiao Yi came back to his senses and smiled: "Indeed, it even brought 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 abides by the rules of the universe can probably bring some inspiration to mathematical problems."

"I really can't imagine what the world looks 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 amusement facilities in the amusement park, and then pointed to the very obvious Ferris wheel over there and said: "Then why don't 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."

He went to the Ferris wheel and quickly lined up. Coincidentally, the people in front of them were Qian Wanli's family.

However, Xiao Yi and his friends did not greet Qian Wanli, and Qian Wanli did not notice them either, 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, compared to the track of the roller coaster, looks more complicated, but in fact it seems more regular.

After all, it looks like a standard prototype.

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

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

Then he put the data in a coordinate system, and put this coordinate system in the "model curve amusement park" he just thought of.

The thinking in his mind inadvertently came to a new level.

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

"So we need an object that is more 'rich' than modular curves..."

A brainstorm started in my mind.

But it soon stopped.

Because it was their turn to go up.

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

As the facility's mechanical arm slowly turned, they began to rise gradually.

Luo Mingya looked at Xiao Yi sitting opposite him.

Xiao Yi was looking at the structure outside.

At this time, the scene in his mind was still the image of perfect engineering, a circular structure, and they were 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 strictly defined. If we simply talk about the parametric equation of an ellipse, a circle can still be called a special ellipse, with its two foci coinciding and the focal length equal to zero.

So if we express it in this way, a circle is like a zero point of an ellipse.

What if we combine it with a modulus curve?

Xiao Yi's thoughts seemed to be traveling through an endless fog at this time, but until a certain moment, when their cabin rose to a higher altitude, the sunlight that passed through the clouds just passed through the porthole and shone on his cheek.

So, his thoughts emerged from the layers of fog.

And then a brand new idea came into being.

"If I can construct a higher-dimensional analogue of modular curves, what would it look like?"

He raised his head and looked up at the sun.

His eyes were particularly bright.

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

"So..."

He took out a pen and a small notebook from his pocket and started writing on it.

He has thought of a brand new analogue of modular curves.

A higher-dimensional modular curve.

He named it the generalized modular curve.

Now it is recorded as X_f^(n), which is itself an n-dimensional complex manifold. It parameterizes a special type of n-dimensional Abelian clusters. These Abelian clusters have some modular properties, similar to the usual elliptic curves.

Hmm...

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

So his mind leaped again, and he recalled the mathematical knowledge he had mastered.

Shimura clusters, Siegel modular forms...

The Ferris wheel gradually rose, and the sun had 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 in her heart, 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, except for this small space 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 likes this feeling more.

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

She didn't know when she had such emotions. Obviously, she used to particularly favor the life of her cousin, and felt that she was probably the same single.

But it changed after all.

Seeing the man's eyes in front of her gradually being solved because of the blockage problem, she also smiled and was happy for him from the bottom of her heart.