Section 345

"In traditional theory, for an elliptic curve E, we usually consider its Hasse-Weil L-function, defined as..."

[L(s, E)=∏(p) 1/(1-a_p p^(-s)+ p^(1-2s))]

"where p takes all prime numbers, and a_p is the trace of E in the reduction modulo p."

"However, this L-function does not fully capture all the arithmetic properties of E, especially some special geometric structures, which are the structures I need."

"The cyclic class must be included..."

"Hmm..."

Xiao Yi closed his eyes.

Perhaps the relaxation of his brows brought some changes, and he suddenly had some ideas.

"For each prime number, if we consider the p-adic etale cohomology group H^i(E, Q^p)(i=0,1,2) of E..."

"Well, these cohomology groups can carry a lot of geometric information of E. In addition, H^1(E, Q^p) also contains the Tate module T^p(E) of E as a subspace."

Xiao Yi's eyes suddenly lit up, and the problem he encountered now suddenly had a solution.

Then he began to deduce on the draft paper, mainly using the p-adic etale cohomology group.

As for the p-adic etale cohomology group, he knew it too well.

After all, the self-conservation theory of etale algebraic varieties that he first came up with was based on this theory, and until now, the self-conservation theory of etale algebraic varieties is still a popular theory in the mathematics community. As for his original paper, the number of citations has exceeded 200.

For other disciplines, more than 200 citations may be considered a lot, but it is not particularly high.

But for mathematics, more than 200 citations are definitely quite a lot in mathematics. Even a classic paper from decades ago may only have one or two hundred citations now.

This is because the threshold of mathematics is too high, and these classic mathematics papers are necessarily quite abstract. There may not be many people who can understand them later, let alone write a paper that can use this theory.

Xiao Yi's paper can have more than 200 citations in less than ten years, mainly because his paper is relatively easy to understand and can be used in many places. When his paper was first published, it caused a lot of discussion in the mathematics community, and many people's topics were solved because of the Etale algebraic variety self-conservation theory.

Therefore, it has such a high number of citations.

"So... now define the extended L-function of E as..."

[L(s, E,)=∏(p) 1/det(1-Frob_p p^(-s)| H^1(E, Q)_sp)]

Finally, Xiao Yi successfully created this new L-function and began to analyze it.

Soon, his eyes lit up.

"Yes, this new L-function can also be extended to the entire complex plane."

Of course, at this time, there is no need to continue to use the elliptic inverse analysis method to extend this thing analytically.

Xiao Yi directly used the most traditional analytical extension method and successfully extended this L-function analytically.

Afterwards, he began to observe the extended L-function.

Undoubtedly, this new L-function satisfies some special functional equations.

However, at the first glance, Xiao Yi first observed that this new L-function is closely related to the Weil conjecture of the elliptic curve E.

"It seems feasible?"

He narrowed his eyes slightly and simply deduced it in his mind.

But when he was getting closer to the Weil conjecture, he encountered some technical obstacles.

"Technical obstacles, this will take some time."

Technical obstacles in mathematics are like a bunch of mixed math problems, which require a combination of various methods to solve.

However, this does not seem to be so difficult for Xiao Yi.

However, just when he was about to take the next step, the door of the room was knocked.

"It's time to eat." Mom's voice sounded.

Xiao Yi's helpless voice followed.

"I know!"

Chapter 281 Inspiration from the Amusement Park

After a meal, as Xiao Yi expected, it turned into a gossip dinner.

He had nothing to say about this.

In fact, anyone who pays attention to him will gossip about him.

Whether it is the people around him or many people on the Internet, the main reason is that they generally cannot understand the work they are doing, so they can only turn to this kind of thing that everyone is more interested in.

While practicing Tai Chi, he just perfunctorily passed it by.

But in the end, Xiao's mother and father still saw Luo Mingya's photo from the old lady, and recognized that this was the girl he went out to see early in the morning when it snowed during the Chinese New Year.

In the end, there was some teasing, but he just ignored it.

While eating, he was thinking about the next proof of Arting's conjecture in his mind.

That is, using Weil's conjecture to solve the current problem.

Weil conjecture is Weil conjecture, which was proved by Pierre Deligne in the last century. The proof of this conjecture is also considered to be one of the most important achievements in mathematics since then, and later provided a closer connection between algebraic geometry and number theory, which is a milestone in modern mathematics.

What Xiao Yi wants to use is the first part of Weil conjecture: for a smooth projection cluster X defined on a finite field, its Zeta function ζ(X, s) satisfies a function equation, replace s with 1-s, and then multiply by some Gamma factors and constant terms, the result is equal to ζ(X, 1-s).

"Obviously, the extended L-function I made has very similar properties in this regard..."

While eating, Xiao Yi was thinking about this matter in his mind. If it was not inconvenient, he would have wanted to start drafting and calculating next to him.

In this way, time passed quickly.

After the meal, Xiao's father had a friend who asked him to go fishing, so he went fishing directly; as for Xiao's mother, she went shopping with the old lady next door. The old lady next door had a Teddy bear, commonly known as the evil fleece, so it was a good opportunity to take the dog for a walk.

So Xiao's mother is now considering whether to raise a dog or a cat.

In short, Xiao's father and mother are now very idle. They are both in their fifties. Although they have not retired, their workplaces basically do not arrange too much trouble for them. They are all very leisurely jobs, which can be regarded as allowing them to take good care of their health in the future.

The country does not want Chief Engineer Xiao to lose interest in research because of his parents' health.

Of course, Xiao Yi is also happy that the two can relax a little.

"Okay, Professor Xiao, I'm going to go back here too. I haven't finished playing today's game yet."

The last person left, Sun Lina, also said goodbye to Xiao Yi.

"Okay."

Xiao Yi nodded.

"But I'm still very curious, Professor Xiao, how is your relationship with my cousin now?"

Before leaving, Sun Lina asked again.

Xiao Yi shrugged. Facing Professor Sun, he didn't feel as awkward as he did when facing an elder. He said, "Professor Sun, you should take care of yourself. You are already thirty-five or thirty-six. I don't know why Auntie doesn't urge you."

Sun Lina was not angry. She smiled and said, "After all, I have been hurt because of this matter. Of course, my mother and father will not say anything."

Xiao Yi rolled his eyes and said, "So, you are suggesting that I find a bad woman, and then get hurt badly, so that I can be as worry-free as you?"

Sun Lina spread her hands and said, "I didn't say that."

"But..." She raised her eyebrows and said, "You can try this. In this way, there will probably be more than just you who are single in the future. My cousin will definitely be one of them."

Then she waved her hand and said, "Okay, I won't say much. Goodbye."

Then she left.

Xiao Yi shook his head with a smile, then returned to his room and continued to attack Ating's conjecture.

In this way, time passed quietly.

…

After the New Year, people returned to their posts, and so did Xiao Yi.

Teaching students every week, and then continuing to study the Artin conjecture.

Now he spends most of his time on these two things.

Especially now that he has found the angle of the Weil conjecture to solve the current problem.

However, the technical difficulties encountered before are indeed extremely difficult.

But what he knows is that as long as the current problem is solved, the subsequent difficulties will be solved!

…

Three months have passed in a blink of an eye.

The difficulty of this problem finally exceeded Xiao Yi's imagination.

After so long, he is still stuck on this problem.

And it is still at his current buff level.

It can only be said that although he is now very strong, surpassing all humans in history, he is not yet strong enough to reach the top level.

In the Science Island Laboratory.

Patting his head, another draft paper was thrown out by Xiao Yi.