The top student must be diligent.

Chapter 78 Mysterious Functor

After opening his mailbox, Xiao Yi saw that he had received a new email, and the sender's email address was Professor Maynard's.

The reply was quite fast!

Go to Mail.

[Hello, Mr. Xiao!
I am very happy to receive your letter. I am very interested in the ideas you mentioned in your letter, especially the possibility of combining far Abelian geometry and automorphic forms that you described. I must say that this seems to be a very bold move, but it has to be admitted that it is in line with the ideas of the Langlands Program.

So, I think there is a good chance.

My specific thoughts are mainly presented in the attachment. It should be noted that I don’t know much about far Abelian geometry, so it is difficult to discuss with you from the perspective of far Abelian geometry. However, I do have some understanding of p-adic Hodge theory. Grothendieck once mentioned a conjecture that since the crystal homology H1(X/W(k))Qp of special fibers and the p-adicétale homology H1(X, Qp) contain the same information, it is believed that there exists a "mysterious functor". Although I cannot find out whether this mysterious functor is helpful to your ideas for the time being, I always have a hunch that it should have some relationship.

Trust the intuition of a number theorist :)
Finally, I look forward to your reply. 】

The attachment is also a PDF file with a title called "Kuznetsov trace formula and p-adic Hodge theory".

After reading the other party's email, Xiao Yi immediately fell into deep thought.

"Mysterious Functor?"

He really hasn't understood anything about this thing.

But after seeing Maynard bring it up, he suddenly had an intuition in his mind that this thing might really be of great use to his current research!

Mathematicians' intuition is sometimes very accurate, and this is especially true for particularly outstanding mathematicians.

During the discussion at the Max Planck Institute for Mathematics, Xiao Yi was able to sensitively perceive the problems in Mochizuki Shinichi's proof, which was inseparable from his intuition.

He immediately downloaded the attachment and started reading it.

In the attachment, Maynard mainly talked to him about the Kuznetsov trace formula, which was the main method he used in the three papers in June.

Then we talked about the connection between this trace formula and p-adic Hodge theory, which basically involves the forefront of mathematics.

At the same time, it also brought a lot of inspiration to Xiao Yi.

This made Xiao Yi feel a little emotional. Although he had read many books and papers, such a simple communication could bring him a lot of new things.

It seems that the knowledge he possesses is still a little insufficient.

Make a plan and then read more books.

Soon, Xiao Yi read to the end of the attachment which had a total of more than 10 pages. At the end, Maynard finally gave Xiao Yi a complete introduction to the mysterious functor mentioned in the email.

"Frobenius homomorphism and Hodge filtering, K tensor, the role of Galois group with K is equivalent to the Barsotti-Tate group associated with X?"

"For all clusters with good reduction on the p-adic domain, there should be a way to go directly from p-adicétale cohomology to crystallographic cohomology?"

Seeing these words, Xiao Yi's eyes suddenly lit up.

"Yes... that's it! From p-adicétale cohomology to crystal cohomology, as long as I can achieve this, I can connect it to tele-Abelian geometry from the perspective of Frobenius homomorphism!"

What a coincidence!

Xiao Yi's understanding of Frobenius homomorphism comes from Shinichi Mochizuki's IUTT theory. The Hodge Cinema proposed in this theory involves something called "Frobenius-like object".

All of a sudden, many inspirations came into his mind.

Then, he immediately took out a piece of draft paper and began to carry out a long deduction along this line of thought.

Perhaps, this deduction will be the key to ending all these problems!
Time passed quietly.

……

The news that Xiao Yi would give a report back to China on September 9th spread quickly, at least, the mathematics circles in Beijing basically knew about it. This kind of report back to China was actually quite normal.

People who go abroad to attend important academic conferences will write reports about what they saw and heard after returning and tell their colleagues in China. After all, knowledge needs to be shared.

However, the person giving the report this time was rather special. He was the "ABC dispute terminator" who pointed out Shinichi Mochizuki's most critical mistake at the Max Planck Institute, and he was also a talented Chinese boy who was only seventeen years old and was highly praised by a number of Fields Medal bigwigs. Therefore, this report became popular before it even started.

In the 2020 College Mathematics Class Joint Examination Group.

Qinghua-Jiang Long: [Screenshot]

Qinghua-Jiang Long: [Oh my god, guess what I saw?! ]

Everyone in the group quickly looked at the screenshot Jiang Long sent.

This picture is a poster, and it is easy to see that it is a promotional picture for an academic report.

And on it is a photo of the reporter.

It’s Xiao Yi!

Suddenly, this picture was like a bomb, blowing out all the people who were lurking in the group.

Shanghai Jiao Tong-Luo Xiao: [Oh my god! This person looks like Xiao Shen! ]

[Shanghai Jiaotong University-Luo Xiao was banned from speaking for 10 minutes by the administrator]

Danfu-Xu Kangcheng: [Ah? Is Xiao Shen going to issue a report? Is Xiao Shen returning to China?]

Jingda-Chen Muhua: [What the hell? Is this true? Xiao Shen went abroad and he can give an academic report when he comes back? Oh my god! We haven’t even finished our military training yet! ]

HKUST-Ye Cheng: [@HKUST-Xiao Yi, Brother Xiao, you promised to come back and participate in the last few days of military training with us, why do you want to stay outside for another ten days! ]

……

There was a heated discussion in the group, but Xiao Yi, the main person involved, did not show up in the group despite being tagged almost a hundred times.

People in the group guessed that Xiao Yi was actually preparing the draft of the report, so he had no time to read it.

Unfortunately, they only guessed half of it and didn't know that what Xiao Yi was working on was something that could be considered a big deal in the mathematics community.

Not long after the group had a heated discussion, Chen Muhua suddenly appeared in the group again.

Jingda-Chen Muhua: [Oh my god, brothers! I just received a notice that on the day when Xiao Shen gives his report, students in our class can sign up to watch it! ]

This news once again blew up the others.

Other Kyoto University students came out to confirm the news.

At this time, students from Qinghua University and Jinghang University also stood up to say that they had also received the news.

All of a sudden, students from other schools became envious and jealous.

[Why? We want to go too!]

Beijing University - Chen Muhua: [Maybe it's because Xiao Shen's report was given at a location where the Institute of Mathematics of Huazhong University of Science and Technology is just across the street from Beijing University? Qinghua and Beijing Aviation University are also quite close. ]

【Damn it! 】

……

(End of this chapter)