God-level Xueba system

"what? What!"

"Did you hear what Yan Xin said?"

"Today he will answer three millennium problems in public!"

"What's left of the Millennium Dilemma?"

"Yang-Mills Theory, Navel-Stokes Equations, and the BSD Conjecture!"

"If Yan Xin really speaks out all three of these, then the term Millennium Problem will no longer exist in the future!"

"Hurry up and call your friends!"

"Let them watch the replay!"

"You know what! What's the point of replaying?"

"People in the fan group, hurry up and mobilize!"

For a while, Yan Xin's fans, Weibo, WeChat, QQ, Space, etc., all social software went crazy!
early morning.

Weibo hot search list.

In less than 10 minutes, Yan Xin reached the top of Weibo No.1!
And the heat is.
burst!
at this time.

A certain male singer surnamed Wang looked at the hot search list on Weibo and sighed.

Damn!

I just want to push a song, it's just a hot search!

Is it so difficult!
And Yan Xin's popularity in the live broadcast room has directly increased from 300 million to 5000 million!
The numbers are still going up!

Those genius math teenagers who were still asleep were all awakened by the sound of the notification!

Everyone opened Yan Xin's live broadcast room under the covers, and then took out their notebooks and waited for Yan Xin to start speaking.

There is one thing to say, since Yan Xin opened this learning area on Gouya Live, it has also driven the rhythm of learning for the whole people!
This is a good trend!
Yan Xin watched the popularity soar in the live broadcast room, and nodded in satisfaction.

He really didn't expect the fans to be so enthusiastic.

Long Xia is already early in the morning, and everyone is still watching his live broadcast!
"Okay! Now let's explain today's first millennium problem: the BSD conjecture!"

As usual, Yan Xin pointed the main camera at her face, and the secondary camera at the draft paper in her hand.

What is different from the past is that there are nearly 1 live audiences in front of me now!
Yan Xin's draft paper was also projected on the big screen in the Arts Center!

"Old rules! I still want to popularize what BSD conjecture is!"

BSD conjecture.

Full name: The Porch and Swinnerton-Dayal Conjecture.

So what is its conjecture overview?

Given an Abelian variety over a global field, conjecture that the rank of its Modal group is equal to the 1-point order of its L function at 0.And the prime coefficient of the Taylor expansion of its L function at 1 has an exact equational relationship with the finite part size of the Modal group, the free part volume, the period of all prime positions and the sand group.

The first half is the weak-BSD conjecture in the usual sense. The BSD conjecture is a generalization of the formula for class numbers divided into circles!

I believe that many people know that Gross proposed a detailed BSD conjecture!

After that, Bloch and Kato proposed a more general Bloch-Kato conjecture for motifs.

The statement of the BSD conjecture relies on Modal's theorem: the rational points of an Abelian variety on a global field form a finitely generated commutative group.The exact part depends on the finiteness conjecture of sand groups.

So what are the results that are known now?

For the case where the analytic rank is 0, Coates, Wiles, Kolyvagin, Rubin, Skinner, Urban et al. proved the weak BSD conjecture, and the exact BSD conjecture is valid beyond 2.

For the case of analytic rank 1, Gross, Zagier et al. proved the weak BSD conjecture, and the exact BSD conjecture is valid outside 2 and derivation.

"Fuck? What the hell is this?"

"I do not know!"

"What is Modal's theorem?"

"What is an Abelian cluster?"

"Fuck!"

"Brothers, how do I feel, this BSD conjecture is quite different from the previous Hodge conjecture and the Riemann hypothesis!"

"I think so too! I haven't heard of many terms here!"

"Oh my god! I just woke up in the middle of the night, and I couldn't understand what Yan Xin was saying!"

"Maybe you haven't woken up yet?"

"Hahahaha!"

Yan Xin looked at the students in the audience with a dazed expression.

It seems that before I answer the BSD conjecture, I have to popularize Modal's theorem and Abelian varieties.

"I believe that many of you may not know what Abelian varieties and Modal theorem are! Since you don't understand, I'd better explain it to you! After all, if you don't understand what these two are, the next proof process may be Everyone can't understand!" Yan Xin said with a smile.

The professors also nodded.

Although they understand, the students must not have studied these things!

Lin Hui listened obsessively.

Looking at the young man on the stage who was chatting and laughing happily and behaved elegantly, she was a little dazed.

He is so outstanding now!
Yan Xin picked up the pen and began to introduce to everyone what Abelian varieties and Modal theorem are on the draft paper.

An Abelian variety is an algebraic group that is also a complete algebraic variety.The condition of completeness implies a strict restriction on Abelian varieties.

Therefore, Abelian varieties can be embedded in projective space as closed subvariety; non-singular varieties, Abelian varieties, and every rational mapping are regular, and the group laws on Abelian varieties are commutative.

The study of automorphisms of Abelian varieties, especially the effect of Frobenius automorphisms on Tide modules, makes it possible to prove the BSD conjecture.

Other problems related to Taylor modules include the study of the role of the Galois group of the basis field closure on the module.This leads to the Tate conjecture and the Tate-Honda theory, which uses the language of Tate modules to describe Abelian varieties over finite fields.

Then there is Modal's theorem.

Given any Abelian variety over a global field, its rational points form a finitely generated Abelian group.

The so-called overall field refers to the algebraic number field (that is, the finite expansion of the rational number field) or the function field of the curve on the finite field.

So what is the relationship between Abelian varieties and Modal's theorem?

it's actually really easy.

Yan Xin picked up the pen and simply drew a few relationship lines on the draft paper.

Elliptic curves refer to smooth projective curves whose genus is 1, and Abelian varieties are high-dimensional extensions of Modal's theorem!
That is to say, it forms an exchange group at a point above a certain fixed domain.

All the students at the scene and the audience in the live broadcast room were listening carefully to Yan Xin's explanation.

"Now that everyone has a general understanding of what Abelian varieties and Modal's theorem are, I will continue to talk about it."

Mathematicians have always been fascinated by the problem of characterizing all integer solutions to algebraic equations such as x^2+y^2=z^2.Euclid once gave a complete solution to this equation, but with more complex equations, this becomes extremely difficult.

In fact, as Yu.V.Matiyasevich pointed out, Hilbert's tenth problem is unsolvable, ie, there is no general way to determine whether such a method has an integer solution.When the solution is a point of an Abelian variety, the Bech and Swinton-Dale conjecture holds that the size of the group of rational points is related to the behavior of a related Zeta function z(s) around the point s=1.

In particular, this interesting conjecture states that if z(1) is equal to 0, then there are infinitely many rational points (solutions), and conversely, if z(1) is not equal to 0, then there are only finitely many such points.

"To sum up, the BSD conjecture is possible to crack. So what should be the summary of its conjecture content?"

Suppose E is an elliptic curve defined on the algebraic number field K, and E(K) is a set of rational points on E. It is known that E(K) is a finitely generated commutative group.Note that L(s,E) is the Hasse-Weil L function of E.

The conjecture says that the rank of E(K) is exactly equal to the order of the zero of L(E,s) at s=1.And the first non-zero coefficient of the Taylor expansion of the latter can be precisely expressed by the algebraic properties of the curve.

"Next, I will combine the relevant knowledge of Introduction to Advanced Mathematics, Introduction to Super Mathematics, and Quantum Physics to give you a systematic answer to the BSD conjecture!"

The students in the audience carefully wrote down everything that Yan Xin wrote on the draft paper.

This is the BSD problem-solving steps!

It is quite valuable for research!