I Just Want to Be a Quiet Top Student

Chapter 629 Rank of elliptic curve

In the field of mathematics, Shen Qi’s name is everywhere.

Shen Qi explained the BSD conjecture in the “History of Number Theory”. The BSD conjecture is inextricably linked with many other theoretical issues. Researching the BSD conjecture is actually a review of the history of modern number theory.

In the development history of modern number theory, 1995 was a key node.

This year, Wiles proved Fermat’s Last Theorem by establishing a connection between elliptic curve and model theory.

This year has also had a significant impact on the BSD conjecture. Before that, mathematicians could not be 100% sure whether the BSD conjecture was meaningful.

Wiles proved the Taniyama-Shimura conjecture in the process of proving Fermat’s Last Theorem. While he proved these two conjectures, he also made the mathematical significance of the BSD conjecture confirmed by the mathematics community.

So what is the mathematical meaning of BSD?

What effect will this conjecture have been proved?

Including Shen Qi, the mathematics community unanimously believes that if the BSD conjecture is proved, then the finite theory of sand groups will be proved, and sand groups are one of the cores for understanding the arithmetic properties of mathematical objects.

In other words, if the BSD conjecture is proved, “to what extent can the information in the algebraic number field be glued together by the information in all local fields” will get an exact answer, which has risen to the height of philosophy. This philosophy is called “Principle of Part and Whole”.

Prove a mathematical problem and perfect a philosophical system.

This is the core meaning of the BSD conjecture.

Mathematics and philosophy are both high-cold subjects, and the CP of Mathematics + Philosophy is too cold to have friends.

There are very few scholars who devote themselves to studying the BSD conjecture. They are lonely fireworks, blooming at a height of 10,000 feet.

Up to now, the BSD conjecture proof scheme closest to the truth comes from Gong Changwei, Skinner, Bhargava and Shankar.

The research results of these four mathematicians spent more than ten years turned into a paper, which is a staggering 6,098 pages, which can fill the trunk of a car.

Four mathematicians Gong Changwei, Skinner, Bhargava, and Shankar proved a conclusion: at least two-thirds of the elliptic curve satisfies the BSD conjecture.

The achievements of these four mathematicians on the BSD conjecture are equivalent to Chen Jingrun’s proof of Goldbach’s conjecture 1+2.

The four mathematicians, Gong Changwei, are Chinese. He was Ou Yeh’s mentor when he was studying at Columbia University.

Zhao Tian looked at the mathematical formula on the whiteboard and asked: “I have a question. Professor Shen has analyzed the past and present of the BSD conjecture in “History of Number Theory” so thoroughly, why didn’t he prove the BSD conjecture?”

The only person who can answer this question is Ou Ye. She said: “Because Professor Shen has a limited level.”

“Hahaha!”

“Slightly.”

“…”

After hearing Sister Ye Zi’s answer, the three students had different expressions.

Dare to say that Professor Shen’s level is limited, I am afraid that Ye Zijie is the only one in the world.

The whole world only allows me to beep you, no one else is qualified.

This is also an alternative show of affection.

Since Professor Shen’s level is limited, let the BSD conjecture be done by a team with unlimited level.

Ou Ye is good at analytic number theory, which is the hardest branch of number theory.

If you compare algebraic number theory to soft science fiction, analytic number theory is equivalent to hard science fiction written by Clark.

Ou Yeh is probably Clark among the number theorists.

Shen Qi was also very Clark. He proved Riemann’s conjecture using pure analytical number theory, which is invincible.

After the Riemann conjecture was settled, Shen Qi had some changes in academic behavior. He became less rigid. When dealing with some academic problems, he preferred to combine soft and hard. This is also the mainstream trend of future development of mathematics. The crossing is more and more frequent and close.

The subtle changes in Shen Qi’s academic thinking more or less affected Ou Ye, after all, the two slept on the same bed.

Ou Ye realized that the pure number theory method could not handle the BSD conjecture, and he could not handle it with the once invincible Shen Qi.

So on the question of BSD conjecture, Euye chose the combination of number theory + elliptic curve +… and followed the trend.

If the mainstream research method of combining software and hardware is adopted, then Professor Shen, who has a limited level, has made some indirect contributions to the BSD conjecture.

In the BSD conjecture, the larger r is, the more rational points mathematicians hope to see. r is the rank of the curve, which is a very important parameter in this problem.

Although mathematicians all over the world have made remarkable progress in the research of elliptic curve theory in recent years, rank is still a mystery.

Even the basic problem of how the rank should be calculated, or whether the rank can be infinite, has not been solved.

Shen Qi wrote in “History of Number Theory”: “…In order to facilitate your better understanding of the BSD conjecture described in this chapter, I suggest you read another book written by myself, “The Proof of the Riemann Conjecture”. ”

The main purpose of Shen Qi’s writing is to increase the sales of “Riemann Conjecture Proof”.

Of course, if readers understand Riemann’s conjecture, the interpretation of the BSD conjecture will be helpful.

Readers only need to understand a little knowledge of the Riemann zeta function to know that the Hasse-Weil function in the elliptic curve is actually the Euler product.

Shen Qi’s real contribution to the BSD conjecture came from a manuscript of his unpublished paper.

In this manuscript, Shen Qi drew a picture casually.

He originally wanted to draw a flounder, then look at the picture and tell Nophie a story.

As a result, Shen Qi drew the fish into a coordinate system and a curve.

Ou Ye has seen this very ugly “fish”. Shen Qi tried to explain the rank in the elliptic curve with the idea of ​​group theory.

But Shen Qi didn’t fully explain the law of rank in the elliptic curve and the calculation principle. After he painted the “fish”, there was no more text.

On the contrary, Ou Ye was deeply inspired, and she realized a new way of thinking from this “fish”.

Ou Ye wrote on the whiteboard:

E(Q)≡Z^r×E(Q)f

E(Q)={(-d, 0), (0, 0), (d, 0)…

The E(Q) here is actually a commutative group that is, the Abel group. Z is an infinite set of integers under addition.

The definition of the BSD conjecture is not difficult to understand, but the difficult part is the proof and derivation process.

The derivation of the proof of the BSD conjecture is a very complicated and cumbersome thing, requiring a lot of knowledge.

Number theory, group theory, elliptic curve, Riemann zeta function, Euler product, Hassel-Wei function and even the Gaussian conjecture of the second degree… The amount of knowledge required is too much.

Fortunately, Zhao Tian, ​​Xiaoyun, and Zeng Han are the elites among the students, and their three knowledge reserves are pretty good.

Scientific research shows that scumbags spend far more time on learning than scumbags.

Zhao Tian, ​​Xiaoyun, and Zeng Han spend more time studying than the scumbags. They are super hardworking scumbags, so they are qualified to conquer the BSD conjecture with Ye Zijie here.

The smart Xiaoyun quickly understood Ou Ye’s strategic intentions: “So, we should use group theory as a breakthrough?”