I Just Want to Be a Quiet Top Student

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

Shen Qi explained the BSD conjecture in "History of Number Theory". The BSD conjecture is inextricably linked with many other minority theory issues. The study of the BSD conjecture is actually a review of the history of modern number theory.

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

This year, Wiles proved Fermat's last theorem by establishing a connection between elliptic curves and model theory.

This year also had a major impact on the BSD conjecture. Before that, mathematicians were not 100% sure whether the BSD conjecture made sense.

In the process of proving Fermat's last theorem, Wiles conveniently proved the Taniyama-Shimura conjecture. While proving these two conjectures, he also made the mathematical significance of the BSD conjecture affirmed by the mathematical community.

So what is the mathematical significance of BSD?

If this conjecture is proved, what effect will it play?

Including Shen Qi, the mathematics community agrees that if the BSD conjecture is proved, then the finite theory of sand groups will also 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, the exact answer to "to what extent the information on the algebraic number field can be glued together by the information on all the local fields" will be obtained, which has risen to a philosophical level. This philosophy is called "Principle of partial 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 so cold that I have no friends.

There are very few scholars who have worked hard and devoted themselves to studying the BSD conjecture. They are lonely fireworks blooming in the sky ten thousand feet high.

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

The research results of these four mathematicians who spent more than ten years turned them into papers, a total of 6098 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 curves satisfy 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.

Among the four mathematicians, Gong Changwei is Chinese, and he was Ouye's mentor when he was a graduate student at Columbia University.

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

The only person who can answer this question is Ouye. She said, "Because Professor Shen's level is limited."

"Hahaha!"

"Slightly slightly."

"..."

After hearing Miss Ye Zi's answer, the three students had different expressions.

Sister Yezi is probably the only person who dares to say that Professor Shen's level is limited in the whole world.

I am only allowed to beep you in the whole world, others are not eligible.

This is also a different kind of show of affection.

Since Professor Shen's level is limited, let the BSD conjecture be handed over to a team with unlimited level.

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

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

Euler is probably the Clarke of number theorists.

Shen Qi was originally very much like Clark. He used pure analytic number theory to prove the Riemann Hypothesis, which can be described as invincible.

After the Riemann conjecture was solved, Shen Qi’s academic behavior has undergone some changes. He has become less rigid. When dealing with some academic problems, he prefers a combination of soft and hard methods. This is also the mainstream trend in the future development of mathematics. The crossover is more and more frequent and close.

The subtle changes in Shen Qi's academic thinking more or less affected Ouye, after all, they slept on the same bed.

Ouye realized that the BSD conjecture could not be solved by pure number theory methods, and Shen Qilai, who was once invincible, could not be solved either.

So on the issue of BSD conjecture, Euler chose the combination of number theory + elliptic curve + ... and followed the trend.

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

On 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 around the world have made remarkable progress in the study of elliptic curve theory in recent years, rank remains a mystery.

Even the basic problem of how to calculate the rank, 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 "The Ins and Outs of the Riemann Conjecture Proof". "

Shen Qi's main purpose of writing this is to increase the sales of "The Ins and Outs of the Proof of the Riemann Hypothesis".

Of course, if readers understand the Riemann conjecture, it will also be helpful to interpret the BSD conjecture.

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

Shen Qi's real contribution to the BSD conjecture came from an unpublished thesis.

In this thesis manuscript, Shen Qi drew a picture casually.

He originally wanted to draw a flounder, and then look at the picture and talk to Nofi.

As a result, drawing and drawing, Shen Qi drew the fish as a coordinate system and curves.

Ouye has seen this extremely ugly "fish". Shen Qi tried to use the idea of ​​group theory to explain the rank in elliptic curves.

But Shen Qi didn't fully explain the law of rank in the elliptic curve and the calculation principles. After he finished drawing the "fish", there was no more content.

On the contrary, Ouye was deeply inspired, and she realized a new way of thinking from this "fish".

Ouye wrote on the whiteboard:

E(Q)≡Z^r×E(Q)f

E(Q)={(-d,0),(0,0),(d,0)...

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

The definition of the BSD conjecture is not difficult to understand, but the difficulty is to prove the derivation process.

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

Number theory, group theory, elliptic curve, Riemann zeta function, Euler product, Hassel-Weil function and even the Gauss conjecture of the quadratic domain...the amount of knowledge required is too much.

Fortunately, Zhao Tian, ​​Xiao Yun, and Zeng Han are the elite among the students, and their knowledge reserve is not bad.

Scientific research has shown that low-achieving students spend far more time studying than high-achieving students.

Zhao Tian, ​​Xiaoyun, and Zeng Han spent more time studying than the scumbags. They are super hardworking academic masters, so they are qualified to overcome the BSD conjecture with Sister Yezi.

The clever Xiaoyun quickly understood Ouye's strategic intention: "So, we need to use group theory as a breakthrough?"

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