Elliptic Curve Rank

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

In his book "History of Number Theory", Shen Qi elaborates on the BSD Conjecture. The BSD Conjecture is closely related to many other number theory problems. Studying the BSD Conjecture is, in fact, a review of the History of Number Theory.

1995 was a pivotal year in the history of modern number theory.

That year, Wiles proved Fermat's Last Theorem by establishing a connection between Elliptic Curve and model theory.

This year also had a significant impact on the BSD Conjecture; prior to this, mathematicians could not be 100% certain whether the BSD Conjecture was meaningful.

In proving Fermat's Last Theorem, Wiles also proved the Taniyama-Shimura conjecture. In proving these two conjectures, he also affirmed the mathematical significance of the BSD Conjecture in the Mathematics Community.

So what is the mathematical significance of BSD?

What effect would proving this conjecture have?

Including Shen Qi, the Mathematics Community unanimously believes that if the BSD Conjecture is proven, then the Finite Group Theory will also be proven, and sand groups are one of the core elements for understanding the arithmetic properties of mathematical objects.

In other words, if the BSD Conjecture is proven, then the question of "to what extent information over Algebraic Number Field can be glued together from information over all local fields" will be answered definitively. This has risen to a philosophical level, and this philosophy is known as the "Local-Global Principle".

To prove a mathematical problem can lead to the refinement of a philosophical system.

This is the core significance of the BSD Conjecture.

Mathematics and philosophy are both highbrow subjects, and the combination of mathematics and philosophy is so highbrow that it has no friends.

Very few scholars dedicate themselves to the study of the BSD Conjecture; they are like lonely fireworks, blooming high in the sky.

To date, the closest possible proofs of the BSD Conjecture come from Gong Changwei, Skinner, and Bhargava and Shankar.

The research findings of these four mathematicians, which took more than a decade to complete, were translated into a paper totaling an astonishing 6,098 pages, enough to fill the trunk of a car.

Four mathematicians, Gong Changwei, Skinner, Bhargava, Shankar, proved that at least two-thirds of Elliptic Curve satisfy the BSD Conjecture.

The achievements of these four mathematicians in BSD Conjecture are equivalent to Chen Jingrun proving Goldbach Conjecture 1+2.

Gong Changwei, one of the four mathematicians, is Chinese. He was Ou Ye's advisor when she was a graduate student at Columbia University.

Looking at the mathematical formulas on the whiteboard, Zhao Tian asked, "I have a question. Professor Shen analyzed the history of the BSD Conjecture so thoroughly in 'The History of Number Theory,' why didn't he prove the BSD Conjecture?"

Only Ou Ye could answer this question, saying, "Because Professor Shen's level is limited."

"Hahaha!"

"Hehehe."

"..."

Upon hearing Sister Yezi's answer, the three students reacted in different ways.

I'm afraid Sister Yezi is the only person in the whole world who would dare to say that Professor Shen's abilities are limited.

I'm the only one allowed to criticize you; nobody else has that right.

This is also a unique way of showing affection.

Since Professor Shen's expertise is limited, let's leave the BSD Conjecture to the team with unlimited expertise.

Ou Ye's expertise lies in Analytic Number Theory, which is the most challenging branch of Analytic Number Theory.

If Algebraic Number Theory is likened to soft science fiction, then Analytic Number Theory is equivalent to hard science fiction written by Arthur C. Clarke.

Ou Ye is probably the Clark of number theorists.

Shen Qi was originally very similar to Clark; he proved the Riemann Hypothesis using purely analytic Number Theory Methods, which was incredibly difficult.

After solving the Riemann Hypothesis, Shen Qi's academic behavior underwent some changes. He became less rigid and tended to use a combination of hard and soft approaches when dealing with academic problems. This is also the mainstream trend in the future development of mathematics, with interdisciplinary collaboration becoming more frequent and closer.

Shen Qi's subtle changes in academic thought more or less affected Ou Ye, after all, the two slept in the same bed.

Ou Ye realized that pure Number Theory Methods couldn't solve the BSD Conjecture; even the once invincible Shen Qi couldn't solve it.

Therefore, on the BSD Conjecture problem, Ou Ye chose to combine number theory, Elliptic Curve, and other methods, following the prevailing trend.

If mainstream research methods combining hardware and software are adopted, then Professor Shen, despite his limited expertise, has still made some indirect contributions to the BSD Conjecture.

In the BSD Conjecture problem, 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 significant progress in the study of Elliptic Curve theory in recent years, the rank remains a mystery.

Even fundamental questions like how to calculate rank, or whether rank can be infinitely large, remain unresolved.

In his book * History of Number Theory , Shen Qi writes, "... To help you better understand the BSD Conjecture explained in this chapter, I recommend that you read another book I wrote, * The Riemann Hypothesis: The Proof and the Journey."

Shen Qi's main purpose in writing this was to increase the sales of "The Whole Story of Proving the Riemann Hypothesis".

Of course, understanding the Riemann Hypothesis will also help readers interpret the BSD Conjecture.

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

Shen Qi's real contribution to the BSD Conjecture came from a paper he had never published.

In this paper draft, Shen Qi casually drew a diagram.

He originally intended to draw a flounder and then tell Nophie a story based on the picture.

As Shen Qi continued drawing, he ended up drawing the fish as a coordinate system and curves.

Ou Ye had seen this incredibly ugly "fish" before. Shen Qi attempted to use Group Theory to explain the rank in Elliptic Curve.

However, Shen Qi did not fully explain the rules and calculation principles of rank in Elliptic Curve. After drawing the "fish", he stopped there.

On the contrary, Ou Ye was deeply inspired, and she gleaned 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)...

Here, E(Q) is actually an abelian group, i. e., Abelian group. Z is the set of infinite integers under addition.

The definition of the BSD Conjecture is not difficult to understand; the difficulty lies in the proof and derivation process.

The proof and derivation of the BSD Conjecture is a very complex and tedious task, requiring a great deal of prior knowledge.

Number theory, Group Theory, Elliptic Curve, Riemann zeta function, Euler product, Hasse-Weil function, and even Gauss Conjecture for Quadratic Number Field... the amount of knowledge required is simply too vast.

Fortunately, Zhao Tian, Xiao Yun, Zeng Han are the elite students, and their knowledge reserves are quite good.

Scientific research shows that students who perform poorly in school spend far more time studying than those who perform well.

Zhao Tian, Xiao Yun, Zeng Han, these three top students, actually spend more time studying than the average student. They are super diligent top students, so they are qualified to be here with Sister Yezi to conquer the BSD Conjecture.

The quick-witted Xiao Yun soon understood Ou Ye's strategic intention. "So, we're going to use Group Theory as a breakthrough point?"

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