I Just Want to Be a Quiet Top Student

Zhao Tian, ​​Xiaoyun, and Zeng Han went to Yanda People's Hospital to visit Ouye.

Ouye, who just woke up, handed over the manuscript to the three students, so and so, and so and so, she taught the students face-to-face tips.

The context of the proof of the strong BSD conjecture compiled by Ouye is very clear, and this proof context adopts a reverse method.

The last step, to prove the strong BSD conjecture, is to prove this sentence: the necessary and sufficient condition for E(Q) to be an infinite set is that the Taylor polynomial of L(E, s) at s=1 has the following form, L(E, s )=c(s-1)^r+higher-order term, where c≠0 and r is the rank of E.

In the penultimate step, to prove the above sentence, it is necessary to count the rational points on the elliptic curve.

In the penultimate step, if you want to count the rational points on the elliptic curve, you need to demonstrate the rank on the elliptic curve first.

In the penultimate step, if you want to demonstrate the rank on the elliptic curve, you can consider the method of group theory.

Through the unremitting efforts of Ouye and her three students, the team has reached the penultimate step.

"Actually, the fourth-to-last step can also be considered as the first positive step. It takes the longest time. If we complete the fourth-to-last step in two years, then the next three steps can be completed within two months... ...Ha...yawn..." Although Ouye is in poor health, her mathematical thinking is very clear.

Ouye just woke up, but he yawned again and again. The three students said: "Miss Ye Zi, please rest, we know what to do! You sleep for a while, we will go first."

The three students carefully packed Ouye's manuscript, then left the People's Hospital and returned to Yanda University.

The small room at the end of the corridor on the first floor of the number courtyard is the war room of three students.

The three first organized Ouyer's manuscripts into electronic data schemas that could be verified by computers.

This job probably requires three people to do it for three consecutive days, and each person's working time will not be less than 12 hours a day.

The three students were very clear about Ouye's train of thought.

Starting from group theory, Euler obtained a hypothesis about the rank of elliptic curves by calculating and proving the rank of typical elliptic curves.

Whether this assumption can be a lemma needs to be verified.

Euler's method is very traditional, inferring a typical theory from typical examples, and then putting the typical theory in all the examples to prove its universality.

The apple fell from the tree and hit Newton on the head. Newton deduced a theory that apples were affected by the gravitational pull of the earth. Is this theory only valid for Apple, or is it universal? This is the universal proof work that Newton will do next, and finally he proved the law of universal gravitation.

Newton is the great light of mankind, but his means of demonstrating great theories is also very traditional, from simple to complex, and then from complex to simple.

In the penultimate step of the strong BSD conjecture set by Ouye, she completed the theoretical construction from simple to complex. Of course, it can only be regarded as a hypothesis at present.

From the complex regression to the simple, it is a huge amount of work to finally prove that the assumption of the rank of the elliptic curve has universality or conditional universality.

This job will be done by Zhao Tian, ​​Xiao Yun, and Zeng Han.

For example, under the condition of prime number p=5, the elliptic curve y^2=x^3-x has seven solutions in total, which are (0, 0), (1, 0), (4, 0), (2, 1 ), (3,2), (3,3), (2,4).

This is easy to calculate. Any one of Zhao Tian, ​​Xiao Yun, and Zeng Han can get the correct solution within 10 minutes through manual calculation.

However, there are infinitely many elliptic curves in theory. Generally, any calculation involving an infinite number of projects cannot be done manually by human beings, and computers must be relied on.

Zhao Tian, ​​Xiaoyun, and Zeng Han will spend three days processing Ouye's manuscript into data that can be verified by a computer.

Based on Ouye's manuscript, if the elliptic curve is verified by computer, it will take no one knows how many days it will take, it may be three days, it may be three years, or thirty years.

Fortunately, Professor Gong Changwei, Ouye's master tutor, made an important contribution to the BSD conjecture.

Professor Gong proved the related theorems of Kolyvagin's inverse proposition, and jointly proved with other mathematicians that at least two-thirds of the elliptic curves satisfy the BSD conjecture.

Professor Gong Changwei is equivalent to helping his disciple Ouye eliminate many checking conditions, so the three students of Ouye only need to verify the elliptic curve that satisfies the Kolyvagin theorem, the Gross-Zagier theorem, and the order of the Shafarevich-Tate group.

Zhao Tian, ​​the eldest of the three students, asked his juniors and juniors with concern: "Summer vacation is coming soon, have you bought tickets to go home?"

Xiao Yun shook her head: "Anyway, my parents are not at home, and I am a single dog with no one to feed me when I go back, so I decided to stay in the capital for a work-study program this summer."

"Xiaoyun, where did your parents go?" Zhao Tian asked.

While sorting out Ouye's manuscript, Xiaoyun said: "My mother went to Germany as a visiting scholar, and my father went to Africa to help African friends build infrastructure. I won't be able to see my parents until next year's Spring Festival."

Zhao Tian knew that Xuemei Xiaoyun's mother graduated from Fudan with a Ph.D. and is currently a professor at East China Normal University. Xiaoyun Xuemei's father graduated from Shuimu University with a master's degree and is currently an engineer of China Construction Eighth Engineering Bureau.

Xiaoyun Xuemei was admitted to Yanda Mathematics Academy when she was in high school. This is not a question of whether she is smart or not, but her genetic inheritance.

"It's okay, it's okay, Xiaoyun, if you don't go home during the summer vacation, just tell me if you need anything." Zhao Tian is an aboriginal resident of the imperial capital, and his native place is Banchengzi Village, Bulaotun Town, Miyun.

Since he is a resident of the imperial capital and a senior, Zhao Tian thinks that he should take care of Junior Xiaoyun.

Xiaoyun nodded and said, "Thank you brother, let's process the data quickly."

Zhao Tian turned to ask the elementary school brother: "What about you, Zeng Han, you should go home this summer vacation?"

Zeng Han concentrated on processing the data and did not raise his head: "I won't go home, I will stay in school."

"Why? Did your parents go abroad too?"

"No reason, it's staying in school anyway."

Zeng Han was recommended to Yanda University at the age of fifteen, and he just turned 18 this year. Zeng Han's parents are both Ph. D., his grandparents, grandparents, and grandparents are all high-level intellectuals. The two families of his parents have a total of six Ph. Ds and seven Masters. The per capita standard for members at the age of thirty-five is an associate professor or an associate researcher.

Zeng Han, who was born in a family of intellectuals, was able to be admitted to Yanda University at the age of 15. It is also not a question of whether he is smart or not, but a genetic inheritance.

Compared with the two juniors from famous families, Zhao Tian, ​​who was born in Banchengzi Village, Bulaotun Town, Miyun, is an inspirational senior.

Zhao Tian often said: "I am not a genius, I have never participated in the Olympiad competition. That year, I was admitted to the Capital No. 4 Middle School with a few blind exams, and three years later, I was admitted to Yanda University. .When I was an undergraduate, I was blind, with a GPA of 4.0, and I was recommended to be a graduate student at the School of Mathematics of Yanda University."

This is Zhao Tian's sincere words. He feels that compared with his juniors and younger sisters, he is just an ordinary person. And Xiaoyun and Zeng Han are geniuses in the true sense.

Creaking, the door of the small room opened.

A man came in. He was not tall, energetic, with a high hairline, and his eyes shone with wisdom. He was not a mortal at first glance.

"Teacher Zhou, why are you here?" The three students were quite surprised.

The person who came was Zhou Yu'an, he, Shen Qi, and Ou Ye were collectively known as the "Three Heroes of the XX Class of Yanda Mathematics Institute".

Zhou Yu'an, Shen Qi, and Ou Ye entered Yanda Mathematics College in the same year. The three of them were classmates. The three of them went to the Department of Mathematics of Princeton to complete their doctoral studies. Their class of undergraduates from the School of Mathematics is also considered to be the strongest class in the history of Yanda.

Can the latecomers of Yanda Mathematics Academy surpass Shen, Ou, and Zhou Jie in the XX class?

It seems to be more difficult at present.

The achievements of Shen Qi alone are almost insurmountable. Not to mention surpassing, even copying is difficult.

Zhou Yu'an, IMO gold medalist, Ramanu Gold Award winner, and director of the mathematics department of Shen Qi Research Center, is second only to Shen Qi in the hearts of Yan University's mathematics students.

Mr. Zhou's establishment is not in the numbering institute, he suddenly drove to the numbering institute, and came in front of the three students, there must be something wrong.