Four Steps
Zhao Tian, Xiao Yun, Zeng Han visited Ou Ye at Yanjing University People's Hospital.
Having just woken up, Ou Ye handed the manuscript to her three students, explaining the instructions in detail.
The outline of the Strong BSD Conjecture Proof, as compiled by Ou Ye, is very clear. This proof uses a reverse reasoning approach.
The final step is to prove the strong BSD Conjecture, that is, to prove the following statement: E(Q) is an infinite set if and only if the Taylor polynomial of L(E, s) at s=1 has the following form: L(E, s)=c(s-1)^r+higher-order terms, where c≠0 and r is the rank of E.
The penultimate step, to prove the above statement, is to count the rational points on the Elliptic Curve.
The third-to-last step, in order to count the rational points on the Elliptic Curve, requires first proving the rank of the Elliptic Curve.
In the fourth-to-last step, to prove the rank of Elliptic Curve, we can consider using Group Theory.
Thanks to the tireless efforts of Ou Ye and her three students, the team has now reached the fourth to last step.
"Actually, the fourth to last step can also be considered the first step from the beginning, as it takes the longest. If we take two years to complete the fourth to last step, then the remaining three steps can be completed in two months... Ha... Yawn..." Although Ou Ye was not in good health, her mathematical thinking was very clear.
Ou Ye woken up, but was yawning repeatedly. The three students said, "Sister Yezi, you should rest. We know what to do! You sleep for a while, we'll be going now."
The three students carefully packed up Ou Ye's manuscript and then left the People's Hospital to return to Yenching University.
The small room at the end of the corridor on the first floor of the Mathematics Department is the war room for the three students.
The three first organized Ou Ye's manuscript into an electronic data format that could be verified by Computer Science.
This job will require three people to work for three consecutive days, with each person working no less than 12 hours a day.
The three students understood Ou Ye's line of thinking very well.
Starting from Group Theory, Ou Ye proved a hypothesis about the rank of Elliptic Curve by calculating and demonstrating the rank of typical Elliptic Curve.
Whether this assumption can be considered a lemma needs to be verified.
Ou Ye adopted a very traditional approach: inferring typical theories from typical examples, and then applying the typical theories to all examples to prove their universality.
An apple fell from a tree and hit Newton on the head. Newton deduced a theory that the apple was affected by Earth's gravity. Was this theory only valid for apples, or did it have universal applicability? This was the task Newton then had to complete to demonstrate its universality, and ultimately he proved the law of Universal Gravitation.
Newton is a great Light of Humanity, but his methods of proving great theories were also very traditional: from simple to complex, and then back to simple.
In the fourth-to-last step of Ou Ye's strong BSD Conjecture, she completed the theoretical construction from simple to complex; of course, it can only be considered a hypothesis at present.
The task of regressing from complexity to simplicity and ultimately proving that the assumption of the rank of Elliptic Curve has universality or conditional universality is an extremely demanding undertaking.
This task will be completed by three students: Zhao Tian, Xiao Yun, Zeng Han.
For example, given a prime number p=5, the Elliptic Curve y^2=x^3-x has seven solutions: (0,0), (1,0), (4,0), (2,1), (3,2), (3,3), and (2,4).
This is easy to calculate; any one of Zhao Tian, Xiao Yun, Zeng Han could arrive at the correct solution within 10 minutes by manually calculating it.
However, there are theoretically infinitely many Elliptic Curve. For any project involving an infinite number of verifications, manual calculations are impossible for humans and must rely on Computer Science.
Zhao Tian, Xiao Yun, Zeng Han will spend three days processing Ou Ye's manuscript into data that can be verified by Computer Science.
Verifying the Elliptic Curve using Computer Science based on Ou Ye's manuscript would take an unknown number of days—it could be three days, or it could be three or thirty years.
Fortunately, Professor Gong Changwei, Ou Ye's master's supervisor, made an important contribution to the BSD Conjecture.
Professor Gong proved the relevant theorems of the Kolyvagin converse and, together with other mathematicians, proved that at least two-thirds of Elliptic Curve satisfy the BSD Conjecture.
Professor Gong Changwei essentially eliminated many verification conditions for his student, Ou Ye. Therefore, Ou Ye's three students only needed to verify Elliptic Curve that satisfy Kolyvagin's theorem, Gross-Zagier's theorem, and Shafarevich-Tate group order.
Zhao Tian, the eldest of the three students, asked his juniors with concern, "Summer vacation is coming soon, have you two bought your tickets home?"
Xiao Yun shook her head. "Anyway, my parents aren't home, and if I go back, I'll just be a lonely single dog with no one to feed me, so I've decided to stay in the capital this summer to work and study."
"Xiao Yun, where did your parents go?" Zhao Tian asked.
While organizing Ou Ye's manuscripts, Xiao Yun said, "My mother went to Germany as a visiting scholar, and my father went to Africa to help our African friends build infrastructure. I won't be able to see my parents until next Chinese New Year."
Zhao Tian knew that Xiao Yun's mother had a doctorate from Fudan University and was currently a professor at East China Normal University. Xiao Yun's father had a master's degree from Shuimu University and was currently Engineer at China Construction Eighth Engineering Bureau.
Xiao Yun was able to get into the Big Mathematics Department at Peking University without taking the entrance exam when she was in high school. This wasn't a matter of whether she was smart or not, but rather a matter of genetics.
"Alright, alright. Xiao Yun, you're not going home for summer vacation. If you need anything, just tell me." Zhao Tian is a native of the capital, from Banchengzi Village, Bulaotun Town, Miyun District.
Since he was a resident of the capital and also a senior student, Zhao Tian felt that he should take care of Xiao Yun, his junior.
Xiao Yun nodded and said, "Thank you, brother. Let's get on with processing the data."
Zhao Tian then asked his junior, "What about you, Zeng Han? You're supposed to go home this summer, right?"
Zeng Han, focused on processing the data, didn't look up. "I'm not going home, I'm staying at school."
"Why? Did your parents go abroad too?"
"For no particular reason, I'm just staying on as a faculty member."
Zeng Han was admitted to Yenching University at the age of fifteen and just turned 18 this year. Both of Zeng Han's parents are PhDs, and his grandparents and maternal grandparents are all Senior Intellectuals. His parents' two families have a total of six PhDs and seven Masters, and five of them are full professors or researchers. The average age of members of the two families at the age of thirty-five is equivalent to that of Associate Professor or Associate Researcher.
Zeng Han comes from an intellectual family, was able to be admitted to Yenching University at the age of 15 without taking the entrance exam. This was not a matter of whether he was smart or not, but rather a matter of genetic inheritance.
Compared to his two younger classmates from prestigious families, Zhao Tian, who comes from Banchengzi Village in Bulaotun Town, Miyun District, is an inspirational senior.
Zhao Tian often says, "I'm not a genius, I never participated in the International Mathematical Olympiad. That year, I just randomly took the exam and got into Capital No. 4 High School. Three years later, I randomly took the exam again and got into Peking University. During my undergraduate studies, I just messed around and got a 4.0 GPA, which allowed me to be recommended for graduate studies in the Big Mathematics Department."
This was Zhao Tian's honest opinion. He felt that compared to his juniors, he was just an ordinary person. Xiao Yun and Zeng Han, on the other hand, were true geniuses.
The door to the little house creaked open.
A man entered. He was not tall, but he was energetic, with a high hairline and a glint of intelligence in his eyes. He was clearly no ordinary person.
"Teacher Zhou, what brings you here?" The three students were quite surprised.
The visitor was Zhou Yu'an, who, along with Shen Qi and Ou Ye, was known as one of the "Three Outstanding Students of the XXth Class of the Big Mathematics Department at Yenching University".
Zhou Yu'an, Shen Qi, and Ou Ye entered Yenching University's Big Mathematics Department in the same year. They were classmates and all three went on to complete their doctoral studies at the Princeton Mathematics Department. Their class of undergraduate students in the School of Mathematics is considered to be the strongest in Yenching University's history.
Can the successors of the School of the Big Mathematics Department surpass the outstanding graduates of the XXth cohort, including Shen, Ou, and Zhou?
It seems quite difficult at the moment.
Shen Qi's achievements alone are almost impossible to surpass. Not to mention surpassing them, even replicating them would be very difficult.
Zhou Yu'an, an IMO gold medalist, Ramanujan Prize laureate, and head of the Mathematics Room at Shen Qi Research Center, is considered by mathematics students at Yenching University to be the second most popular male idol after Shen Qi.
Teacher Zhou's position is not in the School of Mathematics, so his sudden appearance in front of the three students must mean something is amiss.