The Finite Group!
Elliptic Curve appear everywhere in modern mathematics. Mathematicians specializing in other branches have begun to study Elliptic Curve. The pioneer was the mathematician Henri Poincare, a once-in-a-century all-around genius who first observed in 1901 that for every Elliptic Curve, there is a specific group associated with it.
Many mathematicians have combined Group Theory methods to study Elliptic Curve and achieved remarkable results.
Elliptic Curve theory gained attention from mathematicians thanks to the brilliant mathematician Poincare, and its flourishing was due to another diligent genius, Wiles.
Andrew Wiles proved Fermat's Last Theorem using Elliptic Curve, and he should thank Yutaka Taniyama and Goro Shimura for their earlier conjectures.
The Taniyama-Shimura conjecture provides an effective approach, which is an important step in the proof of Fermat's Last Theorem.
Wiles first proved the Taniyama-Shimura conjecture, and soon after, he and Richard Taylor successfully solved Fermat's Last Theorem.
Yutaka Taniyama, one of the proposers of the Taniyama-Shimura conjecture, although he had foreseen the secret earlier than Wiles, struggled to prove his own conjecture.
Yutaka Taniyama, only 31 years old, committed suicide by jumping into the sea on the eve of his wedding. A month later, his fiancée, with whom he was engaged, also committed suicide. This is perhaps the most tragic event in the History of Mathematics.
Elliptic Curve bring not only success and joy, but also failure and sorrow.
After more than a century of development, contemporary scholars studying Elliptic Curve have experienced a mix of joy and sorrow.
The BSD Conjecture that this number theory problem is actually transformed into Elliptic Curve problem.
Ou Ye was not the first person to think of using Group Theory and Elliptic Curve to prove the BSD Conjecture.
Others have tried this before, but without success.
The Computer Science screen displays:
L(E,1)≠0→r(E)=0,#Sha(E)<∞... r(E)=1,#Sha(E)<∞→ords=1L(E,1)=1...
#Sha(E) denotes the order of the Shafarevich-Tate group of the Elliptic Curve E. This group is very mysterious; even its proposers, Shafarevich and John Tate, have not figured out whether it is a finite group.
Of course, if you accept the BSD Conjecture, then the Shafarevich-Tate group must be a finite group.
Zhao Tian, Xiao Yun, Zeng Han were extremely excited. The Computer Science verification results convincingly showed that, based on Ou Ye algorithm, the Shafarevich-Tate group is a finite group!
This is undoubtedly strong evidence to prove the BSD Conjecture and an important preliminary step!
"This is one small step for mankind, one giant leap for mankind!"
"Sister Yezi Ye is awesome!"
"Hahaha, that's really awesome!"
Zhao Tian and Xiao Yun hugged each other in celebration, and even the usually taciturn Zeng Han was filled with excitement.
Why were they so excited? Because, in their view, the research results their team had achieved were very close to Andrew Wiles proving the Taniyama-Shimura conjecture before proving Fermat's Last Theorem.
"We've solved the problem that even the greats Shafarevich and John Tate couldn't! Long live Sister Yezi! Little Zeng, save the verification results and send them to Sister Yezi." Zhao Tian couldn't wait to tell Ou Ye the good news.
Shafarevich, a renowned mathematician from the former Soviet Union, wrote textbooks on Algebraic Geometry, Algebraic Number Theory, and algebra, which were among the earliest mathematics textbooks introduced to Chinese universities after the founding of the New China. Ou Ye's undergraduate professors all used Shafarevich's textbooks.
John Tate, an American mathematician, was also a Princeton alumnus of Shen Qi, Ou Ye, and Zhou Yu'an. John Tate is a double recipient of the Wolf Prize and the Abel Prize. He is a master in the field of Algebraic Number Theory. The only thing he lacks is a Fields Medal.
The Shafarevich-Tate group problem, which the two great predecessors Shafarevich and John Tate failed to prove clearly, was demonstrated clearly by several manuscripts of Ou Ye.
However, the problem that Birch and Swinnerton-Dyer failed to prove, Ou Ye has not yet been able to prove completely either.
"It's a bit of a shame that the Computer Science couldn't verify the case where the analytic rank is greater than or equal to 2; it's an invalid condition. What does that mean...? But we've definitively proven that Elliptic Curve with analytic rank equal to 0 and 1 satisfy the BSD Conjecture. Zeng, have you saved the data? Send it to Sister Yezi quickly!"
Yanjing University People's Hospital.
Shen Qi stood with his hands behind his back, in front of the window of the ward.
Outside, the rain was pouring down, and the floodwaters had submerged his car.
Since I bought Ping An car insurance, I guess I'll just let it flood.
"This weather is awful," Shen Qi complained, then turned and went to the bedside, gazing at the sleeping Ou Ye.
In this clinical trial of the new drug, Ou Ye was assigned to the lowest dose group.
The subjects in the lowest dose group were the first to undergo tolerability testing, and this group has already completed it. After a while, the tolerability tests for the other dose groups will be completed and confirmed to be safe. Ou Ye will then return to the hospital to continue undergoing the Phase I Clinical Trial for pharmacokinetic/pharmacodynamic effects.
Shen Qi had originally planned to pick up Ou Ye and take her home in the afternoon, but a violent storm had flooded the car, so the couple couldn't go home for the time being.
Beep beep.
Ou Ye's phone beeped.
Beep beep.
The sound continues.
Shen Qi glanced at Ou Ye's phone; it was probably filled with messages or emails.
He didn't dare look, he didn't dare ask; unlocking his wife's phone without permission was an ironclad rule.
"Waaah..." Ou Ye's lips moved; she was awake.
"Yezi, you're awake."
"Hmm... it's raining so hard outside."
"It's a heavy downpour, really intense. We'll go home after the rain stops."
"Okay." The first thing Ou Ye did after waking up was to reach for her phone. She swiped the screen to check for new emails and messages.
The message from Zhao Tian reads, "Sister Yezi, our solution has been verified in the Computer Science lab at Shen Qi Research Center. The Shafarevich-Tate group is a finite group, and the Elliptic Curve with analytical rank equal to 0 or 1 match our conjecture! Cases with analytical rank or algebraic rank greater than or equal to 2 require further verification. Please check the new email Zeng Han sent you! It would be even better if you could come to the Computer Science lab! Wishing you a speedy recovery!"
"Wow!" Ou Ye exclaimed softly, looking quite happy.
Shen Qi glanced at Ou Ye's phone screen and roughly guessed why she was happy.
"Rank 0 and 1 are sorted. The Shafarevich-Tate group is indeed a finite group. The two seniors have very sharp intuition." Ou Ye waved her phone and conveyed the good news to Shen Qi.
"That's great! You're amazing, my wife, absolutely incredible!" Shen Qi was genuinely happy for Ou Ye. A patient's mood can affect their health, and he was naturally happy to see his wife happy every day.
Ou Ye opened the new email. The sender was Zeng Han: "Sister Yezi, please check the attachment. Wishing you a speedy recovery."
The attachments were full of graphs and tables, Ou Ye carefully reviewed.
After reading it, Ou Ye suddenly wasn't so happy anymore. She handed her phone to Shen Qi: "For cases where rank is greater than or equal to 2, there isn't a single byte of verification data. I think I might have overlooked an important step."
Shen Qi took the phone and quickly performed calculations and analysis. He said, "According to your settings, the initial Computer Science verification should at least yield verification data after Taylor expansion for cases with a rank greater than or equal to 2. You may have overlooked a crucial step, but you've done a great job. Oh, the rain has stopped. Let's go home and study it slowly when we get there."
(July 1st, the Party's birthday, please vote for me!)