I just want to be a quiet scholar

Chapter 654 Finite Groups! (seeking a monthly ticket)

Elliptic curves appear everywhere in modern mathematics. Mathematicians who focus on other branches have studied elliptic curves. The forerunner is Suan Poincaré. This century-old all-round genius scholar first observed in 1901: For each Elliptic curves have a specific group associated with them.

Many mathematicians have combined group theory to study elliptic curves and achieved fruitful results.

The elliptic curve theory attracted the attention of mathematicians thanks to Poincaré, a genius calculator, and its prosperity benefited from another diligent genius, Wiles.

Andrew Wiles used elliptic curves to prove Fermat's last theorem, and he should be grateful for the conjectures made earlier by Toyo Taniyama and Goro Shimura.

The Taniyama-Shimura conjecture provides an effective way, which is an important step in the proof scheme of Fermat's last theorem.

Wiles first proved the Taniyama-Shimura conjecture, and soon after that Fermat's last theorem was successfully solved by him and Richard Taylor.

One of the proponents of the Taniyama-Shimura conjecture, Taniyama Toyo, although he had insight into the secret earlier than Wiles, was unable to prove his own conjecture.

Gu Shanfeng, who was only 31 years old, committed suicide by jumping into the sea on the eve of the wedding. A month later, his betrothed bride committed suicide and died in love.This may be the worst tragedy in the history of mathematics.

Elliptic curves bring not only success and joy, but also failure and sorrow.

After more than 100 years of development, contemporary scholars who study elliptic curves are also full of sorrow and joy.

BSD conjectures that this number theory problem is actually transformed into a problem of elliptic curves.

Ouye is not the first person who thought of using the method of group theory + elliptic curve to prove the BSD conjecture.

This has been done before, but without success.

The computer screen displays:

L(E, 1)≠0→r(E)=0, #Sha(E)<∞...r(E)=1, #Sha(E)<∞→ords=1L(E,1)=1 ...

#Sha(E) indicates the order of the Shafarevich-Tate group of the elliptic curve E. This group is very mysterious. Mathematicians, including Shafarevich and John Tate, who proposed this group, do not even know whether this group is Finite groups don't understand at all.

Of course, if you admit the BSD conjecture first, then the Shafarevich-Tate group must be a finite group.

Zhao Tian, ​​Xiao Yun, and Zeng Han were very excited, and the results of the computer calculations showed convincingly: based on Euye's algorithm, the Shafarevich-Tate group is a finite group!
This is undoubtedly a strong evidence to prove the BSD conjecture, and it is an important pre-step!

"That's one small step for us, one giant leap for mankind!"

"Sister Yezi is a big deal!"

"Ahahaha, really awesome!"

Zhao Tian and Xiao Yun hugged and celebrated, and Zeng Han, who had always been taciturn, was also excited.

Why are they so excited, because in their view, the research results their team has achieved at the moment are very close to Andrew Wiles proving the Taniyama-Shimura conjecture before proving Fermat's last theorem.

"The problem that Safarevich and John Tate failed to solve, we solved it! Long live Sister Ye Zi! Brother Xiao Zeng, save the result of the calculation and send it to Sister Ye Zi." Zhao Tian couldn't wait to send it to Sister Ye Zi. Tell Ouye the good news.

Safalevich, a famous mathematician in the former Soviet Union, wrote algebraic geometry, algebraic number theory, and algebraic mathematics textbooks, which were among the first mathematics textbooks introduced into domestic college mathematics majors after the founding of New China.Ouye's teachers at the undergraduate stage all learned the Safarevich version of the textbook.

John Tate, an American mathematician, is also a Princeton alumnus of Shen Qi, Ou Ye and Zhou Yu'an.John Tate is a double winner of the Wolf Prize and the Abel Prize. He is a master in the field of algebraic number theory. What he lacks is only a Philippine Medal.

Shafarevich and John Tate did not discuss the Shafarevich-Tate group problem, which was clearly demonstrated by Ouye's manuscripts.

However, Ouye has not been able to fully prove the problem that Birch and Swinnerton-Dell failed to prove.

"It's a pity that the computer can't verify the situation where the analytic rank is greater than or equal to 2. The invalid condition, what does this mean... But we have thoroughly proved that the elliptic curve with analytic rank equal to 0 and 1 satisfies the BSD conjecture, Xiao Zeng, the data is saved Hurry up and send it to Sister Ye Zi!"

……

……

Yanda People's Hospital.

Shen Qi stood with his hands behind his back, standing in front of the window of the ward.

It was pouring rain outside, and the water flooded his car.

Anyway, I bought Ping An auto insurance, so if it is flooded, it will be flooded.

"The weather is terrible." Shen Qi complained, then turned around and came to the hospital bed, staring at the sleeping Ouye.

In this new drug clinical trial, Ouye was arranged to enter the lowest dose group.

The subjects in the lowest dose group were the first to do the tolerance test, and this group has already been done.After a while, the tolerance test of other dose groups is completed and verified to be safe, and Ouye will return to the hospital to continue the phase I clinical trial of pharmacokinetics/efficacy.

Shen Qi originally planned to pick up Ouye home in the afternoon, but it was stormy outside and the car was flooded, so the couple couldn't go home for the time being.

drip.

Ouye's cell phone beeped.

drip.

Keep ringing.

Shen Qi glanced at Ouye's cell phone, it should be something like messages or emails.

He didn't dare to look at it, and he didn't dare to ask. His wife's mobile phone is not allowed to be unlocked without permission. This is an iron discipline.

"Hey..." Ouye's lips moved and he woke up.

"Leaf, you're awake."

"Well...it's raining heavily outside."

"Heavy torrential rain, very fierce, we will go home after the rain stops."

"Okay." The first thing Ouye did after waking up was to touch the phone. She swiped the screen of the phone to check for new emails and messages.

Zhao Tian, ​​the sender of the message: "Sister Yezi, our scheme has been verified in the computer room of Shenqi Research Center. The Shafarevich-Tate group is a finite group, and the elliptic curve with analytic rank equal to 0 and 1 conforms to our guess! analytic rank or algebraic The fact that the rank is greater than or equal to 2 needs to be further verified, so please take a look at the new email Zeng Han sent you! It would be even better if you could come to the computer room! I wish you a speedy recovery!"

"Ah!" Ouye whispered, looking very happy.

Shen Qi saw Ouye's phone screen by accident, and he probably guessed the specific reason why Ouye was happy.

"Rank 0 and 1 are settled. The Shafarevich-Tate group is indeed a finite group. The intuition of the two seniors is very keen." Ouye shook the phone, conveying the good news to Shen Qi.

"That's great, my wife, you are awesome and invincible." Shen Qi was sincerely happy for Ouye. A patient's mood affects patients. Of course he would be happy to see his wife happy every day.

Ouye clicked on the new email, and the sender of the email was Zeng Han: "Sister Ye Zi, please check the attachment. I wish you a speedy recovery."

The attachments are full of graphs and tabular data, Ouye carefully checked them out.

After reading it, Ouye suddenly became less happy. She handed the phone to Shen Qi: "If the rank is greater than or equal to 2, there is no one byte of verification data. I think I should have overlooked an important link."

Shen Qi took the mobile phone, he quickly calculated and analyzed, and said: "According to your settings, the initial computer verification, for the case of rank greater than or equal to 2, at least the verification data after Taylor expansion should be obtained. You may indeed ignore You have missed a certain key link, but you have done a good job. Yo, the rain has stopped, let's go home, and study slowly after we get home."

……

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(July [-]st, the party's birthday, ask for a guaranteed monthly pass!)
(End of this chapter)