Elliptic curves appear everywhere in modern mathematics, and mathematicians who specialize in other branches have studied elliptic curves, the forerunner is the arithmetic master Poincaré, a once-in-a-century all-round genius scholar who first observed in 1901 that for each elliptic curve, there is a specific group associated with it.

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

The theory of elliptic curves attracted the attention of mathematicians due to the genius arithmetic Poincaré, and its prosperity was due to another hard-working genius, Wiles.

Andrew Wiles proved Fermat's theorem using elliptic curves, and he should be grateful to Yutaka Taniyama and Goro Shimura for their earlier conjectures.

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

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

And Yutaka Taniyama, one of the proposers of the Taniyama-Shimura conjecture, although he had insight into the heavenly opportunity earlier than Wiles, he was unable to prove his own conjecture.

At the age of 31, Tataka Taniyama committed suicide by jumping into the sea on the eve of his wedding, and a month later, the bride who was engaged to him committed suicide. This is perhaps the most tragic tragedy in the history of mathematics.

Ellipse curves bring not only success and joy, but also failure and sadness.

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

The BSD conjecture is a number theory problem that actually translates into an elliptic curve problem.

Ouye is not the first to think of using group theory + elliptic curves to prove the BSD conjecture.

Someone has done this for a long time, but it didn't work out.

The computer screen shows:

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, which is so mysterious that mathematicians, including Shaphalevich and John Tate, who proposed it, have not even understood whether this group is a finite group or not.

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

Zhao Tian, Xiaoyun, and Zeng Han were very excited, and the computer calculation results showed very convincingly: based on the Ouye algorithm, the Shafarevich-Tate group is a finite group!

This is undoubtedly a strong evidence to prove the BSD conjecture and an important precursor!

"It's one small step for us, it's one giant leap for humanity!"

"Sister Ye Niu Pi!"

"Aha, it's really a cow!"

Zhao Tian and Xiaoyun hugged each other to celebrate, and Zeng Han, who had always been taciturn, was also enthusiastic.

The reason why they are so excited, because in their opinion, the research results achieved by their team at this moment are very close to Andrew Wiles's proof of the Taniyama-Shimura conjecture before proving Fermat's theorem.

"The problem that Shaphalevich and John Tate didn't solve was solved by us! Long live Sister Ye Zi! Brother Xiao Zeng, save the calculation results and send them to Sister Ye Zi. Zhao Tian couldn't wait to tell Ou Ye the good news.

Shafalevich, a famous mathematician in the former Soviet Union, compiled algebraic geometry, algebraic number theory, and algebra textbooks, which were one of the earliest mathematics textbooks to be introduced into domestic colleges and universities after the founding of the People's Republic of China. Ouye's teachers at the undergraduate level all learned the Shaphalevich 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, a double winner of the Wolf Prize and the Abel Prize, is a master in the field of algebraic number theory, and all he lacks is a Philippine medal.

The Shafarevich-Tate problem, which was not clearly demonstrated by the two great gods of Shafarevich and John Tate, is clearly demonstrated by several manuscripts of Oye.

However, the problem that Birch and Swinnaton-Dyer failed to prove, Oye has not been able to prove thoroughly for the time being.

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

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Yanda People's Hospital.

Shen Qi stood in front of the window of the ward.

Outside, it was pouring rain, and water was flooding his car.

Anyway, I bought Ping An car insurance, and if it is flooded, it will be flooded.

"The weather, terrible. Shen Qi complained, he turned around and came to the bedside, looking at the sleeping Ou Ye.

In this clinical trial of the new drug, Ouye was placed in the lowest dose group.

The first tolerability test was done in the lowest dose group, which had already been done. In a few days, the tolerability trials of the other dose groups have been 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 take Ou Ye home in the afternoon, but there was a violent storm outside, and the car was flooded, and the couple couldn't go home for the time being.

Drip.

Ouye's phone rang with a chime.

Drip.

Keep ringing.

Shen Qi glanced at Ou Ye's mobile phone, it should be something like a message, an email, or something.

He didn't dare to look at it, he didn't dare to ask, his wife's mobile phone was not allowed to be unlocked without permission, this is iron discipline.

"Boom......" Ou Ye's lips moved and he woke up.

"Leaf, you're awake. ”

"Hmm...... It's raining heavily. ”

"Heavy rain, very hard, when the rain stops, we will go home. ”

"Okay. The first thing Ou Ye did when she woke up was to touch her phone, and she swiped the screen of her phone to check for new emails and messages.

Zhao Tian, the sender of the message: "Sister Ye Zi, our plan has been verified in the computer room of Shen Qi Scientific Research Center, the Shafarevich-Tate group is a finite group, and the elliptic curve of analytic rank equal to 0 and 1 is in line with our guess! The situation that analytic rank or algebraic rank is greater than or equal to 2 needs to be further verified, you can take a look at the new email sent to you by Zeng Han! If you can come to the computer room, it will be even better! I wish you a speedy recovery!" ”

"Ahh

Shen Qi inadvertently glanced at Ou Ye's mobile phone screen, and he probably guessed the specific reason why Ou Ye was happy.

"rank0, 1 is done, the Shafarevich-Tate group is indeed a limited group, and the intuition of the two seniors is very sharp. Ou Ye shook his phone and conveyed the good news to Shen Qi.

"That's great, wife, you're awesome, invincible. Shen Qi was sincerely happy for Ou Ye, the mood has an impact on the patient, of course he is happy to see his wife happy every day.

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

The attachments are full of graphs and tabular data, and Ou Ye carefully consulted them.

After reading it, Ou Ye suddenly wasn't so happy, she handed her phone to Shen Qi: "In the case of rank greater than or equal to 2, there is not a single byte of verification data, I think, I should ignore an important link." ”

Shen Qi took the 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's expansion should be obtained." It's possible that you've missed a crucial link, but you've done a good job. Yo, the rain has stopped, let's go home, and study slowly when we get home. ”

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(July 1st, the party's birthday, ask for a guaranteed monthly pass!)