MTL - The Science Fiction World of Xueba-Chapter 20 Q & A

Pang Shao'an and Yao Jianzhong looked at each other. Even if Pang Shao'an knew that his grandson was not the kind of person who ran a train with his mouth full, the status of BSD's conjecture was there. This level of difficulty was proved by a 23-year-old young man. The feeling of magic realism.

For a while, the small courtyard fell into a weird silence.

"Kobayashi, why are you pestering in the yard, come and help grandma choose vegetables."

Grandma Wang Xiufang walked out of the room and could not help seeing the sound of stagnation.

"What other dishes do you choose, Kobayashi, let's go to the study. Have you brought your thesis? I want to take a look first."

Pang Xuelin smiled: "Grandpa, I have uploaded the paper to arXiv, and it is also available in the mailbox, which can be downloaded directly."

"I'll go as well!"

Yao Jianzhong also got up and said.

"What's the matter with you?"

Wang Xiufang looked at Pang Shao'an and Yao Jianzhong who couldn't wait, and was puzzled.

I came to the study, Pang Xuelin boarded the mailbox, downloaded the paper directly, and handed it to Pang Shao'an for review.

Yao Jianzhong sat aside, banging Pang Shao'an and Pang Xuelin from time to time.

He is a theoretical physicist. BSD guesses that this level of mathematical papers, he naturally can not understand, but this does not prevent him from watching the lively sidelines.

Before his wife, his wife wanted to introduce his baby granddaughter to Pang Xuelin.

他 In his opinion, Pang Xuelin is too academically focused. Although his temperament is not a mum, he is not a very talkative person. His granddaughter is lively and with him, they may not be able to get together.

现在 But now, Pang Xuelin's performance has made him look impressive.

If Pang Xuelin really proves the BSD conjecture, then Pang Xuelin will win the Fields Award three years later.

The 26-year-old Fields Prize winner should be a record-breaker. Such a scholar has almost unlimited future achievements.

Although his granddaughter is a flower of Beiwu, to a certain extent, he is not necessarily worthy of Pang Xuelin.

Pang Xuelin and Pang Shao'an naturally did not know Yao Jianzhong's messy thoughts. At this time, all of Pang Shao'an's thoughts were immersed in Pang Xuelin's thesis and slowly turned page by page.

Pang Shao'an's main research area is differential geometry and topology, so this proof of the BSD conjecture is a bit difficult for him.

After watching it for about half an hour, Pang Shaoan took off the reading glasses and shook his head: "It's getting old, Kobayashi, your dissertation, honestly, I can't keep up with Grandpa, but I still understand your proof of thought. You started with the congruence problem, right? "

Xun Pang Xuelin nodded, and gave a thumbs up: "Grandpa, you really look like a torch!"

Pang Shaoan waved his hands and said, "Smelly boy, after studying abroad for a few years, I've learned to make a flatterer. I can't do it. Wait for Xiao Liu to come over and let him see. He has a lot in the field of number theory and algebraic geometry Deep research, he should be able to understand. "

Yao Jianzhong laughed and said, "Lao Pang, I didn't expect you to be unsure."

Pang Shao'an stared and said, "I am now older. If I was young, I would hold it for a few months, and I would definitely understand this paper."

"Hum, you smell bad!"

Yao Jianzhong disdain.

The two old men are normal to each other. Pang Xuelin looked at it with a smile and did not participate.

After a while, Wang Xiufang came in with a middle-aged man with a bald head and glasses.

Tan Lairen is Liu Tingbo, Dean of the School of Mathematical Sciences, Jiangcheng University.

Liu Tingbo laughed: "Professor Pang, Professor Yao, hello."

Then, he turned his eyes to Pang Xuelin and said, "Xiao Pang, you put an atomic bomb in the mathematical world this time!"

Pang Xuelin was curious: "Dean Liu, I only uploaded my thesis to arXiv this morning, how did you know so quickly?"

Liu Tingbo said: "It's not the blessing of your teacher. After reading your thesis, Tao Zhexuan contacted the big brothers such as Deligne, Faltins, and Qiu Chengtong for the first time, saying that you most likely solved the BSD conjecture. And then shared the paper with them. I do n’t know what happened, it was originally a small circle, and it was passed on all of a sudden, and now the entire mathematical community is boiling. Professor Qiu has no contact information for you. Just hit the phone directly on my head, and I'm still stunned! I said that this time, wouldn't you just put me a satellite? "

Xi Pang Xuelin smiled, "Dean Liu, here is the thesis, or should you take a look first?"

Pang Shao'an also said, "Xiao Liu, you are an expert in the field of algebraic geometry and number theory, and you just check Xiaolin!"

Liu Tingbo smiled bitterly: "Professor Pang, I dare not be a guard. If Xiao Pang really solves the BSD conjecture, he can be my teacher!"

Even though he said so, Liu Tingbo sat in front of the computer and looked up carefully.

看 While watching, he still communicated with Pang Xuelin from time to time about some questions in the thesis.

"Xiao Pang, here it is assumed that D has no square factor. Simple elementary considerations show that D is a congruence equivalent to the elliptic curve E_D: y ^ 2 = x ^ 3-D ^ 2x has a rational point on y \\ neq0. It can be proven that such points do not belong to T, so D is a congruence and is equivalent to r_D> 0. (Congruence Problem) Determine all congruences D such that r_D> 0. For a given prime p, (1) p \\ equiv3 (\\ mod8): p is not a congruence but 2p is a congruence; (2) p \\ equiv5 (\\ mod8): p is a congruence; (3) p \\ equiv7 (\\ mod8): p and 2p are both congruences. The tool you use is the height theory of the Heegner point. How did you relate it to L \ '(1, E)? Also, how do you determine that D is a congruence? "

In the three-body world, Pang Xuelin withstood the bombardment of top mathematicians, and dealt with this kind of problem easily and exceptionally. The answer was as follows: "The definition of the Weil-Hasse function L (s, E) for E, A classic result is that a_p has a Hasse upper bound 2 \\ sqrt {p}, which infers that L (s, E) converges on \\ mathrm {Re} \\, s> \\ frac {3} {2}. Then we According to the Gross-Zagier formula, it can be related to L \ '(1, E). In addition, the BSD conjecture holds for E_D. In particular, r_D> 0 if and only if L (1, E_D) = 0. Assume The weak BSD conjecture holds, then (1) theoretically we can determine whether D is a congruence; (2) Tunnell's theorem gives an algorithm to determine whether D is a congruence within a finite step; (3) can prove that D \ equiv5, At 6,7 (\\ mod8) r_D is an odd number, so such Ds are congruent. "

Liu Tingbo thought for a moment, and nodded with satisfaction. After a while, he asked again: "You said here that the Taylor coefficient of L (s, E) at s = 1 is equal to the order of the Tate-Shafarevich group of E. Proportionally, how did you come to such a conclusion? And here, E (Q) (Mordell-Weil group) has a natural commutative group structure. You further asserted that E (Q) was finitely generated according to Mordell's theorem: E (Q) = \\ BbbZ ^ r \\ oplusT, where the torsion group T is a finite Abel group, and r is called the rank of E. Our understanding of T is complete: Mazur determines all 15 possible T . So what about R? Are you missing an effective characterization of R here? "

Pang Xuelindao: "Based on Eichler and Shimura's work on modular elliptic curves and the newly proved Taniyama–Shimura conjecture (modulo theorem), we now know that L (s, E) can be analytically extended to the entire complex plane and the corresponding Riemann conjecture. The BSD conjecture is equal to the order m of zero at R (s, E) at s = 1. In the case that the module theorem has been proved, the BSD conjecture is true for m = 0.1, so L (s, E The Taylor coefficient expanded at s = 1 is proportional to the order of the Tate-Shafarevich group of E. Furthermore, the finiteness of the Tate-Shafarevich group can be derived. "

Liu Tingbo pondered for a long while and gave a thumbs up: "You indirectly proved the weak conjecture of BSD from the problem of congruence, and then extended it to the generalized BSD conjecture. This method is absolutely amazing!"

Uh ...

Next, Liu Tingbo and Pang Xuelin asked and answered, almost every question, Pang Xuelin could give the answer without thinking ~ www.novelbuddy.com ~ Time passed every minute and second, even Wang Xiufang prepared dinner and came up to call them Eating was also fascinated by the question and answer between Pang Xuelin and Liu Tingbo. After watching for a long time, Wang Xiufang quietly exited the study and did not disturb them.

Until ten o'clock in the evening, Liu Tingbo thoroughly reviewed the paper, and the Q & A between them ended.

Although Pang Shao'an and Yao Jianzhong, who were aside, could not keep up with their thinking, their emotions were always in a state of excitement.

They can see that in this question and answer, a world-class problem is slowly being solved from Pang Xuelin.

This process of witnessing a world-class mathematical problem slowly reveals its true beauty, and excites everyone present.

Pang Shao'an looked at Liu Tingbo and said, "Xiao Liu, what is Xiao Lin's proof, do you think he succeeded?"

Liu Tingbo said: "Professor Pang, I dare not say that Xiao Pang has 100% proved the BSD conjecture. I have a 90% certainty of this paper. Xiao Pang, look at this. Your paper was uploaded to arXiv shortly, we wait After a period of time, after the big brothers such as Deligne and Faltings have stated their positions, I will arrange an academic lecture for you at Jiangda University. At that time, it should be able to attract the world's top mathematicians. At this time, You do n’t have to go to class for the time being, you can prepare for the report with peace of mind, and the PPT should be more detailed. ”

Pang Xuelin laughed: "Dean Liu, there is nothing wrong with the class. Anyway, there are still a few days before the official start of the class. I also have classes for one, three, and five every week. Relax."

Liu Tingbo thought for a while and thought, "Okay, then I won't force you, haha, Xiao Pang thank you so much this time. As soon as I returned to my alma mater, I gave my alma mater such a gift."