Pang Shaoan and Yao Jianzhong looked at each other, even though Pang Shaoan knew that his grandson was not the kind of person who ran the train with his mouth full, but the status of BSD's conjecture was there, and this level of problem was proved by a 23-year-old young man, and there was a feeling of magical realism when he thought about it.

For a moment, the small courtyard fell into an eerie silence.

"Xiaolin, why are you in the yard, come, help grandma choose vegetables."

Grandma Wang Xiufang walked out of the house, and when she saw everyone's sluggish looks, she couldn't help but say something.

"What else to choose, Xiaolin, let's go, let's go to the study, did you bring the paper? I'm going to take a look first. “

Pang Xuelin smiled: "Grandpa, I have uploaded the paper to arXiv, and I also have it in my mailbox, so I can download it directly." ”

"I'll go too!"

Yao Jianzhong also got up and said.

"What's wrong with you?"

Wang Xiufang looked at Pang Shaoan and Yao Jianzhong, who couldn't wait, and was a little puzzled.

When he came to the study, Pang Xuelin boarded the mailbox, downloaded the paper directly, and handed it to Pang Shaoan for review.

Yao Jianzhong sat on the sidelines, glancing at Pang Shaoan and Pang Xuelin from time to time.

He was a theoretical physicist, and BSD guessed that he couldn't read a mathematical paper of this level, but that didn't stop him from watching the excitement from the sidelines.

Before, his wife wanted to introduce his baby granddaughter to Pang Xuelin, and he still had some complaints.

In his opinion, Pang Xuelin is too focused on academics, although his temperament is not wooden, but he is not the kind of person who talks a lot, his granddaughter is lively and active, and with him, the two may not be able to get along.

But now, Pang Xuelin's performance has impressed him.

If Pang Xuelin really proved the BSD conjecture, then Pang Xuelin would have won the Fields Medal three years later.

The twenty-six-year-old Fields Medal winner should have broken the record, and the future achievements of such a scholar are almost limitless.

Although his granddaughter is a flower of the Northern Dance, to a certain extent, she is not necessarily worthy of Pang Xuelin.

Pang Xuelin and Pang Shaoan naturally didn't know Yao Jianzhong's messy thoughts, and at this time, all Pang Shaoan's thoughts were immersed in Pang Xuelin's papers, and they slowly flipped through the pages one by one.

Pang Shaoan's main research areas are differential geometry and topology, so this paper proving the BSD conjecture was a bit difficult for him.

After reading for about half an hour, Pang Shaoan took off his reading glasses, shook his head and said: "Old, old, Xiaolin, your paper, to be honest, grandpa, I can't keep up, but I probably still understand your proof idea, you started with the same remainder problem, right?" ”

Pang Xuelin nodded, gave a thumbs up and said, "Grandpa, you are really as wise as a torch!" ”

Pang Shaoan waved his hand and said: "Stinky boy, I have studied abroad for a few years, but I have learned to pat myself on the back, I can't do it, wait for Xiao Liu to come over later, let him see, he has a deep study in the field of number theory and algebraic geometry, he should be able to understand." “

Yao Jianzhong said with a smile: "Lao Pang, I didn't expect you to have a time when you couldn't figure it out." ”

Pang Shaoan glared and said: "That is, I am old now, if I was young, I would definitely be able to figure out this paper if I chewed it for a few months." ”

"Hmph, you stink!"

Yao Jianzhong said disdainfully.

It is normal for these two old men to scold each other, and Pang Xuelin looked at it with a smile and did not participate.

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

The person who came was Liu Tingbo, dean of the School of Mathematical Sciences of Jiangcheng University.

Liu Tingbo smiled and said, "Professor Pang, Professor Yao, hello. ”

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

Pang Xuelin was a little curious: "Dean Liu, I only uploaded the paper to arXiv in the early hours of this morning, how did you know so quickly?" ”

Liu Tingbo said: "It's not thanks to your teacher, after Tao Zhexuan read your paper, he immediately contacted Deligne, Faltins, Yau Chengtong, and other bigwigs, saying that you are very likely to solve the BSD conjecture problem, and then shared the paper with them." I don't know what's going on, this was originally a matter in a small circle, but it turned out all at once, and now the whole mathematical community is boiling. Professor Qiu didn't have your contact information, so he called me directly, and I was still confused! I said that this time, you shouldn't be putting a satellite on me, right? ”

Pang Xuelin smiled and said, "Dean Liu, the paper is here, why don't you take a look at it first?" ”

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

Liu Tingbo smiled bitterly: "Professor Pang, I don't dare to be a gatekeeper, if Xiao Pang really solves the BSD conjecture, he can be my teacher!" ”

Having said that, Liu Tingbo still sat in front of the computer and looked at it carefully.

While watching, he also communicated with Pang Xuelin from time to time on some questions in the paper.

"Xiao Pang, here it is assumed that D has no square factor, and a simple elementary consideration shows that D is the same remainder equivalent to the elliptic curve E_D: y^2=x^3-D^2x has a rational point of y \neq 0. It can be proved that such a point does not belong to T, so D is the same remainder and equivalent to r_D>0. (Same remainder problem) determines all identical remainders D, such that r_D> 0. For a given prime p, (1)p \equiv 3(\mod 8) :p is not the same remainder but 2 p is the same remainder; (2) p \equiv 5(\mod 8):p is the same remainder; (3) p \equiv 7 (\mod 8) :p and 2 p are both congruent remainders. The tool you use is the height theory of the Heegner point, how do you relate it to L'(1,E)? Also, how do you make sure that all Ds are the same remainder? “

When Pang Xuelin was in the three-body world, he withstood the bombardment of those top mathematicians, and it was easy to deal with this kind of problem, and the answer was like a stream: "Regarding the definition of the Weil-Hasse function L(s,E) of E, a classic result is that there is a_p Hasse upper bound 2\sqrt{p}, which introduces the convergence of L(s,E) to \mathrm{Re}\, s>\frac{3}{2}. Then, according to the Gross-Zagier formula, we can relate it to L'(1,E). In addition, the BSD conjecture holds true for E_D. In particular, r_D>0 if and only if L(1,E_D)=0. Assuming that the weak BSD conjecture is true, then (1) theoretically we can determine whether D is the same remainder; (2) The Tunnell theorem gives an algorithm for determining whether D is the same remainder in a finite step; (3) It can be proved that D \equiv 5,6,7 (\mod 8) is r_D odd, so such D is the same remainder. “

Liu Tingbo pondered for a moment, nodded with satisfaction, and after a while, he asked again: "You said here that the Taylor coefficient of L(s,E) expanding at s=1 is directly proportional to the order of the Tate-Shafarevich group of E, how did you come to such a conclusion?" And here, E(Q) (Mordell-Weil group) has a natural commutative group structure, and you further asserted that E(Q) is finitely generated according to Mordell's theorem: E(Q)=\Bbb Z^r \oplus T, where the deflection group T is a finite Abel group, and r is called the rank of E. What we know about the T is complete: Mazur has decided on all 15 possible T's. And what about R? Are you missing an effective portrayal of R here? “

"Based on Eichler, Shimura's work on modular elliptic curves and the newly proven Taniyama–Shimura conjecture (modular theorem), it is now known that L(s,E) can be extended to the entire complex plane and the corresponding Riemann conjecture holds. The BSD conjecture is the order m where R is equal to L(s,E) at the zero point at s=1. With the modulus theorem proved, it is known that the BSD conjecture holds for m=0.1, so the Taylor coefficient of L(s,E) expanding at s=1 is proportional to the order of the Tate-Shafarevich group of E, and further, the finite nature of the Tate-Shafarevich group can be deduced. ”

Liu Tingbo pondered for a long time, gave a thumbs up, and said: "You indirectly proved the weak conjecture of BSD from the same remainder problem, and then extended it to the generalized BSD conjecture, this method is really amazing!" ”

……

Then, Liu Tingbo and Pang Xuelin asked and answered, and Pang Xuelin could give answers to almost every question without thinking.

Time passed minute by minute, even Wang Xiufang made dinner and came up to ask them to eat, but was also attracted by the Q&A between Pang Xuelin and Liu Tingbo, and after watching it for a long time, Wang Xiufang quietly withdrew from the study and did not disturb them.

It wasn't until ten o'clock in the evening that Liu Tingbo completely reviewed the paper, and the Q&A between the two ended.

Although Pang Shaoan and Yao Jianzhong on the side couldn't keep up with their train of thought, 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's hands.

The process of witnessing a world-class math problem slowly unfold itself made everyone present excited.

Pang Shaoan looked at Liu Tingbo and said, "Xiao Liu, how is Xiaolin's proof, do you think he succeeded?" ”

Liu Tingbo said: "Professor Pang, I dare not say that Xiao Pang has proved the BSD conjecture 100%, and I am eighty or ninety percent sure of this paper." Xiao Pang, how do you see this, your paper has only been uploaded to arXiv for a short time, we will wait for a while, and wait for the big guys such as Deligne and Faltins to express their positions one after another, and I will arrange an academic report for you at Jiang University, which should attract the world's top mathematicians to attend the conference. During this time, you don't have to attend classes for the time being, so you can prepare for the report with peace of mind, and the PPT should be more detailed. “

Pang Xuelin said with a smile: "Dean Liu, there is no problem in class, anyway, there are still a few days before the official start of classes, and I only have classes on Monday, Wednesday and Friday every week, and teaching undergraduates is a kind of relaxation for me." ”

Liu Tingbo thought for a while and said: "Okay, then I won't force you, haha, Xiaopang thanked you this time, just returned to his alma mater, and gave his alma mater such a big gift first." ”