The Science Fiction World of Xueba

Pang Shaoan and Yao Jianzhong looked at each other, even though Pang Shaoan knew that his grandson was not the kind of person who talked about running the train, but the status of BSD's conjecture was there. A 23-year-old young man proved a problem of this level. The feeling of magic realism.

For a moment, the small courtyard fell into a strange silence.

"Xiao Lin, why are you stuck in the yard, come, help grandma choose vegetables."

Grandma Wang Xiufang came out of the house, seeing everyone's dull faces, she couldn't help saying.

"What else to choose, Xiao Lin, let's go, let's go to the study, have you brought your paper? I want to see it first."

Pang Xuelin smiled and said: "Grandpa, I have uploaded the paper to arXiv, and it is also in the mailbox, you can download it directly."

"I am coming too!"

Yao Jianzhong also got up and said.

"What's wrong with you guys?"

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 aside, looking at Pang Shaoan and Pang Xuelin from time to time.

He is a theoretical physicist, BSD guesses that this level of mathematics papers, he naturally can't understand, but this does not prevent him from watching the fun.

Before his wife wanted to introduce his precious granddaughter to Pang Xuelin, he still had some criticisms.

In his opinion, Pang Xuelin was too focused on academics. Although his temper was not dull, he was not the kind of particularly talkative person. His granddaughter was lively and active. With him, the two might not get along well.

But now, Pang Xuelin's performance surprised him.

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

The 26-year-old Fields Medal winner should have broken the record. The future achievements of such a scholar are almost limitless.

Although his granddaughter is a flower of Beiwu, to some extent, she may not be worthy of Pang Xuelin.

Naturally, Pang Xuelin and Pang Shaoan didn't know Yao Jianzhong's messy thoughts. At this time, all of Pang Shaoan's thoughts were immersed in Pang Xuelin's thesis, slowly flipping through the pages.

Pang Shaoan's main research fields are differential geometry and topology, so this paper on the proof of the BSD conjecture is a bit difficult for him.

After watching it for about half an hour, Pang Shaoan took off his reading glasses, shook his head and said, "I'm getting old, Xiao Lin, to be honest, I can't keep up with your thesis, but I probably still understand your proof ideas." , you started with the congruence problem, didn't you?"

Pang Xuelin nodded, gave a thumbs up and said, "Grandpa, you really have sharp eyes!"

Pang Shaoan waved his hand and said, "Boy, after studying abroad for a few years, he has learned how to flatter. I can't do it anymore. Let Xiao Liu come over later and let him have a look. He has great knowledge in the fields of number theory and algebraic geometry. In-depth research, he should be able to understand."

Yao Jianzhong smiled and said, "Old Pang, I never thought you would be uncertain sometimes."

Pang Shaoan stared and said: "That is to say, I am old now. If I were young, I would definitely be able to understand this paper after studying for a few months."

"Hmph, you're a stinking fart!"

Yao Jianzhong said disdainfully.

It is normal for these two old men to fight each other. Pang Xuelin watched with a smile and did not participate.

After a while, Wang Xiufang walked in with a middle-aged man with a 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 have dropped an atomic bomb in the mathematics world this time!"

Pang Xuelin said curiously: "Dean Liu, I just uploaded the paper to arXiv early this morning.

How did you find out so quickly? "

Liu Tingbo said: "Thanks to your teacher, Tao Zhexuan immediately contacted Deligne, Faltins, Qiu Chengtong and other big men after reading your paper, saying that you are very likely to solve the BSD conjecture , and then shared the paper with them. Somehow, this was originally a matter in a small circle, but it was spread all of a sudden, and now the entire mathematics world is boiling. Professor Qiu does not have your contact information. You just called me directly, and I was still in a daze! I said, this time, you are not going to release satellites for me, are you? "

Pang Xuelin smiled and said, "Dean Liu, the paper is here, why don't you read it first?"

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

Liu Tingbo smiled bitterly and said, "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 on some questions in the paper from time to time.

"Xiao Pang, it is assumed here that D has no square factor. Simple elementary considerations show that D is a congruent number 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 a congruent number and is equivalent to r_D\u003e0. (Congruent number problem) Determine all congruent numbers D such that r_D\u003e0. For a given prime number p, (1)p \\equiv 3(\\mod 8): p is not a congruence number but 2 p is a congruence number; (2)p \\equiv 5(\\mod 8): p is a congruence number; (3)p \\equiv 7(\\mod 8): p and 2 p are both congruent numbers. The tool you used was the height theory of Heegner points, how did you relate it to L'(1,E)? Also, how did you determine that D is both congruent numbers? "

When Pang Xuelin was in the Three-Body World, he withstood the bombardment of those top mathematicians. It was easy to deal with this kind of problem, and the answer was like a runner: "About the definition of the Weil-Hasse function L(s, E) of E, A classic result is that a_p has a Hasse upper bound 2\\sqrt{p}, which leads to the convergence of L(s, E) to \\mathrm{Re}\\, s\u003e\\frac{3}{2}. Then we according to Gross-Zagier formula, it can be linked with L'(1, E). In addition, the BSD conjecture holds for E_D. In particular, r_D\u003e0 if and only if L(1, E_D)=0. Assuming the weak BSD conjecture holds, Then (1) theoretically we can determine whether D is a congruent number; (2) Tunnell's theorem provides an algorithm to determine whether D is a congruent number within a finite step; (3) it can be proved that D \\equiv 5,6,7(\\ mod 8) when r_D is an odd number, so such D is a congruent number."

Liu Tingbo thought for a moment, nodded with satisfaction, and after a while, he asked again: "You said here, the Taylor coefficient expanded by L(s, E) at s=1 is proportional to the order of the Tate-Shafarevich group of E. Proportional, how did you come to such a conclusion? Also here, E(Q) (Mordell-Weil group) has a natural commutative group structure. You further asserted that E(Q) is finitely generated according to Mordell's theorem: E (Q)=\\Bbb Z^r \\oplus T, where the torsion group T is a certain finite Abelian group, and r is called the rank of E. Our knowledge of T is complete: Mazur determines all 15 possible T .What about R? Are you missing an effective characterization of R here?"

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

Liu Tingbo pondered for a while, then gave a thumbs up and said, "You have indirectly proved the BSD weak conjecture from the congruent remainder problem, and then extended it to the generalized BSD conjecture. This method is absolutely brilliant!"

...

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

Time passed by, and even Wang Xiufang prepared dinner and came up to ask them to eat, but was also attracted by the question and answer between Pang Xuelin and Liu Tingbo. After watching for a long time, Wang Xiufang quietly left the study, not to disturb them .

It was not until ten o'clock in the evening that Liu Tingbo thoroughly reviewed the paper, and the question and answer between the two also ended.

Although Pang Shaoan and Yao Jianzhong on the side couldn't keep up with their thoughts, their emotions were always in a state of excitement.

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

This kind of process of witnessing a world-class mathematical problem slowly revealing its true colors made everyone present excited.

Pang Shaoan looked at Liu Tingbo and said, "Xiao Liu, how about Xiao Lin'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%. I am 80% sure about this paper. Xiao Pang, how about this? Your paper has only been uploaded to arXiv not long ago. We will wait for it." After a period of time, when big men like Deligne and Faltins express their opinions one after another, I will arrange an academic report meeting for you at Jiangda University, which should attract the participation of top mathematicians from all over the world. During this time, You don't need to go to class for the time being, so you can prepare for the report with peace of mind, it's better to make the PPT more detailed."

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

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