It was already midnight, but Ye Qisheng, who was lying on the bed, still had his eyes wide open. He hadn't slept yet.
Thinking that he would no longer be alone and that he would have a lovely and charming girlfriend to accompany him, and that those regrets in his previous life would be officially made up for today, Ye Shengsheng tossed and turned, unable to sleep.
If he really couldn't fall asleep, he would get up and do the math questions.
It just so happened that there were still many problems with the proof of the BSD conjecture. Since he had nothing else to do other than helping Dali move his house tomorrow, he might as well fight it out tonight.
Ye Yisheng got up, put on his clothes, and sat in front of the desk. He spread out one draft paper after another, picked up a pen, and prepared to start.
With his emotions surging, his brain couldn't stop working. It worked rapidly over and over again, as if he had brainstormed many times. Some inspiration actually appeared.
Not to mention, with this flash of inspiration, Ye Shengsheng actually found a new exit.
The BSD conjecture was the full name of the Bayh and Svenaton-Dale conjecture.
Since the 1950s, mathematicians had discovered that the relationship between the two fields was closely related to number theory, geometry, and encryption. For example, Wiles proved Fermat's last theorem. One of the key steps was to use the relationship between the two fields (Taniyama-Shimura conjecture).
The BSD conjecture was related to the oval curve.
In the 1960s, Bayh and Svenaton-Dale of the University of Cambridge, England, used computers to calculate the rational number solutions of some exponential equations and found that these equations usually had infinite solutions.
However, how could he give infinite solutions?
The solution was to first classify. The typical mathematical method was congruence and using it to obtain the congruence class, which was the remainder after being divided by a number.
However, infinite numbers weren't necessary, so mathematicians chose prime numbers. So, to some extent, this problem was related to Riemann's hypothesis about the Zeta function.
After a long period of calculation and data collection, Bayh and Svenaton-Dale observed some patterns and patterns, thus proposing the BSD conjecture: Let E be an oval curve defined on the algebraic number field K, and E(K) be the set of rational points on E. It is known that E(K) is a finitely-generated abelian group, that is, L(s, E) is the Hasse-WeilL function of E, then the rank of E(K) is exactly equal to the order of the zero point of L(E, s) at s=1, and the first non-zero coefficient of the Taylor expansion of the latter can be accurately expressed by the algebraic properties of the curve.
The first half was usually called the weak BSD conjecture, while the second half was the extension of the class number formula of the cyclotomic domain of the BSD conjecture.
At present, mathematicians had only proved the weak BSD conjecture with rank=0 and 1. They were still unable to prove the strong BSD conjecture with Rank > 2.
Previously, Ye Bisheng had followed the route taken by Gros and Coates, trying to deduce the BSD conjecture of rank > 2 based on rank=0 and 1, but he found that he had gradually walked into a dead end.
In the past six months, he had not made any progress.
This time, Ye Qisheng planned to use a different method to prove the BSD conjecture.
Although there were already people in the mathematics community who tried to prove the BSD conjecture through the congruent remainder problem, this path was too difficult and was still in its infancy. There were not many results in the international mathematics community.
Before this, he didn't choose this path, but just now, he suddenly thought that if it was too difficult and there was no result, would it prove that this path wouldn't work? Did the current mainstream thinking mean that it was correct?
He could try to change his train of thought.
Ye Bisheng let his thoughts fly and then started to write.
The first was the proof of the congruent remainder problem, which proved that there were infinitely many congruent numbers with any specified positive number of prime factors.
Then, it was deduced that BSD holds true for such E_D: D is the product of a prime number of type 8k 5 and a number of prime numbers of type 8k 1, as long as the 4-fold map of the class group of BbbQ(sqrt{-D}) is simple.
……
Given a prime number p,(1)pequiv3(mod8): p is not congruent but 2p is congruent;(2)pequiv5(mod8)： p is the congruent remainder;(3)pequiv7(mod8): p and 2p are congruent.
(Weak BSD Conjecture) The BSD Conjecture is valid for E_D. In particular, r_D0 if and only if L(1, E_D)=0.
Assuming that the weak BSD conjecture is true, then (1) we can theoretically determine whether D is congruent;(2)Tunnell's theorem gives an algorithm to determine whether D is congruent in a limited number of steps;(3) It can be proved that when Dequiv5, 6, 7(mod8), r_D is an odd number, so such D is a congruent remainder.
……
According to the height theory of the Heegner point, the famous Gross-Zagier formula, it could be associated with L(1, E).
Based on Eichler and Shimura's work on the module oval curve and the recently proved Taniyama-Shimura conjecture (module theorem), L(s, E) could be extended to the entire complex plane and the corresponding Riemann conjecture was established.
……
Time flew by, and with the development of new ideas, the new proof became smoother and smoother.
After an unknown amount of time, Ye Qisheng finally got a rough result that proved this conjecture and let out a long sigh of relief.
It was thanks to Ye Bisheng's previous efforts that he could get the result so quickly. He had been on this road for a long time, which meant that many roads had been flattened. Now that he was going in another direction, he could cross many obstacles without thinking.
That was why there were results so quickly.
Although there were many steps that were omitted, and many calculations were recorded by Ye Shengsheng with his own special marks, it would take more effort to turn it into a thesis, but the final result was already out.
Just by holding this stack of draft paper, Ye Shengsheng could proudly announce that he had proved the BSD results.
Ye Qisheng let out a long sigh of relief. He stood up and stretched. He walked to the window and opened it, taking a deep breath.
The sky outside was still dark!
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