Chapter 459: Something interesting

Chapter 459
To put it simply, Gushan Zhicun’s conjecture means that all elliptic curves on the rational number field can be modeled.
The problem seems simple, and it is no problem for ordinary undergraduates to understand.
But this conjecture has puzzled mathematicians all over the world for more than 50 years.
Even at the time when Gushan Zhicun’s conjecture was just proposed, the proof process can be described as difficult to describe as an exaggeration.
It was not until 1993 that Wiles announced the proof of Fermat's Last Theorem that the proof of Taniyama Shimura's conjecture took a big step forward.
However, in recent years, as the number of mathematicians devoted to Gushan Zhicun’s conjecture has gradually decreased, the path of exploration of the conjecture has once again become dark.
In fact, the proof of every mathematical conjecture is like a long-distance race.
Generations, a mathematician, ran hard, passing the baton in his hand.
I don't know the end point or the direction. Those who are with me keep falling down, and new ones keep joining.
And now, the baton of Gushan Zhicun's conjecture has been passed to Cheng Nuo.
There are no more people around me.
Ahead, there is no trace of shining lost.
Cheng Nuo could only follow the path taken by his predecessors, groping forward, looking for the light that broke through the darkness, and trying to reach the end of the game.
…………
In order to facilitate communication, the other two professors under Cheng Nuohe's group directly put their office in an office in the Cray Institute of Mathematics.
The general direction of the proof work is controlled by Cheng Nuo.
The two mathematics professors in Denmark and Belgium filled in the details.
For the proof of Gushan Zhicun’s conjecture, Cheng Nuo, like most of his predecessors, took Fermat's Last Theorem as his breakthrough.
In the language of mathematics, Fermat's Last Theorem is a necessary and insufficient condition for Gushan Zhicun’s conjecture.
In other words, after a certain derivation of Taniyama Shimura's theorem, Fermat's last theorem can be proved.
However, the existence of Fermat's Last Theorem cannot prove the correctness of Gushan Zhicun's conjecture.
In a certain sense, Fermat's Last Theorem can only show that Gushan Zhicun’s conjecture holds on semi-stable elliptic curves.
However, Fermat's Last Theorem still has a high reference significance for the proof of Gushan Zhicun's conjecture.
Cheng Nuo also decided to start from this direction and try to prove the method.
Staying in the office alone, Cheng Nuo, who had been in a movement for more than an hour, finally felt that he had caught the trace of inspiration. He took the pen and wrote down the inspiration on the draft paper.
"According to Fermat's theorem n=4, the research object is defined as an elliptic curve E: y^2=x^3-x. Let β be a prime number, the number of solutions of this equation in the finite field Ft is β=1 , 3, 5...respectively..."
"...Next, use the modular group Γ(1):=SL2(Ζ) to act on the complex upper half-plane H={z∈C
Im(z)>0} through fractional linear transformation."
"... In the third step, assuming that E: y2=ax3+by2+cx+d is an elliptic curve in the rational number domain Q, it needs to be considered "reduced" in the coefficient modulus prime. Moreover, the isomorphic elliptic curve may be Give a completely different "reduction": consider y2=27x3-3x and y2=x3-x, the former is not an elliptic curve on F3, and the latter is an elliptic curve on F3. Therefore, the conclusion ①: Isomorphism The elliptic curves should be regarded as equivalent!"
…………
Like Cheng Nuo's certification team, the remaining seven certification teams started their research non-stop under the leadership of their respective team leaders as soon as they got the task.
After all, they are not only racing against the three-year research cycle this time, but also racing against the remaining groups.
The eight topic groups are open at the same time, and the distribution of researchers is proportional to the difficulty of guessing. The starting line for everyone is almost the same.
None of the mathematicians is willing to stay behind.
So this cleaning activity has a hint of racing.
"Geometric Conjecture" Proof Team.
Professor Black, as one of the veteran mathematicians in the field of geometry, was appointed as the team leader.
Like the "Taniyama Shimura Conjecture" proof group, their group has only three members.
In terms of difficulty, the "geometric conjecture" and the "Tanishan Zhicun" conjecture are equally difficult to study.
But one difference is that the two mathematicians under Black are more than a little bit stronger than the two mathematicians under Cheng Nuo.
To put it simply, two of the three members of the Black team have won the Veblen Prize, while Cheng Nuo is the only one on Cheng Nuo.
Therefore, from the beginning to the end, Black did not regard the "Taniyama Shimura Conjecture" research team next door as a person who could face his opponent.
But this kind of thinking was completely changed at the regular progress report meeting held every three months by the Clay Institute of Mathematics for this cleaning activity.
…………
Time enters January 2024.
The proof of Gushan Shimura’s conjecture has been underway for three months.
In the past three months, Cheng Nuo has almost rejected all entertainment activities and devoted all his energy to Gushan Zhicun’s conjecture like an ascetic monk.
Although very tired, survival is very remarkable!
Today, it is the time for the routine progress report once in March.
When Cheng Nuo came to the hall, most mathematicians were already in place.
The so-called routine progress report once in March is to give a brief overview of the subject research during this period, and by the way, talk about the general plan for the future.
According to the guessed difficulty, Cheng Nuo was arranged in the third report.
The first Hodge's conjecture was that there seemed to be a mathematician who was more than 50 years old at that age and talked about it for more than ten minutes, but in a nutshell, it was just four words: no clue!
That's right, Hodge's conjecture has not been solved for a hundred years, and it is also listed as one of the seven major mathematical conjectures. Everyone has no expectation that it can be sorted out in three months.
The second person who went up was Professor Black.
Compared with Hodge’s conjecture, which proved that the group was clueless~EbookFREE.me~ but talked about a lot, Professor Black's content was much more pragmatic.
After three months of research, they have had preliminary ideas for the proof process of the "geometric" conjecture, and they are moving forward steadily. It is expected that this conjecture can be resolved within one year.
In addition, Professor Black also gave a brief description of the specific reasoning content, which was unanimously recognized by everyone.
When he stepped down, Professor Black greeted his applause.
The corners of Black's mouth raised, and he sat back in his seat leisurely.
At this time, Cheng Nuo adjusted his clothes, got up and walked to the stage.
In an instant, Cheng Nuo attracted everyone's attention.
In the recent period, although they are working together at the Cray Institute of Mathematics, Cheng Nuo and his group have been very simple and it is difficult to hear anything about them.
Regarding this group that is obviously not favored by everyone, they are actually also curious about how much they can achieve in three months.
I just hope it's not the cluelessness of Hodge's conjecture group.
Cheng Nuo smiled slightly, without any nonsense, and went straight to the topic, "As we all know, Taniyama Shimura’s conjecture is inextricably related to Fermat’s Last Theorem. To model the automorphic form, you can use Fermat’s theorem to construct simple elliptic curves and polynomial mappings. Description of relationship..."
"... Then, for the curve on the complex number field, we deduce in addition to simple isomorphism groups." At this point, Cheng Nuo paused, showing a mysterious smile, "Then, we found an interesting thing... "
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