Omnipotent Data

Chapter 459

To put it simply, Taniyama Shimura's conjecture means that all elliptic curves above the rational number field can be modeled.

The problem seems to be very simple, and it has no problem for ordinary undergraduates to understand.

But this conjecture has puzzled mathematicians all over the world for more than 50 years.

Even during the period when Taniyama Shimura's conjecture was first proposed, the proof process can be described as difficult.

It wasn't until 1993, when Wiles announced the proof of Fermat's Last Theorem, that the proof of Taniyama Shimura's conjecture took a big step forward.

But in recent years, as the number of mathematicians who have devoted their energy to Taniyama Shimura's conjecture has gradually decreased, the road to explore this conjecture has become dark again.

In fact, the proof of every mathematical conjecture is like a long-distance race.

From generation to generation, a mathematician is running hard, passing on the baton in his hands.

I don't know the end point, and I don't know the direction. People who are traveling with me keep falling down, and new ones keep running and joining.

And now, the baton that Taniyama Shimura conjectured had passed to Cheng Nuo.

There are not many companions around.

Ahead, there is no way to see the slightest light.

Cheng Nuo could only follow the path walked by his predecessors, groping forward, looking for the light that suddenly broke through the darkness, and trying to reach the finish line of the race.

…………

For the convenience of communication, Cheng Nuo and the other two professors in the group directly put their office in an office in the Clay Institute of Mathematics.

The general direction of the proof work is under the control of Cheng Nuo.

Two mathematics professors from Denmark and Belgium filled in the details.

Cheng Nuo, like most of his predecessors, took Fermat's last theorem as his breakthrough point for the proof of the Gushan Shimura conjecture.

In the language of mathematics, Fermat's last theorem is a necessary and insufficient condition for Taniyama Shimura's conjecture.

That is to say, Taniyama Shimura's Theorem can prove Fermat's Last Theorem after a certain derivation.

However, the existence of Fermat's last theorem cannot prove the correctness of Taniyama Shimura's conjecture.

In a certain sense, Fermat's last theorem can only show that Taniyama Shimura's conjecture holds true on semi-stable elliptic curves.

However, Fermat's last theorem still has a high reference value for the proof of Taniyama Shimura's conjecture.

Cheng Nuo also decided to start in this direction and try to prove the method.

Cheng Nuo, who had been alone in the office for more than an hour, finally felt that he had caught the sliver of inspiration. He took a pen and scribbled down the inspiration on the draft paper.

"According to the case of Fermat's theorem n=4, the research object is defined as the elliptic curve E:y^2=x^3-x. Let β be a prime number, and the number of solutions of this equation in the finite field Ft is β=1 , 3, 5... are respectively..."

"...The next step is to use the module group Γ(1):=SL2(Ζ) to act on the complex upper half-plane H={z∈C|Im(z)＞0} through fractional linear transformation."

"...The third step, assuming that E:y2=ax3+by2+cx+d is an elliptic curve on the rational number field Q, you need to consider its "reduction" in the coefficient modulus prime number. And, the isomorphic elliptic curve may Give a completely different "reduction": consider y2=27x3-3x and y2=x3-x, the former is not an elliptic curve on F3, but the latter is an elliptic curve on F3. Therefore, the conclusion ①: isomorphism elliptic curves should be considered equivalent!"

…………

Like Cheng Nuo's proof team, the other seven proof teams, under the leadership of their respective team leaders, began their research work non-stop as soon as they got the task.

After all, this time they not only have to race against the three-year research cycle, but also compete with the rest of the teams.

The eight subject groups started the topic at the same time, and the allocation of researchers was also proportional to the difficulty of guessing. Everyone's starting line is almost the same.

No mathematician wants to be left behind.

So this purge,

It has a hint of racing.

"Geometry Conjecture" proof group.

As one of the veteran mathematicians in the field of geometry, Professor Black was appointed as the group leader.

Like the "Taniyama Shimura Conjecture" proof team, there are only three members of their team.

In terms of difficulty, the "geometry conjecture" and the "Taniyama Shimura" conjecture are equally difficult to study.

But one difference was that the two mathematicians under Black were more than a little better than the two mathematicians under Cheng Nuo.

To put it simply, two of the three members of Black's team had won the Veblen Prize, while Cheng Nuo was the only one on Cheng Nuo's side.

Therefore, from the beginning to the end, Black did not regard the "Taniyama Shimura Conjecture" research team next door as an opponent that could face up to it.

But this idea changed completely during the routine progress report meeting held every three months by the Clay Mathematics Institute for this purge.

…………

Time to enter January 2024.

The proof of Taniyama Shimura's conjecture has been going on for three months.

For three months, Cheng Nuo had rejected almost all entertainment activities, devoting all his energy to Gu Shan Zhimura's conjecture like an ascetic.

Although very tiring, but the survival is very remarkable!

And today is the time for the routine progress report once in March.

When Cheng Nuo came to the auditorium, most of the mathematicians were already in place.

The so-called routine progress report once every three months 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 difficulty of guessing, Cheng Nuo was arranged to report in the third place.

For the first Hodge conjecture, the mathematician who seemed to be in his fifties talked about it for more than ten minutes, but it can be summed up in 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 figured out in three months.

The second person to go up is Professor Black.

Compared with Hodge's conjecture, which proved that the group had no clue, but gave a lot of hype, Professor Black's narration was much more pragmatic.

After three months of research, they have a preliminary idea of ​​the proof process of the "geometry" conjecture, and they are making steady progress. The conjecture is expected to be resolved within a year.

Moreover, Professor Black also gave a brief account of the specific reasoning content, which was unanimously approved by everyone.

When he stepped off the stage, Professor Black received a burst of applause.

The corners of Black's mouth raised, and he sat back in his seat leisurely.

At this time, Cheng Nuo straightened his clothes, got up and walked to the stage.

In an instant, Cheng Nuo attracted everyone's attention.

Recently, although they worked together at the Clay Institute of Mathematics, Cheng Nuo and his research group have been reclusive, and it is difficult to hear any news about them.

For this group that is obviously not favored by everyone, they are actually curious about how far they can do in three months.

I just hope it's not as clueless as the Hodge conjecture team.

Cheng Nuo smiled slightly, without any nonsense, and went straight to the point, "As we all know, Gushan Shimura's conjecture is inseparable from Fermat's last theorem. Automorphic forms can be modeled, and Fermat's theorem can be used to construct simple elliptic curves and polynomial maps. The description of the relationship..."

"...Then, for the curves in the field of complex numbers, we derived simple isomorphism groups." Speaking of this, Cheng Nuo paused for a moment, showing a mysterious smile, "Then, we discovered an interesting thing... "

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