Chapter 459

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

The question seems simple, and it is no problem for the average undergraduate student to understand.

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

Even at the time when Taniyama Shimura's conjecture was first proposed, it is not an exaggeration to say that the process of proof can be described as difficult.

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

However, in recent years, as fewer mathematicians have devoted their energies to Taniyama Shimura's conjecture, the path to explore this conjecture has become dark again.

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

Generations of mathematicians, one after another, are running hard, passing the baton in their hands.

I don't know the end, I don't know the direction, the people who travel with me keep falling, and new ones keep running.

And now, the baton of Taniyama Shimura's conjecture has been passed to Cheng Nuo.

There are no more companions around.

Ahead, there is no light in sight of the lost.

Cheng Nuo could only follow the path that his predecessors had walked, groping his way forward, looking for the light that broke through the darkness, and trying to rush to the end of the race.

............

In order to facilitate communication, Cheng Nuo and the other two professors in the group directly placed their offices in an office in the Clay Institute of Mathematics.

The general direction of the certification work is controlled by Cheng Nuo.

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

For the proof of Taniyama Shimura's conjecture, Cheng Nuo, like most of his predecessors, regarded Fermat's theorem as his breakthrough.

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

In other words, after a certain amount of derivation, Fermat's theorem can be proved.

However, the existence of Fermat's theorem does not prove the correctness of Taniyama's conjecture.

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

However, Fermat's theorem is still of high reference significance for the proof of Taniyama-Shimura's conjecture.

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

Staying alone in the office, Cheng Nuo, who had been in one action for more than an hour, finally felt that he had caught that trace of inspiration, took a pen, and wrote down the inspiration on scratch paper.

According to Fermat's theorem n=4, the object of study is defined as an 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...... ......"

“...... In the next step, the modulo group Γ(1):=SL2(Ζ) is used to act on the complex upper half plane H={z∈C|Im(z)>0}. ”

“...... In the third step, assuming that E:y²=ax³+by²+cx+d is an elliptic curve in the field Q of a rational number, it needs to be considered as a "reduction" of the coefficient modulo prime. Moreover, an 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, it is concluded (1): isomorphic elliptic curves should be regarded as equivalent!"

............

Like Cheng Nuo and his proof team, the remaining seven proof teams started their research work non-stop under the leadership of their respective team leaders as soon as they got the task.

After all, not only did they have to race against the three-year research cycle, but they also had to compete with the rest of the group.

The eight research groups are working at the same time, and the allocation of researchers is also proportional to the difficulty of the conjecture. The starting line is almost the same.

No mathematician is willing to be left behind.

Therefore, this cleaning activity has a hint of racing.

"Geometricized conjectures" proof panel.

Professor Black, one of the oldest mathematicians in the field of geometry, was appointed to the position of group leader.

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

In terms of difficulty, the research difficulty of the "geometric conjecture" and the "Taniyama Shimura" conjecture is comparable.

But one difference is that Blake's two mathematicians are more than a little bit stronger than Cheng Nuo's.

Speaking of a single point, two of the three members of the Blake group have won the Wiblen Award, while Cheng Nuo is the only one on Cheng Nuo's side.

Therefore, from beginning to end, Blake did not treat the "Taniyama Shimura Conjecture" research group next door as a face to face opponents.

However, this idea changed radically at the Clay Institute of Mathematics' regular three-month progress report meeting on the purge.

............

The time is January 2024.

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

For three months, Cheng Nuo refused almost all entertainment activities, and devoted all his energy to the conjecture of Gushan Shicun like an ascetic.

It's tiring, but it's remarkable!

And today, it's the time for the regular progress report in March.

When Cheng Nuo arrived at the synagogue, most of the mathematicians were already in place.

The so-called monthly routine progress report is a brief overview of the research during this period, and by the way, a general plan for the future.

According to the difficulty of conjecture, Cheng Nuo was arranged to report in the third.

The first Hodge guess was that the mathematician, who looked to be in his fifties, spoke on it for more than ten minutes, but it could be summed up in four words: no clue!

That's right, the Hodge conjecture has not been solved for a hundred years, and it is also one of the seven mathematical conjectures, and everyone has little expectation that it will be able to figure it out in three months.

The second person to go up was Professor Black.

Compared with Hodge's conjecture proving that the group had no clue but talked a lot about it, Professor Blake talked about it much more pragmatically.

After three months of research, they have a preliminary idea of the process of proving the "geometrical" conjecture, and they are making steady progress. It is expected that this conjecture will be resolved within a 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 from the stage, Professor Blake was greeted with a round of applause.

The corners of Blake's mouth went up, and he sat back in his seat with a leisurely expression.

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

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

Recently, although they have been working together at the Clay Institute of Mathematics, Cheng Nuo's research group has been living in seclusion, and it is difficult to hear anything about them.

For this group, which is obviously not favored by everyone, in fact, they are also curious about how far they can go in three months.

I just hope it's not the clueless kind of Hodge conjecture group.

Cheng Nuo smiled slightly, without any nonsense, and went straight to the point, "As we all know, Taniyama Shimura's conjecture and Fermat's theorem are inseparable, and the self-defense form is patterned, and a simple elliptic curve can be constructed using Fermat's theorem, and the relationship between polynomial mappings is explained ......"

“...... Then, for curves on complex number fields, we deduce in addition to simple isomorphism groups. Speaking of this, Cheng Nuo paused and showed a mysterious smile, "Then, we found something interesting......"