Chapter 357 BSD Conjecture

Her research cannot be used by anyone who wants to.

The consciousness directly reaches the deductive space, which is a new immersive research. It is a state of deep research and learning, which allows her to concentrate on her studies and ignite more inspiration to help her deduce.

Lines of mathematical formulas gathered under Wu Tong's pen, and then burst out again, projecting the rolling lines around Wu Tong. Gradually, the streams merged into rivers, and the rivers rushed to the sea. The joyful breakthrough sounded in Wu Tong's ears, becoming the drums of victory.

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(4, 127, 131)= log(131)/ log(rad(4·127·131))= log(131)/ log(2·127·131)= 0.46820...

q(3, 125, 128)= log(128)/log(rad(3·125·128))= log(128)/log(30)= 1.426565...

For a triple (a, b, c) that satisfies a, b, c are mutually prime positive integers and a+b=c, c < rad(abc). In this case,

q(a, b, c) < 1, and the case of q>1 is rare. At this time, there are high-power small prime numbers among the factors of these numbers.

Three relatively prime positive integers a, b, c, and c=a+b.

The so-called coprime means that their greatest common divisor is 1. Therefore, 8 + 9 = 17 and 5 + 16 = 21 are a group of numbers that meet the conditions, but 6 + 9 = 15 is not.

Then extract the prime factors of abc. For example, the prime factors of 5, 16, and 21 are 5, 2, 3, and 7. The result of multiplying these prime factors is 210, which is much larger than the original three numbers.

Another example is 5, 27, 32, their prime factors are 5, 3, 2, and the product is 30, which is smaller than 32. But the second situation is extremely rare.

If a and b are both less than 100, we can find 3044 eligible abc combinations, of which only 7 meet the second condition. What the abc conjecture wants to prove is that there are only a finite number of abc combinations that meet the second condition.

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Mathematicians call the product of the prime factors of abc rad(abc). Let's use rigorous mathematical language to express it today. Substituting it into Theorem 1 and Theorem 2, we can be sure that for any ε>0, there are only a finite number of mutually prime positive integer triplets (a, b, c), c = a + b, such that: c > rad(abc)1+ε.

Thus, the ABC conjecture is proved.

After completing the last two words of proof, Wu Tong stared at the freshly written manuscript. It seemed that the numbers and symbols in her eyes condensed into an increasingly deep light. She did not stop her actions, but materialized a piece of draft paper and continued to write. The reflection in the sky switched to the content newly written by Wu Tong, which was a leap from number theory to algebraic geometry.

In the gaps between numbers, Wu Tong caught a glimpse of the connection between the arithmetic and analytic properties of Abelian clusters in the algebra he had been studying.

This extends to the BSD conjecture, one of the world's seven major problems, whose full name is the Bech and Swinnerton-Dyer conjecture.

Given an Abelian variety over a global domain, it is conjectured that the rank of its Modal group is equal to the zero of its L-function at 1, and the leading coefficient of the Taylor expansion of its L-function at 1 is exactly related by equation to the finite part size, free part volume, period of all primes, and sand group of the Modal group.

The first half is often called the weak BSD conjecture, which has been solved. The statement of the SD conjecture relies on Moder's theorem: rational points of an Abelian variety over a global field form a finitely generated commutative group. The exact part relies on the finiteness conjecture of sand groups.

For the case where the analytic rank is 0, Coates, Wiles, Kolyvagin, Rubin, Skinner, Urban and others proved the weak BSD conjecture, and the exact BSD conjecture holds for all cases except 2.

For the case of analytic rank 1, Gross, Zagier et al. proved the weak BSD conjecture, and the exact BSD conjecture holds except for 2 and derivations.

Now the only remaining problem is 2 and the derivation.

Wu Tong did not leave the state of enlightenment. The proof of the ABC conjecture once again added a lot of accumulated power to the Enlightenment Stele, which was about to reach the bottom.

Although this power is not enough to help the Enlightenment Stele advance further, it can be used to support Wu Tong's enlightenment state and maintain it for a certain period of time.

Wu Tong is very skilled in group theory, and is almost unmatched in number theory. Algebra, especially algebraic clusters, is the first time she has stepped into the field of research on a major topic, but it is not a strange area for her. After studying mathematics in depth until now, Wu Tong can confidently say that there is no area in mathematics that is too unfamiliar to her.

Algebra and geometry were originally the next key issues she planned to study, but she suddenly had the idea to work on the ABC conjecture. Based on her research on the proof of the ABC conjecture, she saw the inspiration to work on the BSD conjecture.

I believe no one would refuse the arrival of inspiration. Wu Tong naturally seized it decisively, followed the direction of inspiration, and started to deduce it urgently.

She started with Fourier series calculations, then extended them using continuous functions from functional analysis, and introduced Langlands program transformation group theory...

The so-called Abelian variety is a geometrically complete group scheme on a field, which must be projective, smooth, and commutative. An algebraic group is also a complete algebraic variety.

Because she already had a certain foundation, Wu Tong solved the Tate conjecture before Faltings and generalized the idea and calculation method of using Abelian clusters, and found inspiration for the way forward. Although these inspirations could not allow her to solve the BSD conjecture immediately, Wu Tong was sure that if she continued along this path, she could reach the end, and this was more important than anything else!

Of course, this is undoubtedly the most difficult part, but Wu Tong still wants to try to see if he can complete this problem.

It can be said that she did not start from scratch this time, but made a pleasant breakthrough in the area of ​​optimization and enhancement, which is her forte.

The energy consumption of the enlightenment state is very high and the time is limited. When Wu Tong found the direction of the enlightenment stele, he almost ran out of energy to continue. He was forced to automatically cut off the enlightenment state.

Of course, this did not have much impact on Wu Tong, as it was already July. Wu Tong spent two days to complete the preliminary paper on the ABC conjecture and saved it, and then continued to deduce the BSD conjecture. For her, the ABC conjecture that had been broken through was no longer the most important thing, and the new goal of pursuing the BSD conjecture was her full focus.

Having found a clear solution, Wu Tong did not have a strong desire for the Enlightenment Space. She always trusted what she had learned. It was not that Wu Tong was arrogant and felt that she did not need help. Everyone did not want to take shortcuts. But she understood that help also needed to be based on her sufficient foundation to achieve a good effect of one plus one greater than three, and could not rely on the Enlightenment state.

Wu Tong completed the final proof on the right path at a speed almost as fast as a mad run.

Substituting this into Modal's theorem and BSD conjecture, the proof is true.