[Theorem 7.3: Suppose f is an n-dimensional Siegel modular form, and X_f^(n) is the corresponding Generalized Modular Curve. Then there is a natural Galois representation:ρ_f: Gal(Q/Q)→GL_n(Z_), such that for any prime number p, the characteristic polynomial of Frobenius element Frob_p under ρ_f is equal to Zeta function ζ(X_f^(n), T) of X_f^(n) at p...]
In the office of Xiao Yi, he is writing the last few steps of the proof of Artin Conjecture on the scratch paper.
"Well, this theorem successfully establishes the connection between the geometric properties of Generalized Modular Curve and the arithmetic properties of Galois representation."
"With this result, I can finally convert Artin Conjecture into a question about Galois representation."
"Then, Artin Conjecture under Galois representation is..."
[Theorem 7.4: Let E be an elliptic curve, and L(s, E) be its Hasse-Weil L-function. Then the following two conditions are equivalent: (1) L(s, E) is a holomorphic function on the entire complex plane and satisfies a function equation; (2) There is a modular form f such that Galois representation ρ_E of E is isomorphic to ρ_f.]
Xiao Yi's mouth curled up slightly, as if everything was under his control.
At this point, he successfully transformed Artin Conjecture into another form of problem.
Most conjectures prove that this is basically the case.
The final form that mathematicians need to prove is often very different from the original problem statement. However, by peeling off the various mathematical relationships, symbols representing the equivalence relationship can be drawn between the final form and the description of the conjecture itself.
As for the original description of the problem itself, it is mostly to facilitate people's understanding.
For example, various other problems, such as Hailstone Conjecture, its description seems very simple, but the final proved form is not what it is, but a rather complicated formula.
Including Fermat's Last Theorem as demonstrated by Andrew Wiles, the final form is also completely different.
Therefore, with Xiao Yi now transforming Artin Conjecture, he only needs to prove that Galois representation of each elliptic curve comes from a modular form.
"Then, Theorem 7.5, for any elliptic curve E, there is a Generalized Modular Curve X and a closed embedding i: E→X, such that i induces the isomorphism between Galois representation: ρ_Eρ_Xi_*."
This Theorem 7.5 is the last problem he needs to prove.
Similarly, it did not cause him any difficulties here. He just thought about it for a moment, and then he completely completed his result.
"Then, from Theorem 7.3, we know that ρ_X comes from a Siegel modular form f, that is, ρ_Xρ_f."
"Combining these two results, we have: ρ_Eρ_X i_ρ_f i_."
"This shows that ρ_E also comes from a modular form, the "pull back" of f."
"By Theorem 7.4, this means that L(s, E) is integral and satisfies the function equation."
"To sum up, Artin Conjecture is established."
[Certificate completed.]
After writing these last two words on the scratch paper, Xiao Yi also smiled slightly.
After a long time, he finally successfully cracked this Artin Conjecture.
In this way, he is really one step closer to Riemann Hypothesis.
However, before that, he still needs to derive from the results of Artin Conjecture based on his current results, what does the automorphic representation π that makes each finite-dimensional complex representation ρ equal to their L-function look like.
Only after getting this formula can he start trying to prove Riemann Hypothesis.
Soon, he successfully derived this new automorphic representation π.
"So, we have an equation of function."
[L(ρ_X, s)=ε(ρ_X, s) L(ρ_X^∨, k-s)]
Xiao Yi Start observing this equation.
This is the most important result of Artin Conjecture.
It is this function equation that makes what Artin Conjecture predicts: Each finite-dimensional complex representation ρ: Gal(K/k)→GL(n, C) should correspond to an automorphic representation π, making their L-functions equal: L(s, ρ) = L(s, π), which is established.
Through this result, it is even possible to study the functority conjecture.
Of course, the current research focus of Xiao Yi is not the functority conjecture.
Now, what he wants to see is how to connect this formula with Riemann Hypothesis.
Soon, he smiled slightly and started moving the pen in his hand again.
Now that he has reached this point and Artin Conjecture has been proven by him, the next difficulty can no longer make it difficult for him.
Although he still has to deal with a rather complicated series of derivation, which may take a long time, what is certain is that it is no longer difficult for him.
Time flew by again.
About a month has passed.
In this month, the world is still as it should be, without any changes.
Of course, for China, what is probably more important is that several more nuclear fusion power plants have been built and have been put into operation.
As the initial period of time passes, it is indeed time for the nuclear fusion power station to make dumplings.
Those core economic zones have basically used electricity from nuclear fusion, which has led to A-shares ushering in a grand bull market. This bull market has not stopped almost since the beginning of this year.
At the beginning, investors were still a little unconvinced. After all, recalling the last bull market in 2024, the bull market that lasted for several days was a severe blow to the crazy investors.
However, when this bull market continued to rise for a week, they began to be doubtful, and after it continued to rise for half a month, they had to tentatively start investing in it.
After a month of continuous gains, investors finally fell into madness again.
And up to now, the stock market is still rising and has almost never stopped.
The market has reached an unprecedented level of 9,500, which has already exceeded the 6,124 points set in 2007 by more than one-half, and is just around the corner from breaking through 10,000.
This is mainly because nuclear fusion energy has reduced the operating costs of many companies, especially for industrial companies. What you need to know is that China has the largest number of industrial companies. China has always attached great importance to the development of the industrial economy. It has never vigorously developed the financial economy like those Western countries, while neglecting the development of the industrial economy, especially in industry.
China's industry can be said to be the biggest beneficiary after the realization of nuclear fusion.
As for the mathematicians of Prime Pioneer Program, they still have a headache about which direction they should choose. Regarding this issue, they even had a lot of disagreements at the beginning, and these disagreements also allowed them to make a decision. They first divided into several teams, each conducting research from different directions, and then regularly communicated the results, and then based on these results, judge which direction has more opportunities.
Based on this approach, they have achieved some results.
For example, those mathematicians who want to continue to develop Critical Line Theorem continue to study based on Critical Line Theorem, and now they have successfully come up with a higher number than the 62.5% originally given by Xiao Yi, 66.67%, which is almost two-thirds.
However, by the same token, two-thirds seems to be very close to the final answer, but in fact it is not. There is still a gap like a chasm in the distance.
Even their current results can only be said to have continued to be developed based on the original results of Xiao Yi. It cannot be said that they have achieved results that are very worthy of celebration.
As for mathematicians in other fields, they have more or less achieved certain breakthroughs. However, these breakthroughs cannot be called much. If they want to publish papers, I am afraid they may not be enough to be included in the first section. Perhaps because of their reputation as a group of big guys, the editors may agree to publish their papers in the first section for their sake.
However, for these giants in mathematics, they basically couldn't afford to lose face, so they finally created a website called Prime Pioneer Program, and then published their results directly on this website so that people can see where they have done it.
Of course, it is precisely because of this that the media have been saying all day long that Xiao Yi did not publish his research progress like them, in order to ridicule Xiao Yi for not making any progress, or that he did not publish his research results because he was worried about failure, and then continued to make mistakes.
Although Xiao Yi has basically never seen these taunts from them, and even if he has seen them, he has not cared about them.
In this way, the time came to the early morning of July 15th.
[For any CM elliptic curve E, there is a generalized modular curve X and an embedding i: E→]
[Therefore, substituting Theorem 8.9 and Theorem 9.1, we can determine that all zero points of L(s, E) are located on the straight line Re(s)=1/2.]
[So, all non-trivial zero points of ζ(s) are also located on the straight line Re(s)=1/2.]
[To sum up, Riemann Hypothesis is established.]
The final proof is over.
The pen in Xiao Yi's hand also stopped at the last full stop at this moment, and did not leave for a long time, as if the passage of time had condensed.
Riemann Hypothesis.
Riemann Hypothesis.
Riemann Hypothesis.
Riemann's theorem!
Any famous mathematical conjecture has a different history.
But there is no guess that Riemann Hypothesis will have such an extraordinary status.
At this moment, history and the present intersect, and the problem that countless mathematicians in the past have struggled for, devoted their lives to, and devoted their lives to, just like this, has come to an end in his writing.
Countless images seemed to pass through my mind.
In his office, in order to express his return for the noble honor of becoming Berlin's Academician of the Chinese Academy of Sciences, Bornhard Riemann wrote a letter called "On the Number of Prime Numbers Less than a Given Value." At that time, he probably did not expect that his paper, which was only eight pages long, would become the starting point of Riemann Hypothesis, which almost all mathematicians are obsessed with.
He seemed to also see that generations of mathematicians were thinking, debating and exploring this issue one after another.
Whether it is Euclid thousands of years ago, or Euler, Gauss, Hardy later...
Until now, mathematicians such as Selberg, Bombieri, Faltings, Deligne and so on.
These names have become the guiding light leading to the answer to this question until now.
A young man named Xiao Yi finally illuminated the last light leading to the truth.
The pen in his hand finally stopped standing and was gently put aside by him.
He stood up, stretched, walked to the window, and opened the curtains.
The early morning light shines in.
Last night, he didn't sleep all night.
But finally, this night was not in vain.
"However, the buff level has not been upgraded..."
In response to this, Xiao Yi could only shake his head helplessly. At this point, the buff level was not that easy to upgrade.
As for whether it is possible that his proof is wrong, it is completely impossible. He has full confidence in his proof.
So, he probably needs to solve another problem of similar level before he can upgrade the buff?
The question quickly ran through his mind, and now he didn't want to think about it anymore.
After stretching for a while, he yawned, then returned to his seat and reviewed the various previous proof processes.
Originally it was just a routine observation, but this time, he discovered an unexpected idea from these proof processes.
"Automorphic representation... L-function... and some kind of 'natural' object in geometry?"
"Isn't it... that for every automorphic representation ρ, we can construct a number-theoretic L-function L(s, ρ), and a geometric "natural" object X(ρ)..."
He took out the pen again, then wrote an equation on it, and murmured, "So that they all satisfy such a relationship."
[L(s,ρ)= L(s, X(ρ))]
That is, the L-function of ρ is equal to some "natural" L-function of X(ρ).
He put down his pen again and hugged his head.
If this is true, then it is extraordinary.
This means that he has achieved a great promotion based on Langlands Program.
Langlands Program foresees that every automorphic representation should correspond to a geometric object and an L-function in number theory.
This relationship further predicts that there should be a direct equality relationship between this geometric object and the L-function.
And this relationship is of extremely important significance to mathematics.
It provides a new unified perspective that connects the three major branches of mathematics, algebra, geometry, and analysis, allowing mathematicians to transform problems in algebra into problems in geometry, or problems in analysis!
But is this equation really possible?
Xiao Yi I don't know.
Because this is a brand new problem.
It also requires a long process of proof.
But now Xiao Yi doesn't want to think too much anymore.
For the next week, he just wanted to take a break for himself.
Riemann Hypothesis has proven it for so long, can't we just enjoy it?
sure.
"As for this new question, then..."
"Name it Xiao's Conjecture."
Well, he proved Artin Conjecture and Riemann Hypothesis, and now he will give the mathematical community an even more powerful conjecture.