The top student must be diligent.

Chapter 283 Riemann's Theorem and Shaw's Conjecture

【Theorem 7.3: Let f be an n-dimensional Siegel modular form, and X_f^(n) be the corresponding generalized modular curve. Then there exists a natural Galois representation: ρ_f: Gal(Q/Q)→ GL_n(Z_), such that for any prime number p, the characteristic polynomial of the Frobenius element Frob_p under ρ_f is equal to the Zeta function ζ(X_f^(n), T) of X_f^(n) at p...】

In Xiao Yi's office, he was writing down the final few steps of the proof of Artin's conjecture on a piece of draft paper.

"Well, this theorem successfully establishes the connection between the geometric properties of generalized modular curves and the arithmetic properties of Galois representations."

"With this result, I can finally transform the Artin conjecture into a question about Galois representations."

"So, the Artin conjecture expressed by Galois 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 functional equation; (2) there exists a modular form f such that the Galois representation ρ_E of E is isomorphic to ρ_f. 】

The corners of Xiao Yi's mouth curled up slightly, as if everything was under his control.

At this point, he successfully transformed the Artin conjecture into a problem in another form.

The vast majority of conjectures proved to be basically the same.

The final form that mathematicians need to prove is often very different from the original statement of the problem. However, by unraveling the various mathematical relationships, they can draw symbols representing equivalence relationships between this final form and the description of the conjecture itself.

As for the original description of the problem itself, it is more for the convenience of people's understanding.

For example, in other problems, such as the hailstone conjecture, its description seems very simple, but the final form of the proof is not what it seems to be, but a rather complicated formula.

Including Fermat's Last Theorem proved by Andrew Wiles, the final form is also completely different.

Therefore, with Xiao Yi now transforming Artin's conjecture, he only needs to prove that the Galois representation of every elliptic curve comes from a modular form.

"Then, Theorem 7.5, for any elliptic curve E, there exists a generalized modular curve X and a closed embedding i: E→ X such that i induces an isomorphism between Galois representations: ρ_Eρ_Xi_*."

This Theorem 7.5 is the last problem he needs to prove.

Similarly, there was no difficulty for him here. He just thought about it for a while, and then he completely completed his result.

"Then, from Theorem 7.3, we know that ρ_X comes from a Siegel modular form f, namely ρ_Xρ_f."

"Combining these two results, we have: ρ_Eρ_X i_*ρ_f i_*."

"This suggests that ρ_E also comes from a modular form, namely the 'pullback' of f."

"From Theorem 7.4, this means that L(s, E) is integral and satisfies the functional equation."

"To sum up, Artin's conjecture is valid."

【The proof is complete.】

After writing these last two words on the draft paper, Xiao Yi also smiled slightly.

After such a long time, he finally successfully solved Artin's conjecture.

In this way, he is one step closer to the Riemann hypothesis.

However, before that, he still needs to derive the result of Artin's conjecture based on his current results, that is, what the automorphic representation π that makes every finite-dimensional complex representation ρ equal to its L-function looks like.

Only after obtaining this formula could he begin to try to prove the Riemann hypothesis.

Soon, he successfully derived this new automorphic representation π.

"So, we got a functional equation."

[L(ρ_X, s)=ε(ρ_X, s) L(ρ_X^∨, ks)]

Xiao Yi began to observe this equation.

This is the most important result of Artin's conjecture.

It is this functional equation that makes the prediction of Artin's conjecture true: every finite-dimensional complex representation ρ:Gal(K/k)→GL(n, C) should correspond to an automorphic representation π, so that their L-functions are equal: L(s, ρ)=L(s, π).

With this result, it is even possible to study the functor conjecture.

Of course, Xiao Yi’s current research focus is not the functor conjecture.

Now, he wants to see how to connect this formula with the Riemann hypothesis.

Soon, he smiled slightly and the pen in his hand started moving again.

Now that it has come to this, and Atin's conjecture has been proven by him, the difficulty of the next step will not be too great for him.

Although he still has to deal with a rather complicated series of derivations, which may take a long time, it is certain that it is no longer difficult for him.

……

Time flies by again.

About a month passed.

During this month, the world remained the same and nothing changed.

Of course, for China, perhaps the most important thing is that several more nuclear fusion power stations have been built and put into operation.

As the initial period passes, it is indeed time for nuclear fusion power plants to be produced in large numbers.

Those core economic zones have basically used electricity from nuclear fusion, which has led to a grand bull market in the A-share market. This bull market has almost never stopped since the beginning of this year.

At the beginning, investors were still a little bit unconvinced. After all, thinking back to the last bull market, it was in 2024. That bull market was booming for a few days, and then it dealt a severe blow to the crazy investors.

However, after the bull market continued to rise for a week, they began to doubt, and after rising for half a month, they had to start investing in it tentatively.

After a month of continuous growth, investors finally fell into frenzy again.

And now, the stock market is still rising and has hardly stopped.

The market index has reached an unprecedented 9500, which is more than half of the 2007 points set in 6124. It is just around the corner to break through 10000.

This is mainly because nuclear fusion energy has reduced the operating costs of many companies, especially for those industrial companies. You must know that China has the most industrial companies and has always attached 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 the Prime Number Pioneer Project, they are still troubled by what direction they should choose. They even had a lot of disagreements at the beginning about this issue. These disagreements also made them decide to divide into several teams, each conducting research in a different direction, and then regularly exchanging results. Then, based on these results, they would 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 the critical line theorem continue to conduct research based on the critical line theorem, and have now successfully come up with a higher number than the 62.5% given by Xiao Yi at the beginning, 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, and there is still a distance as huge as an insurmountable chasm.

Even their current achievements can only be said to be developed on the basis of Xiao Yi's original achievements, and cannot be said to be achievements worthy of celebration.

As for mathematicians in other fields, they have also made certain breakthroughs to a greater or lesser extent. However, these breakthroughs cannot be considered significant. If they want to publish papers, they may not be good enough for Area 1. Perhaps relying on their fame, the editors may agree to publish their papers in Area 1 for their sake.

However, these math giants basically couldn't afford to lose face, so they created a website called the Prime Number Pioneer Project, and published their results directly on the website so that people could see where they had come from. Of course, because of this, the media kept saying that Xiao Yi didn't publish his research progress like them, and used this to mock Xiao Yi for not making any progress, or that he didn't publish his research results because he was worried about failure, and then kept repeating what others said.

Although Xiao Yi had basically never seen their taunts, even if he did, he didn't care.

In this way, time came to the early morning of July 7th.

……

【For any CM elliptic curve E, there exists a generalized modular curve X and an embedding i: E→ X, such that i induces an isomorphism between Hecke characters: λ_Eλ_X i_*, where λ_X is the Hecke character of X and i_* is a homomorphism between Galois groups induced by i. 】

[Therefore, substituting into Theorem 8.9 and Theorem 9.1, we can determine that all zeros of L(s, E) are located on the line Re(s) = 1/2. ]

【Therefore, all non-trivial zeros of ζ(s) are also on the straight line Re(s)=1/2. 】

【To sum up, the Riemann hypothesis is established. 】

The final proof is complete.

The pen in Xiao Yi's hand also stopped at the last period at this moment, and did not move for a long time, as if it had frozen the passage of time.

Riemann Hypothesis.

Riemann Hypothesis.

Riemann Hypothesis.

Riemann's theorem!

……

Any famous mathematical conjecture has a different history.

But no conjecture has such an extraordinary status as the Riemann hypothesis.

At this moment, history and the present intersect, and the problem that countless mathematicians in the past struggled for, paid for, and devoted their entire lives to, has come to an end under his pen.

Countless images seemed to flash through my mind.

In his office, Bernhard Riemann wrote a paper titled "On the Number of Primes Less Than a Given Value" to express his gratitude for the great honor of becoming a member of the Berlin Academy of Sciences. At that time, he probably did not expect that his paper of only eight pages would become the starting point of the Riemann hypothesis that almost all mathematicians are obsessed with.

He seemed to see that generations of mathematicians had been thinking, debating and exploring this problem.

Whether it was Euclid thousands of years ago, or Euler, Gauss, Hardy later...

Until now, mathematicians such as Selberg, Bombieri, Faltings, Deligne, etc.

These names have become the guiding lights to the answer to this question until now.

A young man named Xiao Yi finally lit the last lamp leading to the truth.

The pen in his hand finally stopped standing and was gently placed aside by him.

He stood up, stretched, walked to the window and opened the curtains.

The morning light shone in.

He didn't sleep at all last night.

But at least, this night was not spent in vain.

"However, the buff level has not been upgraded..."

In this regard, Xiao Yi could only shake his head helplessly. At this point, the buff level was not so easy to upgrade.

As for whether it is possible that his proof is wrong, that is absolutely 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 passed through his mind, and now he no longer wanted to think about it.

After stretching for a while, he yawned, then returned to his seat and reviewed the previous proof processes.

It was just a routine observation, but this time, he discovered an unexpected idea from the proof process.

"Automorphic representations... L-functions... and some kind of 'natural' object in geometry?"

"Is it true that... for every automorphic representation ρ, we can construct a number-theoretic L-function L(s, ρ) and a geometrically 'natural' object X(ρ)..."

He took out the pen again and wrote an equation on it, muttering, "Make them all satisfy this 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).

Putting the pen down again, he hugged his head.

If this is true, then it would be a big deal.

This means that he has achieved a major extension based on the Langlands Program.

The Langlands program foresees that every automorphic representation should correspond to an object in geometry 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.

Such a relationship is of great significance to mathematics.

It provides a new unified perspective, linking the three major branches of mathematics: algebra, geometry, and analysis, allowing mathematicians to transform problems in algebra into problems in geometry or analysis!

But is it really possible for this equation to hold true?
Xiao Yi didn’t know.

Because this is a brand new problem.

This requires another long process of proof.

But now Xiao Yi no longer wants to think too much.

For the next week, he just wanted to give himself a vacation.

It took so long to prove the Riemann hypothesis, can't we just enjoy ourselves?
of course can.

"As for this new problem, then..."

"Let's name it the Xiao conjecture."

Well, he proved the Artin conjecture and the Riemann hypothesis, and now he has given the mathematical community an even more powerful conjecture.

……

(End of this chapter)