Section 348

Today's team building was not in vain.

In fact, it was she who first proposed the team building. At that time, she saw Xiao Yi frowning because he had not solved the problem, so she suggested taking some time to relax. Maybe relaxing would help him come up with some ideas?

Now, has it been achieved?

The Ferris wheel gradually descended, and the scenery at the top disappeared.

Luo Mingya turned her head and saw the cabin where Qian Wanli and his family were.

Qian Wanli sat with his wife, while Qian Huiyin looked out the window with a happy face.

The breeze slipped into the cabin through the window sill, and Luo Mingya's hair rose with it.

Although she did not pursue the kind of romance with great ups and downs, she still pursued romance after all.

The romance at this time was just right.

...

Finally, the Ferris wheel turned a circle and gradually returned to the bottom.

When he was about to go out, Xiao Yi finally stopped writing and raised his head.

Noticing Luo Mingya's gaze at him, he smiled slightly.

"Is the problem solved?" Luo Mingya asked.

"It's not completely solved." Xiao Yi said: "But now we are not far from the final answer."

Luo Mingya raised her eyebrows: "So, the Riemann hypothesis is about to be solved?"

"That's not the case." Xiao Yi smiled and waved his hand: "What we have solved now is just a stage problem, and we are still a little short of truly solving the Riemann hypothesis."

"But now we can do this..." Xiao Yi's eyes showed confidence, "It's probably almost done."

At this time, the hatch was opened from the outside.

The staff signaled that they could go down.

Luo Mingya and Xiao Yi went down one after another.

At this time, Xiao Yi also put the paper and pen back into his pocket, and then smiled and said to her: "Okay, let's go, I won't continue to study the problem for the rest of the day, just have a good time."

"Okay." Luo Mingya smiled and nodded, and then they walked side by side to other places in the amusement park.

...

A day sometimes makes people feel that it passes quickly, and sometimes it makes people feel that it passes slowly.

When the past is fast, people will hope it can go slower, but when the past is slow, they will hope it can go faster.

For everyone in the Science Island Laboratory, today, they are all the former.

However, time will not wait for anyone. With the closure of the amusement park, all tourists have left, and the day is over. Everyone has reported that they have returned to the dormitory or home and reported safety.

One night passed.

The next day, the sun rose as usual.

Xiao Yi also slept in the lounge in his office last night, so after getting up and washing up, he could start research in the office.

He took out the notebook from the pocket of his clothes yesterday and read the notes left on it. He smiled slightly.

"OK, we can officially start today!"

He contacted Wang Hao and asked him to bring him a breakfast from the cafeteria. Then he sat at his desk, took out draft paper and pen, and started this most critical derivation.

Generalized modular curve, then we must first review the definition of modular curve.

【For a positive integer N, define the modular curve X(N) as the modular space of the complex upper half plane H, modulo the equivalence relation generated by the action of Γ(N). Here, Γ(N) is the principal congruence subgroup of the modular group SL(2, Z), defined as...】

"Next, define the generalized modular curve..."

【Let n be a positive integer, f be an n-dimensional Siegel modular form, i.e., a holomorphic function f:H_n→ C, where H_n is the upper half space of the n×n complex symmetric matrix τ=(τ_ij), which satisfies: for all γ∈Sp(2n, Z), f((Aτ+B)(Cτ+D)^(-1))=det(Cτ+D)^kf(τ), where (A B; C D) are elements in Sp(2n, Z) and k is the weight of f; at each cusp of H_n, f satisfies certain growth conditions. 】

"So, for such f, we can define the generalized modular curve X_f^(n) as the module space of the Siegel upper half space H_n, modulo the equivalence relation generated by the action of Γ^(n)(f)."

[Here, Γ^(n)(f) is a subgroup of the Siegel modular group Sp(2n, Z), which depends on f and is defined as: Γ^(n)(f)={γ∈Sp(2n, Z)|f(γ(τ))=f(τ), for all τ∈H_n}]

"At this point, X_f^(n) successfully parameterizes all n-dimensional Abelian clusters with modular properties described by f."

Writing this, Xiao Yi smiled slightly.

At this point, he has solved the most critical problem.

Although this expanded new geometric concept is named the generalized modular curve, it has become a brand new thing.

It further reflects an important idea in modern mathematics, that is, by introducing new mathematical structures, we can understand the essence of things at a higher level and discover hidden connections.

"Then, next, it's time to return to the extended L-function itself."

Xiao Yi only made a simple observation and easily noticed that for each n-dimensional generalized modular curve X_f^(n), there is a special type of n-dimensional Abelian cluster, whose extended L-function is closely related to the Zeta function of X_f^(n).

Of course, it is not enough to just observe, and a proof is needed.

But since we have come this far, there is no great difficulty.

After spending several draft papers, he finally gave a brand new theorem: suppose E is an n-dimensional Abelian cluster and f is an n-dimensional Siegel modular form; if the modular properties of E are described by f, then the extended L-function L(s, E,) of E is equal to the Zeta function ζ(X_f^(n), s) of the generalized modular curve X_f^(n).

"In this way, the most troublesome step has been successfully completed."

Then, the next thing to do is to move towards the final proof!

Arting conjecture, now there is no stopping him.

By associating each extended L-function with a generalized modular curve, he can use the geometric properties of generalized modular curves, such as dimension, Betti number, Hodge structure, etc., to characterize the characteristics of extended L-functions.

Finally, the answer was finally in front of him.

Half a month later.

…

Chapter 283 Riemann Theorem and Xiao'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 last 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 Artin's conjecture into a problem about Galois representations."

"Then, Artin's conjecture under this 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 functional equation; (2) There exists a modular form f such that the 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's conjecture into a problem in another form.

Most of the proofs of conjectures are basically the same.

The final form that mathematicians need to prove is often very different from the original problem statement, but by unraveling the various mathematical relationships, it is possible to draw a symbol representing the equivalence relationship between this final form and the description of the conjecture itself.

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

For example, other problems, such as the hail conjecture, its description looks very simple, but the final form of the proof is not the original form, but a rather complicated formula.

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

Therefore, after Xiao Yi transformed the Artin conjecture, he only needs to prove that the Galois representation of each 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.

Again, it did not cause him any difficulty 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, 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, that is, the "pullback" of f."

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

"In summary, Artin's conjecture is true."

[Proof completed. 】

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

After such a long time, he finally solved the Artin conjecture.

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

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

Only after getting this formula can he start to try to prove the Riemann hypothesis.

Soon, he successfully derived this new automorphic representation π.

"So, we get a function equation."

[L(ρ_X, s) = ε(ρ_X, s) L(ρ_X^∨, k-s)]