Chen Zhou, who returned to the dormitory, put his backpack still on the chair and reached out to open a page of scratch paper.

On the scratch paper, what was written, if that Notte-senpai was there, would have exclaimed.

This is because this is also the content of the scratch paper, which is the research content of "linear representation of the Artin L-function of the Galois group".

This is also the reason why Chen Zhou was a little hesitant when Professor Atin said that he would assign him a sub-topic for research.

Compared with Professor Artin's sub-project, it would be more interesting to study "the linear representation of the Atin L-function of the Galois group".

"This Noether senior sister is really looking for a topic......

"Perhaps, this is a coincidence, right?"

Chen Zhou picked up the scratch paper, read it back and forth, and shook his head helplessly.

If it weren't for the collision of the subject, Chen Zhou might have thought about it more.

But the topic I am interested in is actually invited to study together.

Then Chen Zhou had no choice but to refuse.

It's not that Chen Zhou doesn't think it's good to cooperate, it's just that he prefers to conduct research independently now.

This is especially true of this topic of interest.

Unless Yang Yiyi studies with him, everyone else, Chen Zhou will not be used to it.

As for this subject, it would have been taken first by Knott and her mentor.

Then Chen Zhou wouldn't care, on the contrary, he would congratulate this senior sister Nott.

After all, nothing is certain about the study of mathematics.

Gently putting down the scratch paper, Chen Zhou took his backpack away and sat on the chair.

Then find a new piece of scratch paper, pick up a pen, and start sorting out the research involved in the topic.

Of course, the priority of this topic is far lower than that of Gochai's research and rubber ball experiments.

Maybe after Gechai solves it, Chen Zhou will raise its priority.

As Noether said, the series of questions here is simply fascinating.

For each unary polynomial, we can define L functions, which are often called Dedekind ζ functions......

After writing this paragraph, Chen Zhou took a pen and drew a circle of the Dedejin ζ function, and habitually took the pen and clicked it a few times beside him.

Then, next to this circle, the Riemann ζ function is written.

The Riemann ζ function is a special case of a univariate one-time polynomial.

However, the Dedekind ζ function, like the Riemann ζ function, can be proved to satisfy the first two conditions of this function by means of elementary proofs.

Thinking of this, Chen Zhou's thinking spread.

A natural generalization of Dedekin's ζ function is to consider the case of multivariate polynomials.

And here, into the realm of algebraic geometry.

The zero point of a multivariate polynomial defines a geometric object, which is an algebraic cluster.

The study of algebraic clusters is called algebraic geometry.

Although algebraic geometry is an ancient discipline, it was only in the 20th century that it underwent a spectacular development.

In the early 20th century, the Italian school made great progress in the study of algebraic surfaces.

However, its lax foundation prompted Oskar Zariski and Andrei Wey to reconstruct the foundations of algebraic geometry as a whole.

Wey pointed out the striking connection between algebraic geometry and number theory and topology.

Later, Grothendieck, known as the emperor of algebraic geometry, further reconstructed the foundations of algebraic geometry with a more abstract approach in order to understand Wey's conjecture, and introduced a series of powerful tools.

In particular, his theory of upper cohomology eventually prompted his student, Professor Deligne, one of Chen Zhou's three reviewers, to prove the Wey conjecture in its entirety.

And for this, won the Fields Medal.

In fact, Grothendieck's theory of upper cohomology is rooted in algebraic topology.

Moreover, Grothendieck also constructs a series of theories of upper cohomology, which have very similar properties.

But it has its origins in a very different structure.

Grothendieck tried to find out what they had in common, and from this he came up with the Motive theory.

This theory is incomplete because it is based on a series of conjectures.

Motive's theory is also known as the standard conjecture by Grothendieck.

If the standard conjecture is proven, then the complete Motive theory is obtained.

It derives all the upper cohomologies while proving a series of surface-independent problems.

For example, the importance of the Hodge conjecture, one of the seven millennial puzzles, is that it can derive standard conjectures.

It has to be said that the proof of the standard conjecture is probably the most important thing in algebraic geometry.

However, the difficulty of proving the standard conjecture is top-notch.

If you really want to compare, from Chen Zhou's point of view, the difficulty of the standard guess has to be one level higher than that of Ge Guess.

Retracting his thoughts, Chen Zhou returned to the scratch paper in front of him, picked up the pen, and began to write:

[Regarding the MotivicL function and the self-defending L function, each MotivicL function is given by Motivic.]

For these functions, it is easy to verify that they satisfy the first condition of the Riemann ζ function, but the second condition does not yet prove the general situation.

A known example is the case of elliptic curves on rational numbers, which is a corollary of the proof of Fermat's great theorem (Taniyama-Shimura conjecture). 】

Chen Zhou remembers seeing in the literature that the complete picture of the Taniyama-Shimura conjecture was proved in 2001 by several students of Professor Wiles.

It has to be said that Professor Wiles' students all have buff bonuses when faced with the corollaries of Fermat's theorem.

Chen Zhou made a mark next to the Gushan-Shimura conjecture and continued to write:

[For almost all L-functions, the third condition, the Riemann hypothesis, is unknown.

The only exception to this is the case of Motive in finite domains, where the L-function satisfies the Riemann hypothesis, which is the Wey conjecture. 】

Chen Zhou wrote the word "Deligne" next to Wei Yi's conjecture.

Although it seems that there are many problems in this, it has been solved a lot.

But in reality, the unsolved problems are really huge.

For the problem of the special values of the MotivicL function, it is now generally believed that a generalization of Motive is needed.

It's a bigger, and more distant dream.

Mathematicians call it mixedmotive.

Its existence is capable of deriving a series of beautiful equations that generalize Euler's formula for Riemann ζ.

The famous Bellinson conjecture and the BSD conjecture, one of the seven millennial problems, are among the ones that can be derived.

In some ways, MixedMotive rivals or even surpasses the standard conjecture.

Because the current mathematical community doesn't know how to construct it.

Of course, although the current mathematical community cannot construct a mixedmotive, it is possible to construct a weakened deformation of it, that is, the derived category.

The Russian mathematician Vladimir Voevodsky was awarded the Fields Medal in 2002 for giving such a construction.

Thinking of this, Chen Zhou's heart is extremely longing, if this solves the standard conjecture, and then constructs the mixedmotive theory.

How many Fields Medals can you win?

Aren't you afraid that you will become the first mathematician to win a prize and become a billionaire?

But soon, Chen Zhou came to his senses.

I didn't even go to bed at night, so I didn't dream yet.

Being honest, down-to-earth, and doing your own research step by step is the most important thing.

Chen Zhou, who didn't think much about it, continued to sort out the research content involved in this topic on scratch paper.

[Each Motive can give a series of representations of Galois groups and Hodge structures in complex geometry, which completely determine the L function, so it is a more fundamental problem to consider them......]

In fact, Motive is a more essential existence than the L-function, but it is difficult to calculate it directly.

An alternative is to consider the different expressions of Motive.

From the existing examples, the class domain theory has solved the case of commutative Galois groups.

That is, a simple, but fundamental idea, is that the representation of a group is more fundamental than the group itself.

It is not the Galois group itself that needs to be considered, but its representation.

In this way, all commutative Galois groups are equivalent to one-dimensional Galois representations, and non-commutative Galois are equivalent to higher-dimensional representations.

Thinking of this, Chen Zhou frowned slightly, he turned on the computer and began to look for literature.

If we look at it this way, we must consider their internal symmetry.

Surprisingly, much of this symmetry comes from a completely different class of mathematical objects, known as self-contained forms.

The origins of the self-defending form can be traced back to the 19th century, and the mathematical god Poincaré was a pioneer in this direction.

Chen Zhou's hand quickly entered what he wanted to find on the computer.

Then download the literature one by one.

Chen Zhou, who originally planned to come back for a while and go to dinner, unknowingly fell into the world of mathematics.