Top Student at Their Peak-Chapter 189 - 101: The Crazy Math Novice_6

The content of Professor Robert’s research is essentially about accurately estimating the upper bound of the number of rational points for a given type of algebraic curves, especially high-dimensional ones, which is closely related to the Diophantine equation.

Finding the number of rational points, and then studying the distribution of these rational points.

In essence, the geometric structure of high-dimensional algebraic varieties tends to be more complex, with more intricate singularities, topological properties, and different homological properties, all of which influence the distribution of rational points.

So the research goal of this type of problem is actually just one: to simplify as much as possible the process of finding rational points and to easily find their distribution. It’s like being given a high-degree Diophantine equation and quickly determining whether it has a solution and solving it.

Alright, that’s how Qiao Yu understands it.

This is the understanding of a Math Novice. If Old Xue were here now and heard Qiao Yu’s thoughts, he might want to teach this ignorant guy a lesson with his fists.

The reason is simple, the research goals are simply too far-fetched.

Simplifying the process of finding rational points and easily finding their distribution on high-dimensional algebraic varieties is nearly impossible; this is mathematical common sense. Now, all anyone is doing is using geometrical and algebraic tools to efficiently estimate the number of rational points and understand their distribution through modern algebraic geometry tools.

As for quickly solving the Diophantine equation?

Solving the elliptic curve or even more complex modular forms-related equations, even if solvable, would make Old Xue just laugh.

Of course, for Qiao Yu, who doesn’t have much reverence for mathematics, all of this is not an issue, and adding that he just learned Peter Schulz’s mathematical thinking yesterday, a bold idea suddenly burst from Qiao Yu’s mind, and he couldn’t stop it.

Why couldn’t he try to solve such problems using the theory created by Peter Schulz?

Regardless of whether it’s feasible or not, he can try introducing the concept of complete space; even if there aren’t suitable tools to handle similar problems, he can create them himself.

Though built upon someone else’s framework, as long as he creates tools within it that adhere to its rules and solve the problems, it surely is feasible.

So now, the problem before Qiao Yu is quite simple: how to introduce the problem of estimating the upper bound of the number of rational points on algebraic curves into the framework of the quasi-complete space theory?

The fearless Qiao Yu sat at the table lost in thought.

A pen also began doodling on the draft paper.

Alright...

This problem doesn’t seem that simple, mainly due to the transformation of the problem.

After thinking for a long time, Qiao Yu came to a conclusion that if the estimation of the upper bound of rational points can be transformed into a problem of homology and geometric properties on complete geometric objects, then it’s logical to use advanced tools of p-adic geometry, such as complete algebraic spaces, the geometrization of modular forms, and p-adic homology theory, to analyze these rational points.

The only issue is whether such transformation might render the problem more abstract and complex.

But it doesn’t matter; after all, he’s just Little Kalami, just playing around. What’s there to lose by trying?

Thus, soon Qiao Yu excitedly wrote down this passage on the draft paper:

"Let X be a high-dimensional algebraic curve defined over a number field K, and X is a closed subset in a p-adic complete algebraic space. Then there exists a constant C that depends on the geometric properties of the curve X, such that the number of rational points on the curve satisfies: N(X)≤C."

Naturally, N(X) represents the number of rational points on the curve X.

This is just an intuition that popped into Qiao Yu’s head, there must be such a constant C. The reason is complex and related to the geometrical configuration of the curve in the complete space, which requires an understanding of Peter Schulz’s theory to comprehend this proposition.

The first step he needs to do now is to prove this proposition.

Because as long as the existence of this constant C is proven, this conclusion will provide a solid theoretical basis for estimating the upper bound of the number of rational points on complex high-dimensional algebraic curves.

After proving the first step, it’s to find the formula for this constant C and prove its correctness.

Then—the problem is solved!

However, just as Qiao Yu was ambitiously preparing to prove this proposition, he suddenly felt that he seemed to have no way to start with the problem he proposed.

He seemed to have fallen into the conundrum of how many steps it takes to put an elephant into a refrigerator.

Step one, open the refrigerator door, step two, put the elephant in, step three, close the refrigerator door.

The only problem is, he seems to have not found a refrigerator as big as an elephant!

Especially when Qiao Yu suddenly realized that if the formula for this constant C really exists, it would not only depend on the geometric properties of the curve but also might depend on the characteristics of the number field K, the modular form structure of the curve, and even other algebraic geometry tools.

Because after racking his brains, Qiao Yu found that the existing algebraic geometry tools seem not to support finding this C.

If a normal mathematician encountered this situation, they might choose to give up, but Qiao Yu is different. After all, he is just a Math Novice, and he has already treated this challenge as a game.

Even if he doesn’t have a clue, what if he succeeds?

Moreover, as the saying goes, if there are no tools, he can completely create them himself.

Think about it, Peter Schulz created such an impressive theoretical framework at the age of 21; there’s no reason why he, at fifteen, can’t create a few practical mathematical tools, especially since the entire theoretical framework is provided by someone else, he just needs to conduct secondary creation within this framework, which is significantly easier.

After all, the rules are already there, and he just needs to prove through rigorous mathematical logic that his tools are not wrong within the framework’s limitations. freēnovelkiss.com

Therefore, the next task can be further simplified: what kind of algebraic geometry tools can help him prove the existence of this constant C?

Qiao Yu frowned and pondered for a long time, then reconfirmed that firstly, he needs a new homology category tool.

Thus a line of text appeared on the draft paper again:

"The homology category QH(Cp) is an enhanced homology category defined on the completion space of the algebraic curve Cp. Its basic objects are the traditional cohomology class H^i(Cp,Zp), but we need to specially process them through a new operator Q, which acts on the cohomology class, making each object in the homology category not only have a topological structure but also possess an additional invariant..."

Phew... Qiao Yu looked at this expression with satisfaction, and with this new homology category, he can more precisely decompose the homology group of the curve, which can significantly simplify the steps to prove the constant C, perfect!

Indeed, researching mathematics is joyful!

Now, a new problem has arisen: how to define this new operator Q, Qiao Yu feels stuck again...

MMPD, whatever! Can’t figure it out, just put it aside for now, after all, a single tool is not enough to prove constant C...

Thus Qiao Yu, who had completely gone mad, began to invent a second tool. Now he needs a new fuzzy measure function to approximate the constant C.

"The algebraic curve p-adic fuzzy measure μfuzzy(Cp) is a new measure function used to describe the fuzzy properties of the algebraic curve Cp in the p-adic geometric environment. It’s defined as follows..."