In Beijing, Qiao Yu put down the phone and excitedly looked at the formulas on the computer, and began to re-examine the geometric background of the parameters and constants in the formulas.

Qiao Xi is right. There are many parameters and complex formulas. Now he needs to find the commonality of these parameters, so what is the commonality behind the current conditions? No, not just the commonality, but the essential commonality!

Otherwise it is not enough to link these parameters together.

So naturally, Qiao Yu put forward a hypothesis: Whether it is modular form, p-adic geometry or quantized homology category, their parameters can be uniformly represented by a single geometric quantity. The most critical part of this hypothesis is to find

A unified geometric quantity that can capture parameters reflecting the complexity of curves in different geometry tools.

Next is the most troublesome and critical step.

Qiao Yu began to analyze the core parameters of each geometric tool one by one, then listed its core parts, and then found the common points of all these parameters.

To put it simply, it is whether these parameters are controlled by a common constraint parameter. Of course, this is only the first step.

After finding the common constraints, you still have to find the connection between these parameters and the common constraints. This is another extremely complex proposition, but how can I put it, it fully meets Professor Zhang's requirements, complicating the proof process and simplifying the results.

.

And it can also make this formula a truly universal formula. When solving the problem of upper bounds of rational number points on curves, everyone does not need to think about anything else, just use his Qiao Yu-Qiao Xi theorem.

Immersed in the proof of the theorem, the sky turned dark before I knew it.

It wasn't until there was a knock on the door that Qiao Yu woke up from his tedious work.

Qiao Yu raised his head and looked out the window, then stretched on the spot, and then went to open the door.

Senior Brother Chen was standing outside the door. When he saw Qiao Yu standing in front of the door, he immediately raised the lunch box he had just packed in the cafeteria and said, "When I went to eat just now, I saw that you were not moving in the room, so I was afraid of disturbing you.

, I didn’t shout. I thought you must not have eaten yet, so I brought you a piece back.”

"Thank you, Senior Brother Chen!" Qiao Yu grinned and gave Senior Brother a sincere smile, then took the lunch box and turned around and walked into the room.

After spending an entire afternoon's brain cells, Qiao Yu felt really hungry.

This lunch box is very timely.

Chen Zhuoyang followed and walked in, still muttering: "Don't forget to sleep and eat all the time when you are studying. No matter how busy you are, you have to eat. No matter what you do, your body is the most important."

"I know, Senior Brother Chen. Today is a special situation!" Qiao Yu explained with a smile, then opened the lunch box and ate hungrily.

Although I didn’t do much exercise in the afternoon, I spent too much mental energy and felt very hungry.

Even more hungry than playing basketball all afternoon.

Chen Zhuoyang curiously looked at the content on Qiao Yu's computer. Oh my god, there were those complicated formulas and all kinds of messy custom symbols. This made the senior brother sigh inwardly.

It can't be compared, it can't be compared.

"Are you still studying your project?" Chen Zhuoyang pointed at the computer and asked casually.

Qiao Yu looked up at the monitor, swallowed the food in his mouth, and then said, "Yes, it's a pity that there are no results yet."

Before, he would have said a few words, but now that he had just found a clue and realized the difficulty of this problem, Qiao Yu became much more cautious.

He didn't want to talk too much before making results.

Chen Zhuoyang said with self-pity: "Don't be impatient. How can the results be so easy? My boss introduced me to a research group before, which mainly focused on the quantization of geometric structures and modulus spaces on high-dimensional algebraic varieties. The main goal was to publish An article exploring patterns

A paper on the relationship between equations and Kahler manifolds. Alas...

"Um? What are you sighing for? You couldn't publish your paper?" Qiao Yu asked curiously.

"Forget it, it took more than a year to work on it, but in the end, a team from Harvard did the same work and published several papers, which were better than ours. Then the project was hastily ended. In the end, the paper was casually published in a second-district water journal. The key is It’s true that I’ve only done three games.” Chen Zhuoyang said melancholy.

Qiao Yu probably understood the pain of Senior Brother Chen. He followed the top tutor and the tutor also gave resources, but he failed to grasp it... Of course, this cannot be entirely blamed on Senior Brother Chen. After all, a research group can be regarded as a whole.

But in other words, if Senior Brother Chen joins in and is able to turn the tide, the ending may be changed.

"Oh, Senior Brother Chen, what was your main job at that time?" Not knowing how to comfort him, Qiao Yu simply asked.

"Classification of modular forms of algebraic curves, research on the relationship with K stability. Hey, let's not talk about this anymore. In fact, I want to tell you that the first draft of my thesis is almost completed. When will you go to Huaqing? I can take advantage of you. Next time I go there and tie up the last bit." Chen Zhuoyang revealed his true purpose time and time again.

Still too thin-skinned.

Qiao Yu felt that if it were him, he had done so many things before, so he would definitely not talk so much at this time, and only then would the real purpose be revealed.

But he still thought about it and said: "Hey, what are you talking about, Senior Brother Chen? For such an important matter, I can just make a trip to Huaqing for you! But Grandpa Senior said that day that he was busy this week, maybe next time You are free this week. If you want Grandpa Shi to take a look at it in person, just give it to me before next Wednesday."

This time point actually has nothing to do with when Mr. Yuan has time. It is simply that Qiao Yu feels that if he is in the right direction, he should be able to almost complete the results before next Wednesday. At that time, he will help his senior brother to take a look first. Is it more convenient to take the thesis to Huaqing and let Grandpa Shi help me with my eyes?

Before he was sure that this direction was correct, he was not in the mood to care about other things.

Chen Zhuoyang said happily: "Next Wednesday? No problem! By the way, I don't ask Mr. Yuan to help me revise my paper. You just need to ask Mr. Yuan to help me see where there are problems and give me some specific comments." Qiao Yu nodded and said affirmatively: "OK! Just remember to give me the paper before Wednesday."

"Thank you so much, little junior brother, I'll go back and work on my thesis first." "Well, it's okay, you can go and do yours!"

"Hey..." After watching Chen Zhuoyang leave, Qiao Yu sighed and suddenly realized that he had more and more things going on now.

Study, read, coax the tutor and the tutor's tutor, do projects, write papers, participate in selections, and then go to IMO to get medals, while beating friends of the same age... Now I still have to worry about my senior's doctoral thesis.

How much does he have to do by himself? This is probably the legendary saying that a capable person works harder! Well, it seems that he must take on the task of revitalizing the Chinese mathematics community! After all, he is now sixteen years old.

He is no longer the fifteen-year-old brat he once was!

Thinking of this, the stuffed Qiao Yu became excited again, and sat down in front of the computer upright, working, working... In order to revitalize Huaxia Mathematics, and to show the tutor, senior master, and senior brother a little bit of Qiao's color

, he must work out Qiao’s upper bound theorem no matter what!

...

Mathematics research often has a very interesting characteristic, and it is a situation that countless mathematicians have encountered, that is, during the research process, it is likely to get stuck on a certain step, or on a certain problem, and be unable to make progress for a long time.

.Yes, it is stuck there alive.

Sometimes I have an epiphany, and when I have passed this hurdle, I feel suddenly enlightened, and the road ahead is smooth.

But it is a pity that for most mathematicians in the world, this obstacle may last a lifetime, so the project ends without any problem, and the previous work and data are stored there, dreaming that one day, they will suddenly have an epiphany and let these

Research will see the light of day one day in the future, but it is more likely that it will never happen again.

Qiao Yu is actually the same, it's just that his talent is a little higher than countless ordinary mathematicians.

When he realized the need to find commonalities in parameters under Qiao Xi's prompts, this problem no longer seemed to be a problem for him.

All the previous reasoning and proof processes have been done. It can be simplified by finding commonalities. The commonalities are hidden in the less obvious connections behind those parameters. As long as the work is detailed enough, Qiao Yu feels that this is definitely the right direction! Facts

And indeed it is.

For three days, Qiao Yu almost stayed at home except for eating. He didn't even read a book. He devoted himself wholeheartedly to this work, and then he really discovered the existence of commonalities. The higher the level of the model form, the steeper the curve.

Complex, so k~curve complexity.

The prime number p controls the local geometric behavior of the curve on the p-adic number field. Different prime numbers correspond to different geometric constraints. The prime number p is also related to the complexity of the curve, so the parameter q in the quantized homology of the local geometric complexity of p reflects the quantization.

The influence of geometric objects on the global complexity of the curve is a further quantification of the geometric complexity of the curve, so q~global geometric complexity. In other words, although different geometric parameters come from different sources, they all reflect the curve’s appearance under different perspectives.

complexity.

What is this? This is the defining condition for parameter unification.

So on Friday night, Qiao Yu designed a unified geometric constraint parameter 0, and put forward a second hypothesis: the geometric constraint parameter 0 is some kind of modular form level, p-adic field prime number and quantized homology parameter. Weighted combinations, which together reflect the global complexity of the curve.

Based on this assumption, it is obvious that Qiao Yu can get a basic structure: 0=f(g,k,p,q). Of course, at this step, it is obviously not enough.

Because the weight of each parameter is different, to make the structure mathematically reasonable, a combination method that can perfectly reflect the weight of each parameter is needed. The next step is the calculation and verification work, which is complicated, but not difficult.

But one night, Qiao Yu came to the conclusion that the growth of k increases logarithmically with the genus g, so: k ~ glog (g); the complexity of local geometry changes exponentially with the increase of genus, so p ~e^g/2; in quantized homology, the relationship between parameter q and genus g grows. Qiao Yu directly calculated an approximate value: q~g^3/2.

The formula comes out naturally: 0=f(g,k,p,q)=g-log(k) g^2.log(p) g·q

After directly bringing in the expressions of the three parameters, it is: 0=g·log(glog(g) g^2.log(e^g/2) g·g^3/2 At this step, there is only a loss. Grid g is an important parameter.

Next is the simplest simplification work: 0=g·(log(g) log(log(g)) g3/2 g^5/2

After working in front of the computer day and night for three days, Qiao Yu finally typed out the final formula for the rational point estimation of the curve on the computer at 11:37 pm on February 21, 2025: N(X)sC( 0)=0^gθ is the geometric constraint parameter he designed, and g is the genus.

This formula...is really beautiful!

After admiring it for a while, Qiao Yu immediately started to verify it. After all, the formula is useless only if it is beautiful. It must be useful. What he has to do is to find out whether it is accurate based on his own formula.

Qiao Yu chose the classic elliptic curve y^2=x^3 x

According to the known conditions of the BSD conjecture, it can be seen that the curve genus is 1, which can be directly introduced into the formula, and then simplified to get the result: 0=5, well, the 1st power of 5 is still 5. The conclusion is obviously correct.

Because this is the classic Hermitian curve. The rational number points on the curve have been calculated by someone more than ten years ago.

Next came the Model curve, special cases of the Fermat curve, various cases of the Kubert curve... Qiao Yu tried them all.

For example, the Model curve: y^2=x^3 k, k is an integer. Qiao Yu verified the cases of k=-1, k=2 and other known finite rational points, and the results were all correct.

Then Qiao Yu opened Professor Robert Green's paper and used his own formulas to conduct comparative calculations with the formulas derived by Robert Green. On certain points, his formulas were mostly the same as Robert's results, but some Not sure, there are still some differences between the two sides' calculations, but not big.

Well, Qiao Yu didn't bother to worry about who was right and who was wrong.

At least at this point, he can start writing the paper. This step is the easiest for him.

Because he had already written down the process of deriving the formula for more than half a month before, and because he had long considered completing a paper, Qiao Yuben had prepared the entire derivation process in detail, and the next step was just to use professional language to describe those The derivation process is integrated.

It's nothing more than lemmas, theorems, and the proof part. You can basically copy and paste it directly. The main thing is the subsequent proof process about the unified geometric constraint parameter 0, which needs to be written now.

Fortunately, Qiao Yu had a whole weekend to complete this paper.

In fact, of course there is no need to be in such a hurry. At Qiao Yu's age, there is no need to seize the day. It doesn't matter if the paper is completed a few days earlier or a few days later.
To be continued...