The top student must be diligent.

Chapter 64 The Nervous Voice

"The abc conjecture can be regarded as a long-standing mathematical dispute."

"Initially, when we learned that the abc conjecture was proved by Shinichi Mochizuki, we were happy. This fully demonstrated that even the most difficult problems can be solved in a mathematical sense."

"But unfortunately, after years of thinking about Professor Mochizuki's IUTT theory, we came to the conclusion that although these four papers have more than 512 pages, and later became more than pages after additions and revisions, they are still difficult for people to read, and the key proofs are always particularly brief and difficult to understand."

"Until now, there are only a dozen people in the world who claim to have understood Professor Mochizuki's papers. Most of them are from Professor Mochizuki's hometown and have a certain relationship with Professor Mochizuki."

"In addition, a few months ago, Professor Mochizuki's proof paper was accepted by the journal PRIMS, which he is the editor-in-chief. It is said that the abc conjecture has become a theorem at Kyoto University."

"I absolutely do not think this is the right approach, and mathematics should not have such a thing that a proposition can be directly turned into a theorem by relying solely on personal influence, as if it were a tyrant."

"So, it's time to end this."

Peter Schultz mercilessly criticized Shinichi Mochizuki and compared him to a tyrant.

However, the vast majority of people in the audience nodded in agreement.

In any case, it is obviously a very overbearing thing to directly regard a conjecture as a theorem in a certain area when it has not yet been generally recognized by the entire mathematical community.

The mathematical community is also made up of individuals, so many people are also dissatisfied with this kind of thing.

Of course, there were also those supporters of Shinichi Mochizuki who did not nod.

That is, the dozen or so people who "understood Shinichi Mochizuki's paper" as Schultz said, all came.

Schultz did not deny these people's applications to attend the meeting, so as not to appear too overbearing.

In fact, among those 16 seats, there are several supporters of Shinichi Mochizuki, all of whom are Japanese.

Next, Schultz began his talk, starting with an introduction to the abc conjecture.

Sitting in his seat, after hearing Schultz's criticism of Mochizuki Shinichi just now, Xiao Yi couldn't help but look at Mochizuki Shinichi on the other side, wondering if this big boss would be angry?

However, looking from the side, Mochizuki Shinichi's expression was very calm and there was no change.

There is a feeling of not showing one's emotions on one's face.

Of course, maybe it’s also because he has already made up his mind?

Xiao Yi continued to listen, and at the same time, he thought about it in his mind in combination with the paper by Mochizuki Shinichi that he had read before, and occasionally flipped through the A4 paper in his hand.

Anyway, although he is sitting here now, since the seat is arranged in the corner, he can be regarded as a small transparent person at the main table.

The reason why Schultz arranged him here was only because his paper served as the "trigger" for this discussion. As for whether he could offer any useful opinions on this discussion, it was estimated that no one present had considered it.

And just like that, time passed quickly.

Schultz also began to raise his formal questions.

"The first point is the Frey curve."

"Mochizuki describes the inequality for all fixed d ≥ 1 that reduces to all number fields k of degree [k:Q] ≤ d and all elliptic curves E/k corresponding to points P∈Mell(k)..."

"…using the local Tate homogenization (rig m → Grig m /qZv′(E×k kv)rig, we obtain the set of local Tate parameters qv∈kv at a finite number of locations v…"

"…but note that the degree of the bound ∞ = (Mell \ Mell) is 1/2, while the degree of the logarithmic derivative is 1/12, which fails to explain the reason for the sixth factor I mentioned above."

……

When the people present heard this, more than 90% of them began to be confused about what Schultz was talking about, but those who could really understand it were enlightened.

This should be the new point of doubt discovered by Schultz!
This point of questioning is different from those he raised two years ago. It is more sharp and goes straight to the essence!
How would Mochizuki Shinichi explain it?
However, before Schultz could continue, Mochizuki Shinichi stood up.

"I'm sorry, Peter, since you intend to raise points one by one, I will also respond to them one by one."

"Regarding the first question you mentioned, the contradiction between the Frey curve and the sixth factor caused by Tate's analysis."

Mochizuki Shinichi left his seat and walked to the big blackboard - Xiao Yi discovered that there was an extra-large blackboard in almost every room of the Max Planck Institute of Mathematics.

Mochizuki Shinichi picked up the blackboard pen from the side and started writing directly on it.

【Grm→Grm/qZv(E×k kv)r】

【deg(qE)=1[k:Q]∑v(qv)log(N (v))】

【h(P)=1/6deg(qE)+O(h(P)^1/2+1)】

【…】

After writing a few simple formulas, Mochizuki Shinichi put down his pen and said, "My explanation is complete."

The people present were all grinning.

Is this the end of your explanation?
Without saying a word, who can understand it all at once? ?

Of course, this is actually Mochizuki Shinichi's character. He acts very casually, rarely explains anything, and is willful and consistent. However, this has aroused the envy of many young students, who envy his "never explaining" character that others can't do anything about.
However, there were still people present who could understand it, and Schultz was one of them.

He frowned and thought, and soon understood the meaning of Mochizuki Shinichi's formulas.

Like him, he borrowed all the analytical methods from Xiao Yi's paper to solve his first problem.

He let out a long sigh. It seemed that today would be another fierce clash of mathematical ideas.

Then, he would no longer restrain himself.

"Yes, Professor Mochizuki, you did give a very good explanation."

"Of course, this is not the only problem I have."

"The second question comes from your paper, IUTT-4, Theorem 1.10."

“Let a natural number d ≥ 1 exist such that there exists a function αd, βd:N→ R depending on d such that αd()→0…”

"So how do you use your IUTT to make your inequality true?"

As for this question, Mochizuki Shinichi was obviously already prepared and started writing on the blackboard.

The two men's confrontation left only their voices in the entire conference hall.

There is no third party involved.

Almost everyone else held their breath as they watched this debate, which could probably be recorded in the history of mathematics.

Almost every point of doubt raised by Schultz can make those who understand him show an expression of sudden enlightenment, and be surprised in their hearts at the high level of this mistake - it is difficult to detect.

However, Mochizuki Shinichi was well prepared. For each problem, he was able to solve it simply by writing down a line of formulas on the blackboard.

Only in response to a few questions would he reluctantly speak out to explain.

In this way, time passed slowly.

Compared to the solemn mood of other audiences, there was one person who had a different feeling.

This person was Xiao Yi. As he listened to the debate between the two big men, he felt that their way of thinking was more and more familiar. At the same time, their questions and answers constantly made him connect with the knowledge in his mind.

till the end.

……

【Gv o×μkv×((qj2v )j=1,.,*)N】

"I've finished explaining."

"Peter, do you have any questions?"

Mochizuki Shinichi once again gave his answer on the blackboard.

This time, however, Schultz remained silent.

Almost all the problems he found for this meeting were cleverly resolved by Mochizuki Shinichi. Only a few of them were still controversial, but he could not raise any further questions.

So, where exactly is the problem?
Could it be that this discussion will end in vain again?

Mochizuki Shinichi...

He is indeed someone that Professor Faltings considers "smart".

It’s still so difficult as expected.

The whole audience remained silent.

The big guys at the main table also remained silent.

The onlookers could see that Schultz had done his best.

Even more than five hours have passed!

No one present had lunch, but no one wanted to leave.

Is the result of this discussion about to emerge?
However, after a long silence.

Suddenly, a slightly nervous voice sounded.

"Professor Schultz, Professor Mochizuki, perhaps we can go back to the beginning and think about this?"

"If, as Professor Mochizuki just explained in the fifth question, in any case, a Hodge Theater is a data abstractly derived from a fixed single-punctured elliptic curve X; the natural functor from the category whose only object is X and whose morphisms are automorphisms of X to the Hodge Theater category is equivalent."

"So, starting from this perspective, let's go back to the third question, which discusses the content of far Abelian geometry."

"As we all know, far-Abelian geometry studies the absolute Galois group of rational numbers, and even the flat fundamental group of any algebraic variety. How their 'far-from-Abelian' parts, that is, the parts that do not conform to the commutative law ab=ba, affect the properties of the corresponding algebraic structure."

"In that case, in the sense that geometry and group theory are equivalent, tele-Abelian geometry will not hold—"

"In other words, it violates Professor Mochizuki's Mochizuki Theorem?"

(End of this chapter)