Chapter 196 - 192: Don’t Tell Me You Don’t Know?
In the audience, Tian Gang frowned when he heard Li Dong say, "I’m going to elaborate a bit."
He had discussed this with Li Dong before.
The first three questions in Zhou Shenzhi’s comment were technical critiques about skipped steps and had to be addressed publicly. There was no issue there.
But the fourth question...
It was about the application path of the zero-point criterion in the case of GL(n) automorphic representations where the ramification index e_v ≥ 3. This question was, in essence, no longer a critique of the paper itself.
It was an exploration of a future research direction.
Strictly speaking, it had nothing to do with this paper on Montgomery’s Conjecture on Association.
Tian Gang had specifically mentioned this point when he spoke with Li Dong earlier.
Li Dong had also said he wouldn’t address it.
So why was the kid planning to elaborate on it now?
He had known Li Dong for some time and was well aware of the young man’s personality.
Li Dong was definitely not some low-key, honest, harmless academic nerd.
When someone like him said he was going to "elaborate," there had to be a motive.
Nearby, Jiang Yubai also frowned slightly.
’What is this kid up to?’
But the people around them didn’t care about such subtleties.
The several hundred people present, from professors to doctoral students, from domestic scholars to international heavyweights, were all suddenly energized.
To be honest, the paper analysis and point-by-point response to the comment during the first half-hour were brilliant, but for many people, that was all just part of the show.
The issue Li Dong was about to elaborate on now, however, was what was truly relevant to them.
The reason was simple.
After Li Dong’s paper on Montgomery’s Conjecture on Association was published, almost every number theorist in the world had immediately begun using it as a brand-new tool.
This was no exaggeration.
If an *Annals* paper merely proved a pretty theorem, it was, at best, a medal—nice to look at but with little practical use.
But Li Dong’s paper was different.
The zero-point criterion he had established was groundbreaking.
The necessary and sufficient connection between the GUE convergence of zeros of automorphic L-functions and local-global compatibility was, in essence, a universal framework.
You could graft it onto almost any research topic involving automorphic representations.
That was why, over the past few months, countless research groups—from Berkeley to the Ecole Normale Superieure de Paris, from Capital City to Tokyo—had been trying to use Li Dong’s zero-point criterion to advance their own studies.
Some projects progressed smoothly.
But many more got stuck.
It wasn’t that there was a problem with Li Dong’s tool.
The tool was a good tool; there was no dispute about that.
The problem was...
They didn’t know how to use it.
This touched upon an awkward reality that was very common in academia but that almost no one was willing to admit publicly.
The use of a mathematical tool depends not just on the tool’s own logical completeness, but even more so on the user’s depth of understanding of its underlying philosophy.
The underlying logic of Li Dong’s zero-point criterion was not a traditional, purely algebraic approach. Instead, it was a completely new framework that had grown from the intersection of analytic number theory and random matrix theory.
If you were accustomed to dealing with problems in the Langlands Program using traditional tools like Algebraic Geometry and p-adic analysis...
Then your first reaction upon getting Li Dong’s zero-point criterion would be to translate it into the algebraic language you were familiar with.
But this was precisely the biggest trap.
Because the core of the zero-point criterion was that it bypassed algebraic language entirely.
So, the reason many research groups were stuck was not a lack of ability, but an excess of mental inertia.
And now, the author himself was about to stand up and explain how this mathematical tool should be used.
How could the hundreds of people in the audience not be excited?
...
Li Dong looked at Zhou Shenzhi.
"Professor Zhou, the core of your fourth question isn’t really about the zero-point criterion itself."
Zhou Shenzhi nodded slightly, saying nothing.
"What you’re really trying to do," Li Dong continued, "is extend the local-global compatibility of automorphic forms from GL(2) to GL(3)."
A faint stir rippled through the audience at these words.
This was hardly a secret within their circle.
Jiang Yubai’s research group had been working on this very direction: generalizing the local-global compatibility from GL(2) to GL(3).
It was the next major milestone on the main path of the Langlands Program.
If they succeeded, it might not be a guaranteed Fields Medal, but they’d at least be one of the most competitive candidates that year.
Although Jiang Yubai was past the age limit, there was still the Abel Prize, wasn’t there?
So it wasn’t surprising that Li Dong had laid this bare.
Zhou Shenzhi’s expression didn’t change.
That was indeed what he was thinking.
And extending the ramification index from 2 to 3 was far more complex than simply answering a question in a comment.
The technical details, the development of new tools, and even the redesign of the entire proof framework involved at least a dozen or more core problems that needed to be solved.
His research group had accumulated years of work in this direction and was undoubtedly in the global vanguard.
So he wasn’t afraid of Li Dong exposing his research direction.
Even if everyone knew where the finish line was, only a handful of people could actually reach it.
Thus, he replied calmly.
"You’re right. That is indeed our research group’s direction."
"So I’d like to ask, what is the specific implementation path for your zero-point criterion when the ramification index e_v=3?"
Li Dong nodded.
Then he said something that no one expected.
"But I don’t think my zero-point criterion is the best tool for you to get from GL(2) to GL(3)."
The entire hall fell silent.
Since ancient times, inventors have always been full of confidence in their own creations.
Edison claimed direct current was safer and more suitable for widespread use than alternating current, waging a protracted "war of currents" with Tesla over it.
Darwin spent his life defending the theory of natural selection, never yielding an inch in the face of Wallace’s doubts.
Even Gauss, after inventing the method of least squares, never missed an opportunity to demonstrate its superiority to his peers.
It was only human nature.
Who would ever think the thing they created wasn’t the best?
But Li Dong was actually saying there was a better tool?
And he was saying it right after he had just spent half an hour hyping up his zero-point criterion?
Penrose’s eyebrows rose, a flicker of interest in his eyes.
Tao Zhexuan stopped writing and looked pensive.
And on Zhou Shenzhi’s face, which had been a mask of calm, a trace of unease appeared for the first time.
He didn’t press for an answer.
But his heart suddenly began to race.
Because he had a very bad feeling.
Li Dong continued.
"The best tool is in the paper you yourselves published in the *Duke Mathematical Journal*."
"’On the Local-Global Compatibility of GL(2) Automorphic Representations in Cases Where the Ramification Index Does Not Exceed 2.’"
"At the bottom of page seventeen of that paper, there’s a core construction regarding the deformation of the p-adic integral path."
"The underlying logic of that construction is to embed the filtration structure of the Hodge-Tate decomposition into the parametrization equation of the integral path. This allows the generic obstruction to be automatically projected onto the zeroth filtration layer during the path deformation process, where it is then precisely canceled out by the first-order weight."
Li Dong paused here, looking at Zhou Shenzhi with curiosity.
"This construction works for e_v=2 because the first-order Hodge-Tate weight and the projection direction of the generic obstruction happen to be perfectly orthogonal."
"But when you move to e_v=3, the second-order weight introduces additional coupling terms, breaking the orthogonality. That’s why your extension stalled."
"Right?"
Zhou Shenzhi said nothing.
His expression could no longer be described as merely "uneasy."
He never expected Li Dong to see that in a single glance.
He and his mentor, Jiang Yubai, had discussed this problem countless times and tried all sorts of methods, only to hit a wall every time.
And this nineteen-year-old kid, just by reading their paper, had identified the very sticking point that had held them back for years.
Seeing his silence, Li Dong didn’t press him, but continued directly.
"But the thing is, you don’t need to restore orthogonality."
"You just need to make a very natural extension to that path deformation scheme: replace the original first-order filtration embedding with a layer-by-layer recursive filtration embedding."
"[...]"
"You don’t need orthogonality, just layer-by-layer compatibility."
He spoke calmly, as if explaining something every undergraduate should know.
"Even without incorporating the GUE convergence condition from my zero-point criterion, you can skip the specific algebraic calculations and directly confirm the completeness of the cancellation at each layer from an analytical standpoint."
He looked at Zhou Shenzhi.
"You didn’t know that, did you?"
Sunshine Hall was utterly silent.
Zhou Shenzhi didn’t speak for a long time.
The idea of a "layer-by-layer recursive filtration embedding," once explained, wasn’t actually that complicated.
One could even say it was the natural extension of that original p-adic integral path deformation scheme.
But he couldn’t do it.
Because he had never truly understood that scheme.
Back then, he had simply walked the final stretch of a path someone else had drawn for him.
And Li Dong, just by reading the paper, had seen through to the very foundation beneath that path.
And now he was asking with genuine curiosity: Your answer is right under your own feet.
Why were you seeking far and wide for what was close at hand?
...
In the audience, Tian Gang, having heard this much, had a good guess as to why Li Dong wanted to elaborate on this question.
’This kid is aiming to crush his spirit.’
He wasn’t helping Zhou Shenzhi solve a problem.
He was dragging the one thing Zhou Shenzhi least wanted to face out into the light of day.
Your core methodology wasn’t something you came up with yourself.
But at the same time, Tian Gang was a little puzzled.
Although Jiang Yubai’s standing in the academic world might not be as high as his, he was by no means someone you could afford to offend lightly.
He was the academic leader of the Capital University School of Mathematical Sciences, supervising over a dozen doctoral and postdoctoral students, and had been cultivating his influence in the field of the Langlands Program for nearly twenty years.
His network of collaborating scholars spanned half the globe.
This wasn’t just one man; it was a school of thought.
Was Li Dong really going to offend the entire Jiang Yubai camp just to respond to one of Zhou Shenzhi’s comments?
Was it worth it?
Or perhaps... the kid’s goal was never just about responding to the comment in the first place?
Tian Gang turned to glance at Jiang Yubai, who was seated in the front row.
Jiang Yubai’s expression was remarkably calm.
...
Seeing Zhou Shenzhi remain silent for so long, Li Dong didn’t press him further.
He just smiled, his tone carrying a hint of goodwill.
"Professor Zhou, the key to this line of thought doesn’t lie with me."
"I’ve only pointed in a direction."
"The person who truly understands the underlying logic of that p-adic integral path deformation scheme should be able to explain it much more clearly than I can."
His gaze shifted to a spot in the back rows of Sunshine Hall.
"Teacher Yang? Why don’t you come up and explain it to Professor Zhou?"
In the front row, a flicker of emotion finally crossed Jiang Yubai’s face.