Chapter 188 - 183: Comment
Capital University, School of Mathematical Sciences.
Zhou Shenzhi hung up the phone, his expression grim.
He had asked a professor from the Yan University School of Mathematics to arrange a private meeting with Li Dong, under the pretext of discussing an academic matter.
His reasoning was perfectly sound.
The zero-point criterion for the local-global compatibility of automorphic representations of GL(n) was a brand-new tool in the mainstream of the Langlands Program, and he had some specific technical details he needed to consult with Li Dong about in person.
However, Yan University had refused.
The reply was polite.
"Student Li Dong is quite busy with his studies recently and is not available for off-campus academic meetings at this time."
Busy with his studies?
Zhou Shenzhi cursed inwardly.
’A guy like Li Dong? If he were at our Capital University, the dean would be itching to just hand him a diploma. Busy with his studies, my ass!’
Of course, he knew this wasn’t Li Dong’s own decision.
It was Yan University that was unwilling.
After all, although that incident from years ago had never been fully exposed, plenty of people in their circle knew the inside story.
Yan University was protecting Li Dong far too well.
Anyone who wanted to contact Li Dong through private channels had to get past Yan University first.
But Zhou Shenzhi was genuinely out of options.
Pushing the ramification index from 2 to 3.
Just that one step, and he was stuck.
Theoretically, there were two paths to solving this problem.
The first was to find that person from back then.
The one who had proposed the p-adic integration path deformation scheme. He was the one who truly understood the underlying logic of this method.
Only he would know how to modify the scheme for higher ramification indices.
But that was a path Zhou Shenzhi could not take.
’That person would never talk...’
The second path was to use Li Dong’s new tool.
In his paper on Montgomery’s Conjecture on Association, Li Dong had established a completely new zero-point criterion.
A necessary and sufficient connection between the pair correlation statistical properties of zeros of automorphic L-functions and the local-global compatibility of automorphic representations.
The power of this tool lay in its ability to completely bypass traditional algebraic methods.
You didn’t need to brute-force the Hodge-Tate coupling structure for e_v=3. As long as you could verify that the pair correlation function F_π(α) of the zeros of the automorphic L-function converged to the GUE predicted value within the corresponding interval, you could directly determine whether local-global compatibility held.
In other words, Li Dong’s zero-point criterion could directly leapfrog the very step that had held Zhou Shenzhi back for years.
But the problem was...
Li Dong’s paper had not yet been the subject of any formal, public academic exchange.
In the world of mathematics, after an important paper is published, it typically goes through a process of academic seminars or public inquiries.
The author would accept questions and challenges from their peers at international conferences, specialized workshops, or public discussions organized by the journal, providing further explanations and clarifications on the paper’s key steps.
But Li Dong’s paper was published in the *Mathematical Annals*. It had long since passed peer review, its logic was entirely self-consistent, and it had no glaring flaws.
So, while the academic community was widely citing and studying the paper, no formal academic conference had yet organized a public discussion specifically centered on it.
In other words, Zhou Shenzhi had no public channel through which to ask Li Dong his specific questions.
His private meeting was blocked by Yan University.
And there were no public channels.
Both paths were completely blocked.
Zhou Shenzhi recalled the fleeting look of disappointment in his professor Jiang Yubai’s eyes.
He gritted his teeth.
Finally, he opened his computer and logged onto arXiv.
He was going to post a Comment.
In mathematics, a "Comment" on arXiv is a very particular form of academic exchange.
Simply put, it’s a way to raise questions, add supplements, or even challenge a published paper.
A Comment in itself isn’t an attack.
Many famous mathematical theorems underwent repeated Comments and responses before finally gaining widespread acceptance.
For instance, before Perelman’s preprints proving the Poincaré Conjecture were formally confirmed, numerous mathematicians wrote detailed Comments and verification notes.
But the impact of such a thing depends on who posts it, how it’s posted, and the effect it has afterward.
If the questions raised in the Comment truly hit a critical point in the paper, the paper would face immense pressure, and the original author would have to respond publicly.
But if the Comment only raised some minor, trivial questions, it usually wouldn’t cause much of a stir.
Regardless of the outcome, a formal Comment on a *Mathematical Annals* paper being posted to arXiv would attract attention throughout the entire number theory community.
Because it meant someone was publicly challenging the integrity of a paper in a top-tier journal.
And as the paper’s author, Li Dong would be obligated to step forward and respond.
This was exactly the effect Zhou Shenzhi wanted.
To force Li Dong out into the open.
Who understood Li Dong’s paper on Montgomery’s Conjecture on Association the best?
Besides Li Dong and Riemann...
It wasn’t Tian Gang, nor was it Liu Ruochuan.
It was Jiang Yubai and Zhou Shenzhi.
Jiang Yubai’s research group had always focused on the Langlands Program, and Li Dong’s zero-point criterion had directly revolutionized their entire line of research.
Therefore, Jiang Yubai had made Li Dong’s paper a core reference for his group, studying it line by line, sentence by sentence.
And Zhou Shenzhi was even more obsessed.
He could almost recite it from memory.
Thus, he had found... not flaws, precisely, but what you might call... "leaps in logic" in Li Dong’s paper.
It was a habit of Li Dong’s.
During his derivations, Li Dong often skipped intermediate steps.
Not because those steps were unimportant, but because, in Li Dong’s eyes, the derivations were so obviously true that it was completely unnecessary to write them out.
For example, in Chapter 4, Section 2 of the paper, when transitioning from the interval |α| ∈ [1, 2] to |α| ∈ [2, 3], Li Dong used the phrase "by natural analogy with the first-order case for the normalized contribution of second-order prime powers" and jumped directly to the conclusion.
But to Zhou Shenzhi, this "natural analogy" was not natural at all.
The way the second-order prime power p² contributed to the explicit formula was completely different from the first-order prime p.
The transition from the linear contribution of p to the non-linear contribution of p² involved the transformation of a key convolutional identity.
This transformation required the use of the precise expansion of the Ramanujan sum at higher-order prime powers, as well as the re-parameterization of the corresponding normalized weight function.
Li Dong had skipped all of this.
Presumably, in his mind, all the derivations between the two steps were as self-evident as walking from the first floor to the second.
But to an outsider—no, even to many top professors—it was like a ten-story gap.
They couldn’t keep up with Li Dong’s train of thought.
Zhou Shenzhi found a total of three such leaps.
None of these three leaps affected the correctness of the paper.
Because if you meticulously followed Li Dong’s line of reasoning to fill in the gaps, every step could be worked out.
But the process of filling them in was, for the vast majority of mathematicians, a technical paper in its own right.
Zhou Shenzhi organized these three leaps into the first three questions of his Comment.
Then, he added a fourth point.
This was his true objective.
The content of the fourth point was not about the correctness of the paper itself, but about the subsequent application of its results.
He wrote:
[When applying the zero-point criterion from Theorem 1 of this paper to determine the local-global compatibility for automorphic representations of GL(n) in the case of ramification index e_v ≥ 3, it is necessary to explicitly compare the convergence of the pair correlation function F_π(α) of the zeros of the automorphic L-function L(s, π) in the interval |α| ∈ [0, 2/n] with the higher-order coupling structure of the Hodge-Tate weights.]
[However, the paper does not provide a concrete implementation path for this comparison. Could the author provide a supplementary explanation regarding the technical feasibility in this application scenario?]
Translated into plain language, it meant:
’Li Dong, I want to use your tool, but I’m stuck when it comes to e_v=3. Can you tell me how to proceed?’
He had disguised the core problem he couldn’t understand or solve as an "academic inquiry into the paper’s future applications," mixing it in with three technical questions about the leaps in logic.
It was flawlessly executed.
Zhou Shenzhi knew what the consequences would be once this Comment was posted on arXiv.
Li Dong would almost certainly have to respond publicly.
Because it was a formal academic challenge to a paper in the *Mathematical Annals*. Not responding would be tantamount to admitting the paper had problems, something no mathematician could accept.
When the time came, Li Dong would either have to fill in the three leaps or provide a more concise alternative proof.
Either way, the academic community would benefit.
This was also the insurance policy he had left for himself.
As for the fourth point... if Li Dong actually responded to it, the problem that had plagued Zhou Shenzhi for years would be solved in an instant.
But he would be under immense pressure.
From Yan University.
No, not just that...
Once this Comment was out, everyone in their circle would wonder, ’Does Zhou Shenzhi have a problem with Li Dong?’
’Is he just trying to stir up trouble because Li Dong’s achievement made his professor’s research path obsolete?’
The public opinion would be ugly.
Zhou Shenzhi hesitated for a long time...
’For my professor!’
His fingers descended.
And began to type.
...
Yan University, in the counselor’s office.
Li Dong was grinning sheepishly.
Director Liu was truly speechless.
He was starting to feel a sense of dread whenever he saw Li Dong now.
"Come on, Brother Dong, winter break is almost here. You’re asking for leave *now*?"
Li Dong chuckled again, sliding a leave request form in front of Director Liu.
"Director Liu, my mom is coming to Kyoto, and I want to show her around the area near campus."
Director Liu paused for a moment, then broke into a smile.
"Well, that’s a wonderful thing."
He readily signed the form.
"Li Dong, your mother did an incredible job raising you."
Li Dong took the form, his smile fading slightly.
"She really is incredible."