After Luo Yu finished his question, he stared straight at Li Dong.
But Li Dong didn’t move.
The conference hall fell silent for a second or two. A few old professors in the front row exchanged glances.
They all understood Luo Yu’s question.
The multiplicative decomposition of Proposition 3.1 was predicated on the assumption that the pole of every L(s, π?χ) at s_0 is simple.
This assumption wasn’t actually explicitly stated in the paper.
If two different characters both produced a pole at the same s_0, the uniqueness of p(s_0) would immediately fall apart.
The key could open the lock from the front, but it could also open it from the side.
This was a question that could truly shake the foundations of the entire presentation.
And it was a question that only Li Dong himself could answer clearly.
At least, that’s what most people in the audience thought.
But now, as Li Dong looked at Luo Yu, his eyes held a hint of... should one call it impatience, or exasperation?
’I already fucking said this is Fu Chen and Gu Ming’s presentation.’
’Why are you asking me?’
Li Dong shifted his position in his chair but still said nothing.
Luo Yu waited for a moment, but Li Dong remained silent.
At this moment, a sense of smugness began to well up inside Luo Yu.
He genuinely felt he had hit a critical point.
’That kid Li Dong doesn’t dare to answer my question.’
Luo Yu shook his head, smiled faintly, and was just about to deliver his victor’s declaration.
Just then, Gu Ming, who was on the stage, took half a step forward.
"Hello, Teacher Luo."
"There’s no need to trouble Teacher Li with the question you’ve raised."
"Fu Chen and I can answer it for you."
Gu Ming was actually quite irritated as he said this.
He knew who Luo Yu was.
In that social media post from the start of the year, this specially-appointed researcher from Capital University was mentioned in the same breath as East God.
But he had also read the man’s work and, to be honest, it was very mediocre—all of it just riding on his advisor’s coattails.
’So, him being ranked alongside East God? That was an international joke.’
Moreover, after asking his question, Luo Yu hadn’t even glanced at him or Fu Chen, which made Gu Ming feel very disrespected.
’All airs and no substance.’
At this point, Fu Chen also caught on and spoke up.
"Teacher Luo."
"Gu Ming and I will answer your question together."
He nodded to Gu Ming. Gu Ming walked to the laptop and clicked a couple of times, flipping the slides backward.
The image on the screen reversed all the way from Section 5 back to the introduction.
Definition 1.1 reappeared on the large screen.
"Teacher Luo, in our definition, m_anm is not written in terms of the order of the residue."
"We wrote it as ord_{T=1}(T - a(s_0))."
"This line right here."
The pointer tapped beneath the word "ord."
"This, in itself, is our answer to your question."
The hall grew quiet. Most people were confused. ’What does he mean, "it itself is the answer"?’
Fu Chen looked at the confused faces in the audience and continued.
"If you were to calculate the residue by multiplying term-by-term according to the Jacquet-Shalika formula, the product would become undefined when two characters χ simultaneously produce a pole at s_0."
"This is the ’opening the lock from the side’ problem you just mentioned."
"If we had initially defined m_anm in terms of the order of the residue, then this correspondence would indeed fail at the position of a common pole."
"But we didn’t."
"What we wrote is the order of the zero of a polynomial at T=1."
"To use an analogy..."
Fu Chen said very seriously.
"Teacher Luo’s analogy just now was a key and a lock."
"You were worried that for the same lock, one key could open it from the front, while another key could open it from the side."
"But the door we’re using doesn’t have that kind of lock."
"We installed an electronic lock."
"Whether it’s χ1 or χ2, when you come and press the keypad, the door either opens or it doesn’t. There’s no way to get in from the side."
"In the case of a common pole, when applied to the quantity ord_{T=1}, it simply causes the order of the zero of T - a(s_0) at T=1 to increase neatly by one step."
"It doesn’t become undefined."
"It just becomes ’heavier’."
Fu Chen paused and glanced at Gu Ming.
Gu Ming continued.
"And this technique with polynomial multiplicity isn’t something we came up with."
"It’s the polynomial multiplicity method from Kisin’s 2009 paper in Annals, used for constructing the stack of finite flat group schemes."
"Kisin’s original purpose for constructing this stack was to translate the geometric multiplicity on the deformation ring into polynomial multiplicity in the sense of Hilbert-Samuel."
"All we did was apply the same translation, but in reverse."
"We didn’t touch the geometric multiplicity on the deformation ring side."
"What we changed was the L-function residue side."
"We used the same language of polynomial multiplicity to ’digest’ the issue of common poles."
After Gu Ming finished speaking, the hall fell completely silent.
An old professor in the front row, who had worked on local representations of GL2 for thirty years, first froze for a moment, then his brows slowly furrowed.
He was very familiar with Kisin’s Annals paper.
He himself had used the polynomial multiplicity calculation on that stack in the last two years.
But he had used it in the direction of the deformation ring.
He had never thought of turning the method around and applying it in reverse to the L-function residues.
Similar reactions were occurring almost simultaneously throughout the hall.
Several professors working on the p-adic Langlands program lowered their heads and began sketching diagrams in their notebooks.
Those who worked on automorphic representations exchanged glances, but no one spoke.
As for Luo Yu...
The expression on his face had turned ugly.
He had been debated to this point by a first-year grad student and a third-year undergrad. If he didn’t say something now, he would lose all face today.
His mind raced, and he finally managed to grasp at a straw.
"Of course I’ve read Kisin’s Annals paper."
Luo Yu tried to keep his tone as steady as possible.
"The method is elegant, yes, but in Kisin’s paper, polynomial multiplicity is used on the local rings of the deformation ring side, not the L-function side. For you to use it in reverse, this step..."
He thought for a moment, searching for the right words.
"This step... so far, you two are the only ones who have done this."
"Until a second group has independently verified it, one has to question whether this kind of cross-application is truly valid."
"Is it really appropriate to bring up an unverified method in a setting like this?"
A low murmur of discussion rose from the audience.
Many professors actually thought Luo Yu had a point.
Kisin’s polynomial multiplicity had originally only been established on the deformation ring side. Applying it in reverse to the L-function side truly did require another round of verification from a peer-review perspective.
On the stage.
Gu Ming glanced at Luo Yu, then at the doubtful faces in the audience.
Suddenly, for a brief moment, he felt himself channeling the badass vibe of East God.
The corner of Gu Ming’s mouth tilted up slightly.
"Teacher Luo."
"Kisin’s exact words regarding our reverse application were..."
He deliberately spoke slowly.
"’Exquisitely ingenious.’"
All the whispering in the hall came to an abrupt halt.
Luo Yu’s mind went blank with a BUZZ.
’Kisin?’
’Kisin has already seen this work?’
Before Luo Yu had a chance to speak, Fu Chen had already started.
"Let me add something to Teacher Luo’s comment about it being ’unverified’."
"The content of our presentation today is from a paper we submitted to Compositio at the end of this year."
"It’s being published in the next issue."
From the audience, someone could be heard sharply drawing in a breath.
Fu Chen, however, continued.
"This reverse application method that Gu Ming and I presented today—the entire paper—was accepted after major revisions following two rounds of peer review."
"That phrase just now, ’exquisitely ingenious,’ came directly from the reviewer’s comments."
"And one of the reviewers... was Kisin."
The hall erupted in an instant.
’A first-year grad student and a third-year undergrad, publishing in Compositio?’
There wasn’t a single person present who didn’t know what tier of journal Compositio was.
Many of the people sitting there had spent half their lives without ever getting their name published in it.
And now, these two students from Li Dong’s research group were already ahead of them?
Only then did Li Dong stand up, nodding at Fu Chen and Gu Ming.
’Not bad, you two. That flex was at least half as good as one of mine.’

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