When the title of the presentation appeared on the projector, the conference hall fell silent.
Those who understood knew the significance of this topic within the Langlands program.
Those who didn’t understand remained quiet as well.
At a conference of this caliber, it was unwise to rashly comment on a subject one couldn’t follow.
Fu Chen stood before the microphone.
He could feel his hands trembling slightly.
But the more nervous he felt, the more he forced himself to slow down.
He knew that if he spoke too quickly, he would stumble over the convoluted terminology that followed.
The slide advanced to the next page, one he was familiar with.
Taking a deep breath, he began to speak.
"Good morning, professors. I will begin with a brief overview of our paper’s main approach."
"Everyone has likely read the paper Teacher Li published in *Annals* last year. On the final page, he left an open conjecture..."
A few people in the audience nodded, almost subconsciously.
’The Li Clan conjecture. Who hasn’t heard of it?’
Fu Chen continued.
"Teacher Li’s work on the universality of automorphic functions followed an analytic path."
"The zero-point statistics under the Riemann spectral operator brought the local-global compatibility of GL(n) automorphic representations to a stage where numerical verification was possible."
"But the conjecture itself cannot stand on the analytic leg alone."
He paused for a moment.
"Our paper addresses the other leg."
The slide changed again, this one reading: Cyclic Base Change—Anomalous Residues—Deformation Rings.
"Under cyclic base change, the L-factor, according to the factorization in Proposition 2.1, decomposes into a product of a finite number of character twists."
"We focus on a class of residues among them that do not appear as predicted by the Jacquet-Shalika formula."
"We call these ’anomalous residues.’"
As he spoke, Fu Chen noticed several professors specializing in number theory in the audience involuntarily leaning forward.
In that instant, the nervousness in his heart began to fade.
"Under a natural map, these anomalous residues correspond to the four-dimensional minimal prime ideals of the deformation ring R□(ρˉ,τ) on the Galois side."
"The correspondence is functorial—it strictly preserves multiplicative multiplicity."
"We view this correspondence as the dual of the Breuil-Mézard type correspondence on the L-function side."
"It is also a pillar we are attempting to extend into the algebraic side of the Li Clan conjecture."
The slides continued to advance.
Fu Chen himself couldn’t say when he stopped feeling nervous at all.
He was still on stage.
Facing a row of professors whose names he couldn’t all recall.
Among them were individuals whose academic achievements surpassed even his own advisor’s.
But the more he spoke, the clearer the picture in his mind became.
Local root numbers, classification of inertial types, the special fiber of the framed deformation ring, the lemma he had modified from under Shotton’s main theorem...
Every step connected back to the sleepless nights he had endured.
When he reached the section on §4, he paused slightly at the transition in the operator construction.
He had gotten truly stuck on this part before.
The final workaround that solved it was provided by Gu Ming.
Gu Ming, standing to the side, noticed a flicker of hesitation in Fu Chen’s eyes. Without a word, he walked over and added two lines to the whiteboard, completing the endpoint conditions for the operator.
Fu Chen glanced at it and continued smoothly.
Gu Ming then returned to his spot.
From start to finish, the two hadn’t exchanged a single word.
Time slowly ticked by. Thirty minutes had passed.
The professors in the audience had split into two groups.
One group was only half-understanding what they were hearing.
Some of them began to jot down keywords, planning to look them up after the session.
Others simply put down their pens; since they couldn’t take down everything, they decided to just absorb the general atmosphere.
The other group consisted of those who were truly following this line of research.
Their expressions were grim.
It wasn’t displeasure, but shock.
And Luo Yu’s expression was the grimmest of all.
From the moment the title appeared until now, he had barely said a word, the smile long vanished from his face.
He worked on the p-adic Langlands program.
The main thread of this presentation—from the L-factors of the cyclic base change, all the way to the geometric multiplicity of R□(ρˉ,τ), weaving from Henniart’s inertial types to Shotton’s 2018 paper...
...was all material he was intimately familiar with.
It was also the area he had been trying to tackle himself for the past two years, but without producing anything substantial.
He had listened for thirty minutes.
By now, he had to admit to himself: ’This presentation is fucking brilliant.’
But how was that possible?
’They’re a first-year master’s student and a junior undergraduate.’
’Where was I as a first-year master’s student? At G?ttingen, attending seminars, and for the first two months, I barely understood a thing.’
’And as a junior? I was just an unknown face at Princeton.’
He had never even heard the names of the two on stage now.
Luo Yu’s gaze shifted from Fu Chen and his partner on stage, finally landing on a man with his arms crossed over his chest.
Li Dong.
’This kid hasn’t moved an inch since the presentation began.’
’There’s no way these two students came up with this. This has to be Li Dong’s work!’
’...But then again.’
’Would Li Dong really just toss a presentation of this magnitude to two students and let them present it?’
’If it were him, Luo Yu, he would never be willing to part with it.’
’If this were his own work, he’d be the one on stage. Why on earth would he let two students take the spotlight for something like this?’
Fu Chen and Gu Ming, unaware that someone now saw them as puppets, continued with their presentation.
At the fifty-minute mark.
Fu Chen displayed the final summary slide and bowed to the audience.
"That concludes the main points of Gu Ming’s and my presentation. If any of the professors have questions, please feel free to ask."
The hall was silent for a second or two.
Then, the first hand went up.
It was Professor Zheng Wenbing from Nankai University.
He worked on automorphic L-functions and, while not at the absolute top, was a well-known figure in the domestic number theory community.
Zheng Wenbing’s tone was polite.
"Student Fu, let me first ask a question about the general direction of your work."
"Your correspondence’s main theorem is currently only written down for GL2. For the conjectural generalization to GLn in Section 7, are you planning to take a global approach, or will you continue to build upon local results?"
Fu Chen paused for half a second after hearing the question.
This kind of question wasn’t hostile, but it was difficult to answer.
Answering with the local approach would make them seem to lack ambition, but claiming a global approach would immediately invite follow-up questions about whether they had the necessary tools.
Fu Chen smiled faintly.
"At this stage, we are inclined to first clearly compute the explicit deformation rings for GL3 locally. Shotton’s path is not viable for a general n, and our group currently doesn’t have any more suitable tools at hand."
"As for the global path, we can, in theory, leverage Teacher Li’s work."
"But how much to borrow, and when, are things I haven’t fully considered yet."
Zheng Wenbing nodded and sat down.
Immediately after, a second person raised their hand.
It was Associate Professor Qiu Jinnan from the University of Science and Technology of China, who specialized in Galois representation deformations.
He had published two papers in *Compositio* in the last two years and had made quite a name for himself among the mid-career generation of mathematicians.
Qiu Jinnan’s question wasn’t a trap either.
"In your Theorem 1.3, the correspondence, specifically property (P2)—the total multiplicity of anomalous residues equals the total multiplicity of the four-dimensional cyclic Z? of R□(ρˉ,τ)."
"My question is, has this equality currently only been verified for a few specific residual representations, or has it been proven for all inertial types?"
It was a very solid question.
Fu Chen wasn’t flustered; he answered with confidence.
"(P2), in the case where ρˉ|PeF is non-scalar, we followed the direct deduction in §4."
"In the several sub-cases where ρˉ|PeF is scalar, we used the explicit computations from §5.2 to §5.4 of Shotton’s original paper and checked them one by one. The verification for this part was mainly done by Gu Ming."
Hearing his name, Gu Ming, who was standing beside him, nodded but didn’t add anything.
Qiu Jinnan grunted in acknowledgment and also sat down.
Hearing this, Li Dong also gave a slight nod to himself.
’Fu Chen’s last sentence subtly gave Gu Ming credit, both mentioning his junior’s name and avoiding a position where he seemed to be taking all the glory.’
’Li Dong had been worried they might run into trouble, but it seemed his fears were unfounded.’
At this point, the presentation was already in its concluding phase.
Li Dong glanced around again.
No one else seemed to want to raise their hand.
’Okay, we’re in the clear.’
Just then, a hand shot up.
It was Luo Yu.
Fu Chen had obviously noticed this hand as well.
He didn’t recognize Luo Yu, only noting that the person who raised his hand didn’t seem as old as the other professors.
He nodded toward the moderator.
A microphone was passed over.
Some people in the conference hall recognized Luo Yu.
After all, the article from the beginning of the year, "Two New Stars of the Huaxia Mathematical Community," wasn’t exactly obscure in domestic circles.
The p-adic Langlands guy, a specially-appointed researcher from Capital University...
Luo Yu nodded at the stage, his tone exceptionally calm.
"Student Fu, Student Gu, hello."
"First, let me say that I listened to your presentation on this correspondence very attentively, and I’m very impressed."
"My question might be a little more on the technical side."
Fu Chen nodded politely.
Only then did Luo Yu begin to speak, slowly and deliberately.
"Your anomalous residue at point s_0 is determined by factorizing BCL(π) into a finite number of L(s, π?χ) according to Proposition 2.1, with each χ corresponding to one term. The natural residue, r_nat, is the product of the term-wise residues from this factorization."
"I have no issue with this step."
"You also wrote in Proposition 3.1 that a(s_0; π, L) = ∏χ a_χ(s_0; π)."
"The premise for this multiplicative decomposition is that for every L(s, π?χ) at s_0, the pole is simple."
Hearing this, Fu Chen’s expression shifted slightly.
Luo Yu didn’t look at him, continuing on his own.
"Let me put it another way."
"Your functorial correspondence, in essence, pairs each anomalous residue point s_0 with a unique four-dimensional minimal prime ideal p(s_0)."
"A one-to-one correspondence."
"To use an analogy, it’s like a key and a lock, a lock and a key. One-to-one."
At this point, Luo Yu’s pace slowed ever so slightly.
"But... what if two different χ simultaneously produce a pole at s_0?"
The scattered, hushed whispers in the hall vanished in an instant after that sentence.
"In this ’co-pole’ situation, your r_nat, strictly speaking, is not cleanly defined."
"In that scenario, the value of a(s_0) would be inflated by the order of the pole. I can concede that your m_anm might still hold."
"But what about p(s_0)?"
"The key I have can open the lock when inserted from the front, but it can also be inserted from the side."
"Because χ1 corresponds to one prime ideal, and χ2 corresponds to another, but they are both ’captured’ by the same s_0."
"How do you prove that p(s_0), in this case, corresponds to only that single, unique prime ideal, and not several?"
Having said this, Luo Yu tilted his head and glanced at the projector screen.
"Let’s take it a step further."
"In step three of §5, you claim that the m_0-primary decomposition of the patched module allows you to read off e(s_0)."
"What I want to know is, for that class of ’co-pole’ points, does this primary decomposition read out e(s_0) itself, or the sum of several components?"
"If it’s the latter, then doesn’t that step in §5, along with your main Theorem 5.1, become not quite as ’clean’ as you’ve written it?"
When Luo Yu finished speaking.
The hall fell deathly silent.
Even those who understood the least about this field understood the implication.
This lock could be opened from the front, but also from the side. It was no longer unique.
In mathematics, this sort of thing is called a hole.
After asking his question, Luo Yu didn’t look at the two students on stage.
His gaze went past Fu Chen and Gu Ming, landing on Li Dong.

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