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Chapter 244: Chapter 235: Li’s Conjecture

This conjecture.

It was like watching a man walk to the edge of a cliff, without a rope or a harness, and simply point into the air...

And say, ”You can cross here.”

For a moment, Langlands felt a little dazed.

He had seen countless conjectures in his life.

Some conjectures were beautiful; you had to squint to see the subtle structures hidden within.

Some conjectures were clumsy, cobbled together with heaps of evidence, their intentions obvious at a glance.

But one like this...

This was the first he’d ever seen.

Those seventy-six pages were the foundation Li Dong had laid for it.

And the few lines of text on this single sheet of A4 paper...

...allowed him to vaguely glimpse a skyscraper.

The building was immense, towering.

He could only look up at it.

He couldn’t make out its silhouette.

”Almost everywhere equal...”

Langlands seemed to be murmuring these words to himself.

Almost everywhere equal.

In real analysis, it was the most unassuming of phrases.

But here, those words carried immense weight.

For the pair correlation function, they carried the statistical information of the zeros.

And the statistical information of the zeros was the deepest, most recently discovered aspect of automorphic L-functions.

’Could two automorphic L-functions with different Euler products really have sets of zeros that coincide almost everywhere?’

Langlands’s first reaction was...

’Impossible.’

But he didn’t rush to put the paper down.

He looked at the A4 sheet in his hands again.

Frank sat across from him.

He said nothing.

He simply placed a fifth cup of coffee gently by the old man’s hand.

Langlands subconsciously reached for the fountain pen on the desk.

He wanted to test it.

’This kind of thing... a conjecture... surely one can perform some small verifications?’

Langlands wasn’t sure.

But he had to probe it to know whether it would shatter at the slightest touch or stand firm.

He pulled out a blank sheet of paper and unscrewed the cap of his fountain pen.

The first thing he wrote down was a case that everyone knew inside and out.

Cyclic base change.

The base change for GL(2) over a cyclic extension E/F—something his own students, Arthur and Kloze, had completed back in 1989.

Let π be a cuspidal automorphic representation of GL(2, A_F).

E/F is a cyclic extension, with the Galois group generated by a character χ.

The L-function of the base change of π, π_E, can be written as the product of the L-functions of π twisted by the powers of χ.

L(s, π_E) = ∏ L(s, π⊗χ^k)

Langlands’s pen paused on the ”∏” symbol.

He wanted to verify the necessity part of the necessary and sufficient condition.

’In this already-proven special case, the conclusion on Li Dong’s paper should be self-consistent...’

Since π_E is a transfer of π, their pair correlation functions should be almost everywhere equal.

The old man calculated slowly on the paper.

The zero set of L(s, π_E) is the union of the zero sets of the L(s, π⊗χ^k)s.

The pair correlation function of π_E, F_{π_E}(α), should formally be split into two parts.

One part is the internal pair correlation of the zeros within each L(s, π⊗χ^k).

These have the same form as F_π(α), because twisting does not change the GUE universality.

The other part consists of the cross-correlation terms between the zeros of the different L(s, π⊗χ^k)s.

Langlands’s pen stopped.

This cross-term.

’According to Li Dong’s criterion, it should dissipate in the interval [0, 4/n] into...’

He slowly continued his calculation.

Halfway through his calculation, his brow furrowed slightly.

Frank watched his furrowed brow.

His own heart leaped into his throat.

After a few more minutes, Langlands’s tightly furrowed brow slowly relaxed.

The part of the cross-term that had seemed wrong was strongly suppressed by Li Dong’s ramification index constraint, e_v ≤ n.

Suppressed to the point of being almost everywhere zero.

Langlands let out a soft ”Mm.”

The necessity direction held up in this special case of cyclic base change.

But that wasn’t enough.

The necessity direction was too easy.

If functoriality holds, the L-functions are equal, so the zeros are equal, and the pair correlation functions are naturally equal as well.

What he really wanted to test was the other half—the converse.

’If two cuspidal automorphic representations have pair correlation functions that are almost everywhere equal, are they necessarily related by functoriality?’

Langlands picked up another sheet of paper.

He was going to look for a counterexample.

A counterexample that would debunk this conjecture with a single touch.

The first thing that came to mind was two Galois-conjugate automorphic representations.

Their L-functions look very similar at first glance, but the transfer between them does not correspond to any L-homomorphism within Langlands functoriality.

Langlands smiled.

’I should be able to shatter this seemingly perfect conjecture with a single stroke.’

He lowered his head, his pen flying across the paper as he deconstructed and calculated the pair correlation functions of the two representations step by step.

In less than ten minutes, the pen in the old man’s hand fell gently onto the paper.

The result was completely unexpected.

Under Li Dong’s zero criterion, the pair correlation functions of this seemingly perfect pair of conjugate representations couldn’t achieve ”almost everywhere equal” at all.

In an extremely narrow yet crucial interval, the two sets of values would completely diverge, the difference too clear to be ignored.

It didn’t even satisfy the conjecture’s core premise; it didn’t qualify as a counterexample.

Langlands took another blank sheet of paper.

He tried the second-trickiest weapon in the field for finding loopholes: CAP representations.

This thing was a master of disguise.

It looks almost identical to a qualifying cuspidal automorphic representation and can easily slip into the preconditions, but it is essentially a ”pseudo-cuspidal” representation arising from a residual representation on a smaller group, and by its very nature, it does not satisfy the requirements of Langlands functoriality.

The work of countless colleagues had been undone because they failed to guard against this imposter.

But this time, Langlands stopped after writing only a few lines.

He didn’t even need to finish the calculation; he had already reached a conclusion in his mind.

Li Dong’s conjecture had placed an iron gate at the very first step.

”Both satisfy the zero criterion for the local-global compatibility of automorphic representations.”

This imposter couldn’t even pass this first security check. It was stopped at the door, denied even the chance to touch the conjecture’s core conclusion.

Frank sat opposite him, watching all this in silence.

In truth, he himself had done the exact same thing with that A4 sheet four days ago.

He had chosen a few cases he was most familiar with, trying to debunk the conjecture.

He ended up spending the entire afternoon trying.

After he was done, he sat in his office and stared out the window in a daze for a full half hour.

Only then did he decide to buy a plane ticket to Princeton.

Now, Langlands switched to yet another sheet of paper.

This time, he tried an even trickier scenario...

...a case where, under a nontrivial L-homomorphism, two representations match locally at almost all places but fail at a finite number of bad places.

This kind of thing was the most troublesome for the traditional trace formula method.

But Li Dong’s conjecture didn’t use the trace formula.

It used the pair correlation of zeros.

The pair correlation of zeros is a global property; it doesn’t depend on any single bad place.

Langlands stared at the few lines he had calculated.

He didn’t move for a long while.

Finally, he set down his pen.

”Remarkable,” the old man said in a low voice.

”Truly remarkable.”

He looked up at Frank.

”Frank.”

”On my end, I find no issues.”

Frank froze.

He had actually anticipated this.

But hearing those words from Langlands himself was different.

It meant...

...that at his level, at least, the conjecture hadn’t shattered. It had stood firm.

From now on, this conjecture could be called ”Li’s Conjecture.”

Frank nodded silently and stood up.

He glanced at the now-cold cup of coffee on the desk, then at the old clock on the wall.

Since arriving this morning, the old man had barely stood up at all.

He had spent four hours just reading the paper.

Then he spent another half an hour or more doing calculations based on that A4 sheet.

Frank desperately wanted to keep talking with him.

To talk about the A4 sheet, about the entire paper, about the young man named Li Dong.

But he couldn’t.

Langlands’s body couldn’t handle such long hours of work.

”Professor,” Frank lowered his voice.

”Then I won’t disturb your rest any longer.”

Langlands nodded.

”I’ll hold on to the paper.”

”I’ll write up my review in a few days.”

”Alright.”

Frank gave a slight bow, picked up his briefcase, and left quickly.

He knew he had to get back to Chicago immediately.

The paper needed to officially enter the peer review process for the *Mathematical Annals*. Besides Langlands, letters had to be sent out as soon as possible to the other external reviewers who could handle this subject.

One was at IHES, one at MIT, and another at Oxford.