Chapter 256 - 246: No Results Within 100 Years

Chapter 256: Chapter 246: No Results Within 100 Years

However, Wiles completely failed to notice their expressions and continued.

"The core of your conjecture is what you might call the ’Functoriality Zero-Point Equivalence Criterion.’"

"Are you trying to establish the almost-everywhere equality of the zero-pair correlation functions of automorphic L-functions as a necessary and sufficient condition for Langlands functoriality to hold?"

The question cut straight to the heart of the matter.

A bit of explanation is needed here.

To put it in layman’s terms, the conjecture Li Dong proposed at the end of his paper was this:

If two automorphic representations satisfy the Langlands correspondence, then the zero-pair correlation functions of their automorphic L-functions must be equal almost everywhere.

Conversely, if the zero-pair correlation functions are identical and local-global compatibility is added to the mix, then functoriality becomes automatic.

In short, Li Dong wanted to use zero-point statistics from analytic number theory to create an equivalent translation of the purely algebraic functoriality conjecture within the Langlands Program.

For decades, all research into functoriality had followed the purely algebraic path of the Arthur-Selberg Trace Formula.

The fundamental lemma was only proven by Wu Baozhu in 2010, and Laforgue had used geometric methods to establish functoriality over function fields.

No one had ever approached it from the angle of the statistical distribution of zeros of automorphic L-functions.

Li Dong thought for a moment before speaking.

"My current thinking is to first prove the base change functoriality from GL(n) to GL(m)..."

"Because the Euler product structure of the base change L-function, L(s, π, BC), is well-defined, its zero distribution should be completely identical to that of the original L-function."

"If my zero-point criterion truly holds... the existence of the base change would be an automatic corollary."

Wiles’s eyes narrowed.

"So you want to re-prove the work of Arthur and Clozel in its entirety?"

"More than that." Li Dong didn’t feel the least bit embarrassed.

"I want to generalize it to general reductive algebraic groups."

It was a bombshell.

At that, even Guan An’s expression changed.

The theory of several complex variables and the Langlands Program were inextricably linked, so Guan An understood the implications.

He couldn’t help but interject.

"Li Dong, for functoriality on general reductive algebraic groups... what are you using as the necessary and sufficient condition?"

"The zero-point criterion with local-global compatibility, combined with the almost-everywhere equality of the correlation functions under L-homomorphism."

Li Dong’s reply was swift.

For this answer to be valid, there was an implicit premise.

The zero-point criterion itself would have to be universally applicable at the level of general reductive algebraic groups, not just for GL(n).

This would be like taking the GL(n) result and elevating it to a much higher level.

Wiles’s breathing visibly quickened.

The old professor stood up.

"A whiteboard. Do you have a whiteboard?"

Liu Ruochuan snapped out of his daze and quickly had someone bring in a whiteboard.

...

And then, the atmosphere in the room began to slowly change.

At first, their purpose for being there was quite simple: to host Professor Wiles, have some tea, and exchange pleasantries with the old man.

But now...

Wiles and Li Dong each had a marker and stood before their own whiteboards.

Guan An interjected with a sharp point from the perspective of several complex variables. Liu Ruochuan followed up along the lines of Algebraic Number Theory. Even Tian Gang couldn’t resist, sketching out half a trace formula on the side.

As for the Frenchman... Mateo Levan.

He had long since fallen silent.

His reason for being here today was simple: he and Maxim Raziwill were old friends.

Who was Raziwill? (For readers who may have forgotten, please refer to Chapter 169.)

He was Li Dong’s competitor for the 2026 Fields Medal.

Levan had originally been on his way to Hecheng.

But when he heard that Wiles had made an impromptu trip to Yan University, he changed his plans. He held the old professor in high esteem and wanted to come over to pay his respects.

And, of course, to get a look at the young man who was putting so much pressure on his old friend.

At this very moment...

He was already sighing silently for his old friend.

’Maxim, oh Maxim, just accept your fate.’

He was in applied mathematics, but he also did research in pure math.

He could tell that when it came to the Langlands Program, Wiles was, surprisingly, showing faint signs of being overshadowed.

’Overshadowed’ wasn’t quite the right word. More often than not, it was Li Dong who was guiding Wiles.

And Wiles, like a young man rediscovering his first academic love, followed along with shining eyes, moving forward together with him.

’This time, the Fields is a lost cause.’

Levan sighed to himself.

...

Professor Wiles himself felt more clear-headed than ever.

He felt ten years younger, his mind working faster than ever before.

He wished this discussion could go on forever, without end.

What he didn’t know, however, was that...

Li Dong, on the other hand, was nearly scared out of his wits.

His Passive Skill, [Inheritance of the Torch (Basic Edition)], had been active ever since he walked through the door.

As the one who had proposed the conjecture, Li Dong was able to guide Wiles in certain directions he had preconceived.

The problem was...

...the moment he brought something up, Wiles would immediately catch on.

He could even extrapolate from it and push the idea one step further.

And that single step often made the edifice in Li Dong’s mind, which had previously been just a silhouette, that much clearer.

Li Dong grumbled internally.

’Damn, this old man’s Basic Attributes are probably even higher than mine, aren’t they?’

His current Basic Attributes were: Logic 0.4, Memory 0.6, and Concentration 0.4.

He was willing to bet that in two areas—Logic and Concentration—Wiles was definitely superior to him.

Memory was harder to say.

After all, he had a Memory Palace.

Comparing just the raw stats, he might not even be a match for the old man who had spent seven years mulling over Fermat’s Last Theorem.