You, You, and You

Zhao Tian chuckled twice, "Hehe, just researching it for fun."

"Then let's study it together." Ou Ye walked to the blackboard and carefully observed the derivation formulas on it.

"Sister Yezi, please sit down." Zeng Han pulled out a chair and placed it behind Ou Ye.

"Hmm." Ou Ye sat down.

"Sister Yezi, have some water." Xiao Yun poured a cup of hot water and handed it to Ou Ye.

"Thank you." Ou Ye took the water glass but didn't drink it; she just used it to warm her hands.

Tick-tock, tick-tock, time flies.

Ou Ye remained silent, and the three students were equally silent.

This may be their research method.

After a long silence, Ou Ye asked, "Xiao Zhao got into our math department for graduate studies, right?"

Zhao Tian nodded repeatedly. "Yes, yes, I really wanted to choose Sister Yezi as my master's advisor, but Sister Yezi 're not taking on any students lately, so I can only choose Professor Liu."

"Oh, Professor Liu is a good teacher; he's very knowledgeable in Group Theory," Ou Ye said, then turned to the other two and asked, "What are your GPAs?"

Zeng Han and Xiao Yun said in unison, "4.0."

"Oh, excellent." Although the club that Ou Ye serves as the advisor for doesn't have many members, every single one of them is an elite student.

After a brief chat, Ou Ye fell silent again, focusing intently on the mathematical symbols on the blackboard.

One teacher and three students, just quietly staring at a blackboard.

After a long while, Ou Ye stood up and erased the chalk writing on the blackboard, leaving only a lonely line of words. "The necessary and sufficient condition for E(Q) to be an infinite set is L(E,1)=0."

This proves the statement, which means proving the BSD Conjecture.

However, the statement "E(Q) is an infinite set if L(E,1)=0" is not in the form that Birch and Swinnerton-Dyer originally proposed. Their initial assumption was mathematically stronger than this statement, which is known as the strong BSD Conjecture.

The line on the blackboard read "Weak BSD Conjecture." The three students felt a chill; had Sister Yezi found a breakthrough through the Weak BSD Conjecture?

Below the weak BSD Conjecture, Ou Ye continued writing:

L(E, s) = c(s-1)^r + higher-order terms

It's obvious which is stronger; the following equation is stronger, as it's the expression for the strong BSD Conjecture.

Regardless of their level of expertise, outstanding Mathematics Department students can easily understand the BSD Conjecture; the challenge lies in proving it.

After hesitating for a moment, Ou Ye finally erased the weak BSD Conjecture and left the strong BSD Conjecture.

"Let's do this." Ou Ye tapped the blackboard.

"Sister Yezi, what do you mean by doing this?" Zhao Tian asked.

Ou Ye: "Prove it."

"Prove the Strong BSD Conjecture?" Zhao Tian was stunned.

Ou Ye: "Yes, we'll prove it."

"Sister Yezi, what do we mean?" Xiao Yun asked.

Ou Ye: "You, you, and you, plus me."

"Sister Yezi, are you serious?!" Zhao Tian thought he had misheard.

Ou Ye was puzzled. "Are you three faking it?"

Zhao Tian stammered, "I... I... I already said it, the three of us were just joking around with the BSD Conjecture! Take me for example, I've even studied P versus NP. I'm sure Professor Shen has studied these problems too. Studying it doesn't mean we have to prove it... We were just looking at it, we never thought about conquering it."

Zeng Han had been silent until now, suddenly spoke up, "Senior Brother Zhao has already been accepted into a graduate program. Senior Sister Yun and I don't have much to do usually, so why don't we follow Sister Yezi and learn the techniques for proving the strong BSD Conjecture?"

With a GPA easily reaching 4.0 and no interest in dating, Zeng Han and Xiao Yun usually had nothing to do. So Xiao Yun got a Hong Kong and Macau travel permit, Zeng Han spent his days writing and erasing on the blackboard, pondering various academic problems.

Ou Ye stared at Zeng Han, this tall, thin, and serious boy seemed to possess a calm and composed demeanor more typical of a science student compared to the boisterous Zhao Tian.

Feeling a little uncomfortable being stared at so directly by Sister Yezi, Zeng Han smiled shyly, his dimples appearing and disappearing on his cheeks.

A moment later, Ou Ye said, "Xiao Zeng, I remember you. You're that child prodigy who was admitted to Peking University at the age of 15."

"I wouldn't dare call myself a genius, I'm just good at solving problems." Zeng Han scratched his head, looking rather silly.

"You're not an adult yet, are you?" Ou Ye asked again.

Zeng Han said, "I'm a few months shy of turning 18."

"Great," Ou Ye nodded.

Whether it was because being admitted to Yenching University at 15 was amazing, or because being under 18 was amazing, Zeng Han was especially happy to receive praise from Sister Yezi.

Ou Ye tapped the blackboard again and said, "If you three want to do it, then do it with me. If you don't want to, then forget it."

Zeng Han was the first to speak up, "I'll do it with Sister Yezi."

Xiao Yun then raised her hand. "I'll do it with Sister Yezi too."

With his juniors all expressing their determination, Zhao Tian, as a senior and the president of the Ou Ye Computer Society, couldn't remain indifferent.

Zhao Tian puffed out his chest and said, "Count me in!"

So on this ordinary day, a frail and sickly female mathematics professor, along with three undergraduate students, launched a formal assault on one of the Millennium Prize Problem, the BSD Conjecture.

It's understandable that Ou Ye is determined to solve the strong BSD Conjecture.

It's understandable that Ou Ye needs an assistant.

It's understandable that the assistant is a doctoral or master's student.

However, Ou Ye's assistants are three undergraduates... well, one of them is a prospective master's student.

Even with Zhao Tian, a prospective master's student in the Mathematics Department, Ou Ye's assistants were still not strong enough.

At least that's how it appears.

What were Ou Ye's considerations?

She spent the whole morning observing.

L(E, s) can be extended to a function that gives a response for any complex number s, and calculus can be applied to this function. This means that L(E, s) can be represented using the well-known Taylor polynomial.

Therefore, the strong BSD Conjecture can actually be stated as: E(Q) is an infinite set if and only if cr ≠ 0, but for n = 0,..., r-1, each coefficient cn is 0, where r is the rank of E.

To put it more intuitively, counting the number of the first few zero terms in Taylor polynomial provides a measure of how much the function is zero at relevant points.

Therefore, another way to put the strong BSD Conjecture is that the rank of E gives an accurate measure of how much L(E,1) is zero.

The playful antics of these three undergraduates on the blackboard have touched upon the core of the strong BSD Conjecture.

This is Ou Ye's observation conclusion.

In mathematics, talent is immediately apparent.

Zhao Tian, Xiao Yun, and Zeng Han may lack a broad perspective, but they are intelligent, talented, and focused on mathematics. These three students are the Boss, yet none of them are in relationships; they find mathematics far more interesting than romance.

Around noon, Ou Ye said to the three students, "Go and have lunch first. We'll meet back here in an hour."