Arrangement

The computer processor ran at high speed. The Taylor expansion of L(E,s) at s=1 had an order equal to the Hasse–Weil rank r(E) of the Elliptic Curve E.

The rational points on the Elliptic Curve danced a mysterious dance under the computer's direction. They seemed chaotic, but in fact, they moved to a rhythm.

Even simple Elliptic Curves of the form y^2=x^3+ax+b had infinitely many rational points, and there were infinitely many such curves.

The conditions for modular curves had been firmly established by the three students, Zhao Tian and the others, because in 1999, Kato and Kazuya had proved that all Elliptic Curves over the rational numbers were modular curves.

It was a grand ball of numbers and symbols. The primes p took the stage, spinning and leaping through a sea of data. They counted the number of rational points on certain Elliptic Curves, turning it into a count of solutions modulo the prime p.

The computer screen had a black background, as profound as the depths of the universe. The dancers in the darkness were a host of white sprites. Under the oppressive force of ordp(L'(E,1)/CE)=ordp(#Sha(E)), the p's could barely control their calculations. Their breathing grew ragged, their steps grew chaotic, and they teetered on the brink of madness.

Watching the numbers and symbols constantly change on the screen, Xiao Yun was fascinated. "This is the beauty of mathematics, the power of mathematics. The secrets of the universe are hidden here."

"Mm." Zeng Han nodded in agreement.

Verification progress: 2%.

Even with a high-performance computer, verifying the theoretical conjecture Ou Ye had constructed would take a week.

To Xiao Yun and Zeng Han, the mathematical world formed by the vast number of Elliptic Curves and equations possessed an unparalleled beauty and sense of power.

Perhaps because their prefrontal