Four Steps

Zhao Tian, Xiao Yun, and Zeng Han went to Yanjing University People's Hospital to visit Ou Ye.

Just after waking up, Ou Ye handed her manuscript to the three students and carefully explained her ideas to them.

Ou Ye had organized a clear outline for the proof of the Strong BSD Conjecture. The approach worked backward, using each step to force the one before it.

The final step was to prove the Strong BSD Conjecture—that is, to prove the following statement: E(Q) is infinite if and only if the Taylor expansion of L(E,s) at s=1 has the form L(E,s)=c(s-1)^r+higher-order terms, where c≠0 and r is the rank of E.

The penultimate step was to prove the statement above, which required counting the rational points on elliptic curves.

The step before that was to count the rational points on elliptic curves, which required first establishing the rank of an elliptic curve.

The step before that was to establish the rank of an elliptic curve, for which one could consider using Group Theory.

After Ou Ye and her three students had worked tirelessly, the team had reached the fourth step from the end.

"Actually, the fourth step from the end can also be considered the first step from the beginning. It takes the longest. If we spend two years on this step, we can finish the remaining three in two months... Ha... yawn..." Although Ou Ye was in poor health, her mathematical thinking was perfectly clear.

Ou Ye had just woken up, but she kept yawning. Her three students said, "Sister Yezi, get some rest. We know what to do! Sleep for a while. We'll get going."

The three students carefully packed up Ou Ye's manuscript, then left the hospital and returned to Yanjing University.

The small room at the end of