The Finite Group!

Elliptic curves appeared everywhere in modern mathematics. Mathematicians specializing in other branches had begun studying them, and the pioneer was the arithmetic prodigy Poincare. In 1901, this once-in-a-century polymath was the first to observe that every elliptic curve was associated with a particular group.

Many mathematicians had studied elliptic curves using methods from Group Theory, achieving significant results.

The theory of elliptic curves had attracted mathematicians' attention thanks to the brilliant Poincare, while its flourishing owed much to another kind of genius: the hardworking Wiles.

Andrew Wiles had used elliptic curves to prove Fermat's Last Theorem. He should have been grateful to Yutaka Taniyama and Goro Shimura for making their conjecture years earlier.

The Taniyama-Shimura Conjecture provided an effective path forward and was an important step in the proof of Fermat's Last Theorem.

Wiles first proved the Taniyama-Shimura Conjecture. Not long afterward, he and Richard Taylor successfully solved Fermat's Last Theorem.

Yutaka Taniyama, one of the people who had proposed the Taniyama-Shimura Conjecture, had glimpsed the truth earlier than Wiles, but had been unable to prove his own conjecture.

At just 31, Yutaka Taniyama jumped into the sea and took his own life on the eve of his wedding. A month later, his fiancée died by suicide as well. It was perhaps the most tragic episode in the History of Mathematics.

Elliptic curves had brought not only success and joy, but also failure and sorrow.

After more than a century of development, scholars studying elliptic curves today were still experiencing both triumphs and setbacks.

The number-theoretic BSD Conjecture could, in fact, be reformulated as a problem about elliptic curves.

Ou Ye was not the first to think of using Group Theory together with elliptic curves to prove the BSD Conjecture.

Others had tried long ago, but without success.

The computer