Thursday, October 29, 2026

Time: 16:10-17:00

venue: Mathematics Building Room 527

Speaker:  Prof. Yao Cheng (Department of Mathematics, National Central University)

Title:An introduction to the Birch and Swinnerton-Dyer conjecture

Abstract:

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. The set $E(\mathbb{Q})$ of rational points on $E$, with the point at infinity as its identity element, forms an abelian group. By Mordell’s theorem, $E(\mathbb{Q})$ is finitely generated, and hence \[ E(\mathbb{Q})\cong\mathbb{Z}^r\oplus T \] for some finite abelian group $T$ and some integer $r\geq 0$, called the rank of $E$. It is known that the order of $T$ is at most $16$; hence, $r$ may be regarded as a measure of the size of $E(\mathbb{Q})$.

On the other hand, one can attach to $E$ a Dirichlet series $L(E,s)$, which initially converges absolutely only for $\Re(s)>3/2$. By the Modularity Theorem, proved by Wiles and others, $L(E,s)$ extends holomorphically to the entire complex plane. It therefore makes sense to consider its order of vanishing $r'\geq 0$ at $s=1$. The celebrated Birch and Swinnerton-Dyer conjecture asserts that $r=r'$.

In this talk, we will provide an elementary introduction to the Birch and Swinnerton-Dyer conjecture and survey some of its recent developments.