Colloquium / Seminars
Topic: An introduction to the Birch and Swinnerton-Dyer conjecture
Speaker:Prof. Cheng, Yao
(Dept. of Mathematics, Tamkang University)Date time:May 17, 2022, 14:00 - 15:00
Venue:SA223
Abstract:
Abstract
Let $E$ be an elliptic curve over $\mathbb{Q}$. Together with the point at infinity, the set of rational points $E(\mathbb{Q})$ on E becomes an abelian group. By Mordell's theorem, the abelian group $E(\mathbb{Q})$ is finitely generated and hence $E(\mathbb{Q})\cong\mathbb{Z}^r\oplus T$ for some finite (abelian) group $T$ and an integer $r\ge 0$ called the rank of $E$. It is known that the order of $T$ is less than $16$ and hence $r$ measures the size of $E(\mathbb{Q})$. On the other hand, to $E$ one can attach a Dirichlet series $L(E,s)$ which originally only converges absolutely for $Re(s)>3/2$.
By the Modularity theorem of Wiles et. al., $L(E,s)$ has holomorphic continuation to the whole complex plane and hence it makes sense to consider the order of vanishing $r'\ge 0$ of $L(E,s)$ at $s=1$. The celebrated Birch and Swinnerton-Dyer conjecture asserts that $r=r'$. In this talk, we will give an introduction to the Birch and Swinnerton-Dyer conjecture and also survey some of its recent developments.
Download:Talk_1110517.pdf
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