AI 中文总结
本文简要介绍Birch和Swinnerton-Dyer猜想，通过回顾丢番图方程、定义椭圆曲线的代数秩与解析秩，提出二者相等的核心猜想。
AI 中文摘要
本文对Birch和Swinnerton-Dyer猜想给出简要且部分非形式化的介绍。在简要回顾基本丢番图方程后，我们定义了椭圆曲线上的自然阿贝尔群结构，进而引出相应代数秩的定义。之后，我们简述如何定义椭圆曲线的解析秩，并陈述该猜想，即这两个秩相等。
英文摘要
We give a brief and partially informal introduction to the Birch and Swinnerton-Dyer conjecture. After a short review on basic Diophantine equations, we define the natural abelian group structure on an elliptic curve, leading to the definition of the corresponding algebraic rank. Afterwards, we sketch how to define the analytic rank of an elliptic curve and we state the conjecture, that is, the equality of these two ranks.
Comments18 pages
Journal refMatemática Contemporânea, Volume 61, pages 153-173 (2025)
DOI：10.1007/s44425-025-00035-2
