The Bloch-Kato conjecture (BKC) is an important and difficult problem in number theory and arithmetic geometry, which came up with Bloch and Kato in 90s, and was then refined by Fontaine and Perrin-Riou later. It is a natural generalization of the Birch and Swinnerton-Dyer conjecture (BSD), which is one of Millennium Problems. However, we only know BKC is true in few cases. 

The seminar will give a brief introduction to the motivation and historical development of BKC by some explicit examples and explain its relation with the BSD conjecture. In the second part, we will talk about some progresses in BKC.