In his celebrated work, Kato constructed the Euler system of Beilinson--Kato elements and proved spectacular results on the Iwasawa main conjecture and the Birch and Swinnerton-Dyer conjecture by using the Euler system. We study his work in this mini-course with emphasis on the case of elliptic curves.

1. Birch and Swinnerton-Dyer conjecture and the Iwasawa main conjecture: an overview

2. Iwasawa theory for elliptic curves: a summary

3. The methods of Euler systems and Kolyvagin systems

4. The construction and properties of Kato's zeta elements: a sketch

5. Further developments (if time permits)

**References:**

Delbourgo, Elliptic curves and big Galois representations, 2008

Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque, 2004

Mazur and Rubin, Kolyvagin systems, Memoir of AMS, 2004

Rubin, Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic Geometry, 1998

Rubin, Euler systems, Annals of Mathematics Studies, 2000

Scholl, An introduction to Kato's Euler systems, Galois Representations in Arithmetic Algebraic Geometry, 1998