Class group, lwasawa theory, $p$-adic field, $L$-functions, elliptic curves, Selmer group, Birch and Swinnerton-Dyer conjecture.
Abstract
This article is a tribute to the rich and enduring contributions of John Coates in lwasawa theory. It is beyond the scope of the current article to give a detailed exposition on the entire range of the work of John Coates, as it would involve deep and technical aspects of number theory. We shall therefore focus only on some parts of his work, aiming to keep the exposition as simple and accessible as possible. The work of Kenkichi lwasawa on the study of class groups in arithmetic deeply influenced John Coates and we shall therefore begin with a review of these ideas.
