Iwasawa theory was introduced around 1960 in the context of class groups of $\mathbb{Z}_p$-extensions of number fields. In the 1970's the ideas of Iwasawa theory were extended to elliptic curves: the Main Conjectures relate certain Galois cohomology groups (the algebraic side) with $p$-adic L-functions (the analytic side). Like the original Main Conjecture of Iwasawa, these Main Conjectures have consequences, via the so-called control theorems, for the related special value formulas and one key tool used to prove them is the theory of Euler systems. In this talk, I’ll give an overview of these ideas in the case of elliptic curves and explain how, making use of the Euler system of Heegner points, these results can be proved and used to deduce the p-part of the Birch-Swinnerton-Dyer conjecture when the rank is at most one.
