Euler systems are one of the most powerful tools for proving new cases of the Birch--Swinnerton-Dyer and Bloch--Kato conjectures. In my lectures, I will give an introduction to the theory of Euler systems, and give some examples of how to construct and where to find them.

One special case of the Birch--Swinnerton-Dyer conjecture

is the statement that if E is an elliptic curve over a number field,

and the L-function of E does not vanish at s = 1, then E has only

finitely many rational points and its Tate-Shafarevich group is

finite. This is known to be true for elliptic curves over Q by a

theorem of Kolyvagin.

Kolyvagin's proof relies on an object called an 'Euler system' -- a

system of elements of Galois cohomology groups -- in order to control

the Tate-Shafarevich group. It has long been conjectured that Euler

systems should exist in other contexts, and these should have

similarly rich arithmetical applications; but only a very small number

of examples have so far been found. In this talk I'll describe the

construction of a new Euler system attached to pairs of elliptic

curves -- or more generally pairs of modular forms -- and its

arithmetical applications to the Birch--Swinnerton-Dyer conjecture.

This is joint work with Antonio Lei and David Loeffler, and it has

recently been generalised by Guido Kings, David Loeffler and myself