Let E be an elliptic curve over the rational numbers: that is, E is a non-singular algebraic curve defined by
equation of the form
y² = x³ + ax + b,
where a and b are rational numbers. Understanding the set of rational points on such curves E is one of the
central problems of algebraic number theory. These rational points form a finitely-generated abelian group,
and the rank of this group (which is a rough measure of the "density" of the rational points on E) is therefore
finite; however, it is not easy to compute, and in fact there is no known algorithm which can be guaranteed to
calculate it.
A breakthrough in understanding these ranks came in the 1960's when Birch and Swinnerton-Dyer
formulated a conjecture (the BSD conjecture)- relating the ranks
of an elliptic curve E to the Hasse--Weil L-function of E.
This is a complex-analytic function of one variable, L(E, s), which is defined as an infinite product, with a
term for each prime p defined in terms of the reduction of E modulo p. The conjecture predicts that even
though this analytic object is built up from purely local information, it in fact encodes global information: its
order of vanishing at the point s=1 should equal the rank of the elliptic curve E.
The BSD conjecture is considered one of the major open problems in pure mathematics, and it was chosen as one of the Clay Millennium Maths Problems. Since its formulation in the 1960s, the BSD conjecture has been vastly generalized; for example, it can be formulated for so-called 'abelian varieties', of which elliptic curves are a special case. The most general version of the BSD conjecture is the Bloch--Kato conjecture, which applies to a very general class of objects called "p-adic Galois representations".
One of the most powerful tools for attacking the BSD conjecture and its generalisations is an algebro-geometric tool called an Euler system. Until recently, only four examples of Euler systems were known to exist. During the project, my collaborators and I have developed a systematic approach to constructing new Euler systems, thereby more than doubling the number of known Euler systems. In order to use an Euler system for deducing new cases of the BSD and Bloch--Kato conjectures, one needs to relate it to values of the L-function of the underlying algebraic object. My collaborator David Loeffler and I succeeded in proving such a relation for one of our Euler systems, thereby deducing many new cases of the Bloch--Kato conjecture and of the Iwasawa Main conjecture which can be seen as a p-adic analogue of the Bloch--Kato conjecture. As a special case of this work, we will obtain new cases of the Birch--Swinnerton-Dyer conjecture for abelian surfaces and for elliptic curves defined over imaginary quadratic fields; this part of the project is still work in progress, which we hope to complete within the next few months.
