Report
Teaser, summary, work performed and final results
Periodic Reporting for period 3 - Euler systems (Euler systems and the Birch--Swinnerton-Dyer conjecture)
Teaser
\"Let E be an elliptic curve over the rational numbers: that is, E is a non-singular algebraic curve defined byequation of the formy² = x³ + ax + b,where a and b are rational numbers. Understanding the set of rational points on such curves E is one of thecentral problems of...
Summary
\"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. I propose to solve new cases of the conjecture and its generalisations via an algebraic tool called an Euler system. Until recently, only four examples of Euler systems were known to exist. My collaborators and I have developed a programme leading to the systematic construction of new Euler systems, which should have a wide range of arithmetic applications, bringing new cases of the BSD conjecture within reach.\"
Work performed
My collaborators and I have made substantial progress towards to goal of the proposal.
1) Together with A. Lei and D. Loeffler, I have completed the construction of an Euler system for the Asai representation attached to Hilbert modular forms. This project was mentioned in the proposal as being in the initial stages; the main new work is the proof of the Euler system norm relations.
2) Together with D. Loeffler and C. Skinner, I have established a criterion for the non-triviality of the Euler system classes in 1). Proving that the Euler system is non-zero is one of the most challenging problems in our programme, and we have given a computable criterion for testing this in concrete examples.
3) Together with D. Loeffler and C. Skinner, I have completed the construction of an Euler system for genus 2 Siegel modular forms. We have developed a new, general approach for proving the Euler system norm relations which will be applicable to other Euler systems.
4) Construction of an Euler system for certain twists of the symmetric square of a modular form, and proof of one inclusion of the Iwasawa Main Conjecture
5) Survey article (joint with D. Loeffler) developing the theory of Euler systems with local conditions
6) In progress, with D. Loeffler, V. Pilloni and C. Skinner: Construction of p-adic L-functions for genus 2 Siegel modular forms and for the Asai representation of quadratic Hilbert modular forms.
7) In progress (J. Rodrigues and A. Cauchi): construction of an Euler system for GSp(6)
Final results
The goals of the project are to prove new cases of one of the major open problems in mathematics, the Birch--Swinnerton-Dyer conjecture. This problem is of central importance in number theory, one of the oldest branches of mathematics, and it is linked to many other mathematical fields. Beyond pure mathematics, number theory also has a range of surprising and important real-world applications, such as in the design of secure communications systems (cryptography). Thus progress achieved in the project has the potential to lead to breakthroughs in several mathematical fields, and to have concrete applications to technology and computing.