## Periodic Reporting for period 1 - Euler systems (Euler systems and the Birch--Swinnerton-Dyer conjecture)

**Reporting period:**2015-07-01

**to**2016-12-31

## Summary of the context and overall objectives of the project

"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."

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 from the beginning of the project to the end of the period covered by the report and main results achieved so far

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) Ongoing work on the construction of an Euler system for the spin representation of a genus 2 Siegel modular form (joint with D. Loeffler and C. Skinner). We have developed a new, general approach for proving the Euler system norm relations which will be applicable to other Euler systems.

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) Ongoing work on the construction of an Euler system for the spin representation of a genus 2 Siegel modular form (joint with D. Loeffler and C. Skinner). We have developed a new, general approach for proving the Euler system norm relations which will be applicable to other Euler systems.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.