Periodic Reporting for period 4 - Euler systems (Euler systems and the Birch--Swinnerton-Dyer conjecture)
Période du rapport: 2020-01-01 au 2020-12-31
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.
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.
My collaborators and I have made substantial progress towards to goal of the proposal.
1) (with A. Lei and D. Loeffler) 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) (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) (with D. Loeffler and C. Skinner) 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) (with D. Loeffler) 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) on a general theory of Euler systems with local conditions
6) (with D. Loeffler, V. Pilloni and C. Skinner) Construction of a p-adic L-functions for genus 2 Siegel modular forms, using the recently developed tool "Higher Hida theory"
7) (J. Rodrigues and A. Cauchi) Construction of norm-compatible families in the motivic cohomology of GSp(6)
8) (with D. Loeffler) Proof of an explicit reciprocity law for the Euler system constructed in 3); proof of the Bloch--Kato conjecture for genus 2 Siegel modular forms in analytic rank 0
9) (with D. Loeffler) Proof of the cyclotomic Iwasawa Main Conjecture for quadratic Hilbert modular forms
10) (with D. Loeffler) Proof of the Bloch--Kato conjecture for the symmetric cube of modular forms in analytic rank 0
11) Survey article (with D. Loeffler) Development of a theory of Euler systems, p-adic L-functions and explicit reciprocity laws for GSp(4) x GL(2) and GSp(4) x GL(2) x GL(2)
12) (with D. Loeffler) Proving a relation between the GSp(4) x GL(2) Euler system and values of a p-adic L-function; this is the crucial step towards proving new cases of the Bloch--Kato conjecture in this case.
13) (A. Pozzi, H. Darmon and J.Vonk) The values of the Dedekind-Rademacher cocycle at real multiplication points.
14) (A. Pozzi, A. Betina and M. Dimitrov) On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve.
1) (with A. Lei and D. Loeffler) 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) (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) (with D. Loeffler and C. Skinner) 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) (with D. Loeffler) 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) on a general theory of Euler systems with local conditions
6) (with D. Loeffler, V. Pilloni and C. Skinner) Construction of a p-adic L-functions for genus 2 Siegel modular forms, using the recently developed tool "Higher Hida theory"
7) (J. Rodrigues and A. Cauchi) Construction of norm-compatible families in the motivic cohomology of GSp(6)
8) (with D. Loeffler) Proof of an explicit reciprocity law for the Euler system constructed in 3); proof of the Bloch--Kato conjecture for genus 2 Siegel modular forms in analytic rank 0
9) (with D. Loeffler) Proof of the cyclotomic Iwasawa Main Conjecture for quadratic Hilbert modular forms
10) (with D. Loeffler) Proof of the Bloch--Kato conjecture for the symmetric cube of modular forms in analytic rank 0
11) Survey article (with D. Loeffler) Development of a theory of Euler systems, p-adic L-functions and explicit reciprocity laws for GSp(4) x GL(2) and GSp(4) x GL(2) x GL(2)
12) (with D. Loeffler) Proving a relation between the GSp(4) x GL(2) Euler system and values of a p-adic L-function; this is the crucial step towards proving new cases of the Bloch--Kato conjecture in this case.
13) (A. Pozzi, H. Darmon and J.Vonk) The values of the Dedekind-Rademacher cocycle at real multiplication points.
14) (A. Pozzi, A. Betina and M. Dimitrov) On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve.
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).
During the last five years, my collaborators and I have developed all the tools necessary to prove many new cases of the Birch--Swinnerton-Dyer conjecture, for a class of objects called 'abelian surfaces'. We are now in the process of putting together all these pieces; we expect the work to be complete within the next few months. Our work has already had applications to a number of other important open problems in mathematics, namely to the Bloch--Kato conjecture and the Iwasawa Main conjecture.
During the last five years, my collaborators and I have developed all the tools necessary to prove many new cases of the Birch--Swinnerton-Dyer conjecture, for a class of objects called 'abelian surfaces'. We are now in the process of putting together all these pieces; we expect the work to be complete within the next few months. Our work has already had applications to a number of other important open problems in mathematics, namely to the Bloch--Kato conjecture and the Iwasawa Main conjecture.