Periodic Reporting for period 2 - ShimBSD (Shimura varieties and the Birch--Swinnerton-Dyer conjecture)
Reporting period: 2022-11-01 to 2023-10-31
The project "Shimura varieties and the Birch--Swinnerton-Dyer conjecture" (ShimBSD) addresses problems in number theory around the Birch and Swinnerton-Dyer conjecture, which is one of the most famous open problems in pure mathematics (selected by the Clay Institute in 2000 as one of their seven Millennium Prize problems). The problem concerns the arithmetic of elliptic curves, a class of equations in two variables, and predicts that the number of solutions of these equations is intimately related to an auxiliary object called an L-function. Any progress towards this conjecture would be a major landmark in pure mathematics, and could also impact other mathematical fields (including those directly relevant to real-world applications, such as cryptography, in which elliptic curves are ubiquitous).
The overall objectives of the project are to use new methods arising from the theory of Shimura varieties and automorphic forms to prove new cases of the Birch--Swinnerton-Dyer conjecture, and its generalisations such as the Iwasawa main conjecture and the Bloch--Kato conjecture, using intricate mathematical constructs known as "Euler systems". The key objectives of the project are to construct and study new Euler systems, and to give criteria ("explicit reciprocity laws") for when these Euler systems are non-zero; if this can be successfully carried out, new instances of BSD and other conjectures will follow.
The overall objectives of the project are to use new methods arising from the theory of Shimura varieties and automorphic forms to prove new cases of the Birch--Swinnerton-Dyer conjecture, and its generalisations such as the Iwasawa main conjecture and the Bloch--Kato conjecture, using intricate mathematical constructs known as "Euler systems". The key objectives of the project are to construct and study new Euler systems, and to give criteria ("explicit reciprocity laws") for when these Euler systems are non-zero; if this can be successfully carried out, new instances of BSD and other conjectures will follow.
Notable new results obtained (published or released as preprints) up to the end of the period:
- New explicit reciprocity laws for GSp(4) x GL(2), generalising earlier work for GSp(4).
- General approach to "vertical" norm relations for Euler systems, based on spherical varieties.
- New cases of BSD for abelian surfaces, and for elliptic curves over imaginary quadratic fields.
- Relations among Euler systems for different groups, via degeneration at critical-slope Eisenstein series.
- New p-adic L-functions for Hilbert modular groups, and explicit reciprocity laws relating these to Euler systems.
Ongoing work as of the end of the report period:
- Study of "boundary contributions" to cohomology of Shimura varieties: this greatly strengthens earlier work of the PI and collaborators by removing the dependence on a conjecture (the "eigenspace vanishing conjecture"), rendering the results unconditional.
- Investigation of applications of the Hilbert modular Euler system to Iwasawa theory for symmetric squares of modular forms.
- New developments towards understanding variation of Eichler--Shimura isomorphisms in p-adic families (mostly via the work of Ju-Feng Wu, a post-doc member of the project team) which will be an important technical input in future applications to the BSD conjecture.
- New explicit reciprocity laws for GSp(4) x GL(2), generalising earlier work for GSp(4).
- General approach to "vertical" norm relations for Euler systems, based on spherical varieties.
- New cases of BSD for abelian surfaces, and for elliptic curves over imaginary quadratic fields.
- Relations among Euler systems for different groups, via degeneration at critical-slope Eisenstein series.
- New p-adic L-functions for Hilbert modular groups, and explicit reciprocity laws relating these to Euler systems.
Ongoing work as of the end of the report period:
- Study of "boundary contributions" to cohomology of Shimura varieties: this greatly strengthens earlier work of the PI and collaborators by removing the dependence on a conjecture (the "eigenspace vanishing conjecture"), rendering the results unconditional.
- Investigation of applications of the Hilbert modular Euler system to Iwasawa theory for symmetric squares of modular forms.
- New developments towards understanding variation of Eichler--Shimura isomorphisms in p-adic families (mostly via the work of Ju-Feng Wu, a post-doc member of the project team) which will be an important technical input in future applications to the BSD conjecture.
I believe that the project is on well on course to prove the Bloch--Kato conjecture in analytic rank 0 for all cohomological automorphic forms for the groups GU(2, 1) and GSp(4), and I am confident that we will also obtain new unconditional results in the non-cohomological cases that are most relevant to the Birch--Swinnerton-Dyer conjecture, including the cases of abelian surfaces over the rationals, and elliptic curves over imaginary quadratic fields. The project has already led to significant new results towards the BSD conjecture, including a proof of the conjecture for analytic rank 0 elliptic curves over imaginary quadratic fields assuming a variety of technical hypotheses, which we hope to ameliorate in future work.