Objective
"The theory of rational points on curves and their Jacobians is distinguished by being both attractive and notoriously difficult. Despite major theoretical advances, explicit methods are of particular importance in this area. For instance, the conjecture of Birch and Swinnerton-Dyer (BSD), one of the Millennium Prize problems, was formulated based on numerical evidence. A proof of the strong version of this conjecture for abelian varieties seems out of reach at present, and even the verification in examples was, until recently, only possible in dimension 1.
Besides being interesting in its own right, the importance of explicit methods for the computation of the rational points on curves stems from the fact that many moduli problems can be reduced to such computations. Therefore, explicit methods can be used to solve theoretical problems, but in the other direction, theoretical advances often lead to improved explicit methods. One example is the recent computation of the rational points on the ""cursed curve"" X_ns^+(13) using the quadratic Chabauty (QC) method, an explicit special case of Kim's non-abelian Chabauty program.
We propose two research projects, connected by height theory, to significantly advance the state of the art in explicit methods for rational points on curves and Jacobians. In the first one, we will develop an explicit theory of heights to compute Mordell-Weil groups of Jacobians of non-hyperelliptic curves of genus 3. We will use it for the verification of the strong BSD conjecture for modular examples, going beyond the hyperelliptic case for the first time. In the second one, we will drastically increase the applicability of the QC method by removing several restrictive conditions, and apply it to Atkin-Lehner quotients of modular and Shimura curves, thereby solving several open moduli problems."
Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
-
HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA)
MAIN PROGRAMME
See all projects funded under this programme
Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships
See all projects funded under this funding scheme
Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) HORIZON-MSCA-2021-PF-01
See all projects funded under this callCoordinator
Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
9712CP Groningen
Netherlands
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.