Objective
In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).
Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:
I. BSD over totally real number fields. I plan to prove new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics algebra
- natural sciences mathematics pure mathematics arithmetics L-functions
- social sciences law
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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.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
MAIN PROGRAMME
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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.
ERC-COG - Consolidator Grant
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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) ERC-2015-CoG
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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.
08034 BARCELONA
Spain
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.