Project description
Efficiently representing Abelian varieties
The intersection of algebraic geometry and number theory encompasses numerous subjects and discoveries that deeply interest the scientific and mathematical community and researchers, with abelian varieties being particularly significant. Unfortunately, Abelian varieties, which are projective varieties whose points form a group, have become challenging to represent with equations already in dimension 2, among other difficulties that hinder their study. Supported by the Marie Skłodowska-Curie Actions programme, the AbVarFq project aims to develop concrete methods for representing Abelian varieties defined over a finite field and to classify them up to isomorphism, including their polarisations. This effort will not only provide more efficient representation methods but also enhance our understanding of these varieties.
Objective
Some of the most extensively studied objects at the intersection of number theory and algebraic geometry are abelian varieties, which are projective varieties whose points form a group.
The main objective of the project is to concretely represent abelian varieties defined over a finite field and classify them up to isomorphism, together with their polarizations.
The main goal has been achieved in previous years by work of the Researcher, Supervisor and collaborators in several cases, which enjoy two properties: the varieties admit canonical liftings and the p-divisible groups do not play a special role in the classification.
The project deals with the cases where these crucial properties do not hold, making them theoretically more complicated to grasp, but also more interesting.
Achieving the objective will have several consequences. Firstly, we will obtain an efficient way to represent abelian varieties over finite fields, overcoming the facts that equations are too cumbersome, already in dimension 2, and that Jacobian varieties give a complete description only in low dimension and with certain kind of polarizations. Secondly, we will fill some important gaps in our current understanding of many invariants attached to the abelian varieties, like the p-rank or the Newton polygon. Third, the project will pave the way to: compute the cohomology of moduli spaces of the abelian varieties by interpolating our point-counts over finite fields; shed light one the set of conjectures connecting automorphic forms and representation theory usually known as the Langlands program; study isogeny graphs of abelian varieties over finite fields, which have the potential of being useful in (post-quantum-)cryptography; understand properties of algebraic-geometric codes via Jacobians. Note that the last two applications could have significant impact on making digital communications more secure and reliable, and hence considerably affect our society and economy.
Fields of science (EuroSciVoc)
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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.
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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)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA)
MAIN PROGRAMME
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Topic(s)
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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
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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
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Call for proposal
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(opens in new window) HORIZON-MSCA-2023-PF-01
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06100 Nice
France
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