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Abelian varieties over finite fields

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.

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Topic(s)

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Funding Scheme

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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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Coordinator

UNIVERSITE COTE D'AZUR
Net EU contribution

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.

€ 195 914,88
Address
GRAND CHATEAU 28 AVENUE VALROSE
06100 Nice
France

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Region
Provence-Alpes-Côte d’Azur Provence-Alpes-Côte d’Azur Alpes-Maritimes
Activity type
Higher or Secondary Education Establishments
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Total cost

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