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Abelian varieties over finite fields

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 conc... ver más

31/08/2026

UNIVERSITE COTE DA...

196K€

Presupuesto del proyecto: 196K€

Líder del proyecto

UNIVERSITE COTE DAZUR No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2024-04-23

HORIZON-MSCA-2023-PF-01-01: MSCA Postdoctoral Fellowships 2023 Expected Outcome:Project results are expected to contribute to the following outcomes:

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1 Participantes

UNIVERSITE COTE DAZUR

195.91K€ | Lider

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Duración del proyecto: 28 meses Fecha Inicio: 2024-04-23
Fecha Fin: 2026-08-31

Líder del proyecto

UNIVERSITE COTE DAZUR No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 196K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto 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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