Abelian varieties over finite fields

Description du projet

Représentation efficace des variétés abéliennes

L’intersection de la géométrie algébrique et de la théorie des nombres englobe de nombreux sujets et découvertes qui intéressent profondément la communauté scientifique et mathématique et les chercheurs, les variétés abéliennes étant particulièrement significatives. Malheureusement, les variétés abéliennes, qui sont des variétés projectives dont les points forment un groupe, sont devenues difficiles à représenter avec des équations déjà en dimension 2, entre autres difficultés qui entravent leur étude. Soutenu par le programme Actions Marie Skłodowska-Curie, le projet AbVarFq vise à développer des méthodes concrètes pour représenter les variétés abéliennes définies sur un corps fini et les classer jusqu’à isomorphisme, y compris leurs polarisations. Cet effort permettra non seulement d’obtenir des méthodes de représentation plus efficaces, mais aussi d’améliorer notre compréhension de ces variétés.

Objectif

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.