Abelian varieties over finite fields

Descripción del proyecto

Representación eficiente de las variedades abelianas

La intersección entre la geometría algebraica y la teoría de números abarca numerosos temas y descubrimientos que interesan profundamente a la comunidad científica y matemática y a los investigadores, en los que las variedades abelianas son especialmente significativas. Desgraciadamente, las variedades abelianas, que son variedades proyectivas cuyos puntos forman un grupo, se han convertido en un reto de representar con ecuaciones ya en dimensión 2, entre otras dificultades que obstaculizan su estudio. Con el apoyo de las Acciones Marie Skłodowska-Curie, el equipo del proyecto AbVarFq pretende desarrollar métodos concretos para representar variedades abelianas definidas sobre un campo finito y clasificarlas hasta el isomorfismo, incluidas sus polarizaciones. Este esfuerzo no solo proporcionará métodos de representación más eficaces, sino que también mejorará nuestra comprensión de estas variedades.

Objetivo

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.