Abelian varieties over finite fields

Projektbeschreibung

Effiziente Darstellung Abelscher Varietäten

Die Schnittmenge zwischen algebraischer Geometrie und Zahlentheorie umfasst zahlreiche Themen und Entdeckungen, die für die wissenschaftliche und mathematische Gemeinschaft sowie für forschende Personen von großem Interesse sind, wobei Abelsche Varietäten eine besonders wichtige Rolle spielen. Bedauerlicherweise sind Abelsche Varietäten, bei denen es sich um projektive Varietäten handelt, deren Punkte eine Gruppe bilden, bereits in der zweiten Dimension nur schwer durch Gleichungen darstellbar, was neben anderen Schwierigkeiten ihre Untersuchung erschwert. Das im Rahmen der Marie-Skłodowska-Curie-Maßnahmen geförderte Projekt AbVarFq zielt darauf ab, konkrete Methoden zur Darstellung von Abelschen Varietäten über einem endlichen Körper zu entwickeln und diese bis auf Isomorphie zu klassifizieren, einschließlich ihrer Polarisationen. Diese Aktivitäten werden nicht nur effizientere Darstellungsmethoden liefern, sondern auch unser Verständnis dieser Varietäten verbessern.

Ziel

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.