Explicit Methods for the Darmon Programme

Informazioni relative al progetto

ExplicitDarmonProg

ID dell’accordo di sovvenzione: 655662

DOI

10.3030/655662

Progetto concluso il 10 Maggio 2017

Data della firma CE 24 Marzo 2015

Data di avvio 1 Settembre 2015

Data di completamento 30 Settembre 2017

Finanziato da

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Costo totale

€ 195 454,80

Contributo UE

€ 195 454,80

Coordinato da

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 1 - ExplicitDarmonProg (Explicit Methods for the Darmon Programme)

Periodo di rendicontazione: 2015-09-01 al 2017-08-31

Sintesi del contesto e degli obiettivi generali del progetto

Elliptic curves are mathematical objects that have been studied since Diophantus in the third century AD. They are now of vital importance in cryptography and cyber-security and are used for securing online bank transactions. In the 1960s, Birch and Swinnerton-Dyer performed massive experimental computations on elliptic curves using one of the earliest available computers (EDSAC II in Cambridge). On the basis of these computations they formulated what is now known as the Birch and Swinnerton-Dyer conjecture, one of seven very famous open problems in Mathematics, for which there is a $1M prize for resolving. All special cases of the Birch and Swinnerton-Dyer conjecture that have been proved rely on a construction due to German mathematician Heegner. In 2000, Canadian mathematician Darmon proposed a wide-ranging generalization of the Heegner construction, which has led to the concepts of Darmon points and cycles, and also to new proofs of special cases of the Birch and Swinnerton-Dyer conjecture. This project is concerned with Darmon points and cycles from an algorithmic point-of-view: it aims to develop constructions and software for these objects.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

"In the first year of the fellowship, the researcher has completed work on the first research project of the original proposal ""Higher Degree Darmon Points"". This consisted of three work packages: ""Describe higher degree overconvergent cohomology"", ""Devise algorithms to compute Darmon homology cycles"" and ""Compute the integration pairing"", and resulted in two publications and two submitted manuscripts, plus two software packages that are freely available to other researchers."

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

The project has yielded the first ever algorithms and software for both Leezarbs and exotic Darmon points. This is expected to provide constructions of points in all rank 1 situations, and give a better understanding Heenger points.

Concepts prevalent in the subject of Darmon points