Explicit Methods for the Darmon Programme

Información del proyecto

ExplicitDarmonProg

Identificador del acuerdo de subvención: 655662

DOI

10.3030/655662

Proyecto finalizado el 10 Mayo 2017

Fecha de la firma de la CE 24 Marzo 2015

Fecha de inicio 1 Septiembre 2015

Fecha de finalización 30 Septiembre 2017

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 195 454,80

Aportación de la UE

€ 195 454,80

Coordinado por

UNIVERSITY OF WARWICK
United Kingdom

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

Período documentado: 2015-09-01 hasta 2017-08-31

Resumen del contexto y de los objetivos generales del proyecto

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.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

"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."

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

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