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Report

Teaser, summary, work performed and final results

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

Teaser

Summary

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.

Work performed

\"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.\"

Final results

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.