The implementation of the action during the whole project has been very successful, thanks to the hard work of all members of the team.
Together we have organized a weekly research seminar on motivic cohomology and reciprocity laws, with lectures delivered by members of the team and prominent experts in the subject from abroad.
We have also organized a number of workshops and advanced courses, with lectures given by some of the most prestigious number-theorists, which have been fundamental for achieving the results we have obtained so far on the goals of the project.
The members of the team have also traveled abroad in order to participate in seminars and conferences, with the triple aim of learning new techniques, collaborating in research with other colleagues and disseminating our results.
Major achievements have been obtained in all projects and beyond. Namely, the achievements obtained may be summarized as follows:
Project I: Construction of overconvergent sheaves of modular forms over Shimura curves employing the methods of Iovita-Andreatta-Pilloni, construction of associated triple-product L-functions, construction of an Euler system of diagonal classes supported over Shimura curves over totally real fields, proof of the main reciprocity law relating the two latter.
Project II: Substantial progress on the elliptic Stark conjecture in new settings by Gatti-Guitart, Gatti-Guitart-Masdeu-Rotger, Darmon-Lauder-Rotger, Betina-Dimitrov-Pozzi by studying new scenarios including:
-the case where the central critical.L-value does not vanish,
-the setting where higher weights are allowed,
-the case where the eigencurve fails to be ètale at the associated weight one forms
-the case where the two weight 1 forms are dual to each other.
-CM cases and their relationship with Bertolini-Darmon Kolyvagin classes
Project III: Completion of a series of articles on the rationality of Stark-Heegner points as part of a volume in collaboration with Bertolini-Seveso-Venerucci, making substantial progress on Darmon's conjecture. Completion of papers on Beilinson-Flach elements, Stark units and Rankin-Hida p-adic L-functions by Rivero and Rotger.
These achievements have been published, or are accepted for publication, in renowned peer-refereed International math journals. Lectures disseminating these results have been delivered in seminars and main conferences at plenty of prestigious institutions.
