Periodic Reporting for period 4 - BSD (Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato) Reporting period: 2021-03-01 to 2022-08-31 Summary of the context and overall objectives of the project The achievements of the project have revolved around the conjecture of Birch and Swinnerton-Dyer (BSD) and its generalizations by Bloch and Kato (BK), using as main tool the theory of Euler systems and p-adic L-functions.During the conduction of the project we have developed innovating strategies for approaching new horizons in BSD and BK. The results obtained have met to a large extent the initial goals of the proposal and have also reached additional results of high interest. More precisely our main results fall in the following research lines:BSD over totally real number fields: Substantial progress towards proving new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, covering many new scenarios that had never been considered before. BSD in rank r=2: p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.Darmon's 2000 conjecture on Stark-Heegner points: Substantial progress towards the resolution of Darmon’s striking conjecture announced at the ICM-2006 by recasting it in terms of the Hida-Harris-Tilouine triple p-adic L-function.Beyond the goals of the original proposal, we have obtained the proof of important cases of the Harris-Venkatesh conjecture. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 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. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) At the end of the project, Victor Rotger has completed a collaboration with Henri Darmon (McGill, Montréal), Michael Harris (Columbia and IHES) and Akshay Venkatesh (Stanford and Princeton), where we have provided a tamely ramified version of the elliptic Stark conjecture of Darmon-Lauder-Rotger and relate it to the conjecture of Harris-Venkatesh on derived Hecke algebras for weight one modular forms. This is a very exciting project, directly related to the proposal, but beyond the state of the art and expected results.Victor Rotger has also conducted a collaboration with Vinayak Vatsal (Vancouver) and Marti Roset (Paris) on L-invariants associated to a weight 1 form.Daniel Barrera has been collaborating in very interesting projects with Mladen Dimitrov, Chris Williams, A. Jorza, R. Brasca which are directly related to the project, but which extend in new and unforeseen directions some of its aspects, mainly concerning the theory of overconvergent modular forms and p-dic L-functions.Francesc Fité and Xavier Guitart have also found exciting applications of the main ideas of our project to the Sato-Tate conjecture and the theory of Stark-Heegner points.Antonio Cauchi has achieved exciting results on algebraic cycles and regulators on Siegel varieties.Armando Gutiérrez has achieved great results on the Siegel-Weil formula and Borcherd products.Guillem Sala has obtained deep results relating p-adic L-functions to higher homotopy groups.Victor Hernandez and Santiago Molina have established new results on the theory of p-adic L-functions associated to modular forms, their factorization and their connection to Darmon points and plectic points.Oscar Rivero and Victor Rotger have found new formulae interrelating the Euler systems of circular units and Kato elements, hinting to a comparison theory for more general Euler systems. imagebsd.jpg
