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Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato

Periodic Reporting for period 3 - BSD (Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato)

Reporting period: 2019-09-01 to 2021-02-28

In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. The Prizes were conceived to record some of the most important challenges with which mathematicians were grappling at the turn of the second millennium. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this project is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this fundamental problem and their generalizations by Bloch and Kato (BK).

Breakthroughs on BSD were achieved by Coates and Wiles (1976), Gross, Zagier and Kolyvagin (1987), and Kato (1990). Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, in the last two years, three independent revolutionary approaches have seen the light: the works of (1) the Fields medallist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. The latter permit to prove BSD for many new number fields under the assumption that L(E/K,s) does not vanish at s=1. This new idea of ours has been highly influential, as it has inspired the subsequent work of mathematicians including M. Bertolini (Essen), A. Lauder (Oxford), S. Dasgupta (UCSC), M. Harris (Columbia), A. Venkatesh (Stanford), K. Prasanna (Michigan), Y. Liu (M.I.T.) F. Castella (UCLA), M. Greenberg (Calgary), M. Longo (Padova), A. Lei (McGill), D. Loeffler (Warwick), S. Zerbes (UC London) and G. Kings (Regensburg).

In spite of that, our knowledge of BSD is rather poor. In my project I conduct innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of three PhD and three postdoc students that the CoG may allow me to consolidate. The results we are pursuing represent a departure from the achievements obtained with my coauthors during the past years, and during this second period we have made substantial progress towards them:

BSD over totally real number fields. INew ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
BSD in rank r=2. Most of the literature on BSD applies when the rank is r=0 or 1. We work on p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
Darmon's 2000 conjecture on Stark-Heegner points. We work on 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.
The implementation of the action during this second period 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.
The major achievements have been obtained in projects I.1 I.2 II.1 II.2 III.1 and III.2 where the main goals marked in our agenda for this first period have been obtained successfully. 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.
-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.
-project III: Nearly completion of the article on the rationality of Stark-Heegner points by Darmon-Rotger as part of a volume in collaboration with Bertolini-Seveso-Venerucci; Completion of papers on Beiulinson-Flach elements, Stark units and Rankin-Hida p-adic L-functions by Rivero and Rotger.
At the end of this second period, Victor Rotger is about to complete a collaboration with Henri Darmon (McGill, Montréal), Michael Harris (Columbia and IHES) and Akshay Venkatesh (Stanford and Princeton), where we hope to provide 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 started a collaboration with Vinayak Vatsal (Vancouver) and Marti Roset (Paris).
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.