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

Total Cost €


EC-Contrib. €






Project "BSD" data sheet

The following table provides information about the project.


Organization address
postcode: 8034

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Spain [ES]
 Project website
 Total cost 1˙428˙588 €
 EC max contribution 1˙428˙588 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-CoG
 Funding Scheme ERC-COG
 Starting year 2016
 Duration (year-month-day) from 2016-09-01   to  2021-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).

Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:

I. BSD over totally real number fields. I plan to prove new 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.

II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.

III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.


year authors and title journal last update
List of publications.
2019 Óscar Rivero, Victor Rotger
Beilinson–Flach elements, Stark units and -adic iterated integrals
published pages: , ISSN: 0933-7741, DOI: 10.1515/forum-2018-0281
Forum Mathematicum 0/0 2019-11-07
2019 Daniel Barrera Salazar, Santiago Molina Blanco
Triple product p-adic L-functions for Shimura curves over totally real number fields
published pages: , ISSN: , DOI:
2017 Daniel Barrera Salazar, Shan Gao
Overconvergent Eichler–Shimura isomorphisms for quaternionic modular forms over ℚ
published pages: 2687-2715, ISSN: 1793-0421, DOI: 10.1142/S1793042117501494
International Journal of Number Theory 13/10 2019-06-18
2019 Daniel Barrera Salazar, Chris Williams
Exceptional zeros and $mathcal {L}$-invariants of Bianchi modular forms
published pages: 1, ISSN: 0002-9947, DOI: 10.1090/tran/7436
Transactions of the American Mathematical Society 2019-04-16
2019 D. Casazza, V. Rotger
On the elliptic Stark conjecture at primes of multiplicative reduction
published pages: , ISSN: 0022-2518, DOI:
Indiana University Mathematics Journal 2019-04-16
2018 Daniele Casazza, Victor Rotger
Stark points and the Hida–Rankin p-adic L-function
published pages: 451-473, ISSN: 1382-4090, DOI: 10.1007/s11139-016-9824-y
The Ramanujan Journal 45/2 2019-04-16
2017 Francesc Fité, Xavier Guitart
On the Rank and the Convergence Rate Toward the Sato–Tate Measure
published pages: , ISSN: 1073-7928, DOI: 10.1093/imrn/rnx234
International Mathematics Research Notices 2019-04-16
2019 X. Guitart, M. Masdeu, S. Molina
An automorphic approach to Darmon points
published pages: , ISSN: 0022-2518, DOI:
Indiana University Mathematics Journal 2019-04-16
2019 Santiago Molina
Anticyclotomic $p$-adic $L$-functions and the exceptional zero phenomenon
published pages: 1, ISSN: 0002-9947, DOI: 10.1090/tran/7646
Transactions of the American Mathematical Society 2019-04-16
2018 Francesc Castella
On the $p$-part of the Birch–Swinnerton-Dyer formula for multiplicative primes
published pages: 1-23, ISSN: 2168-0930, DOI: 10.4310/cjm.2018.v6.n1.a1
Cambridge Journal of Mathematics 6/1 2019-04-16
2019 Adel Betina
Ramification of the Eigencurve at Classical RM Points
published pages: 1-32, ISSN: 0008-414X, DOI: 10.4153/cjm-2018-029-4
Canadian Journal of Mathematics 2019-04-16
2018 Francesc Fité, Elisa Lorenzo García, Andrew V. Sutherland
Sato–Tate distributions of twists of the Fermat and the Klein quartics
published pages: , ISSN: 2522-0144, DOI: 10.1007/s40687-018-0162-0
Research in the Mathematical Sciences 5/4 2019-04-16

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