In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. These 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 seven problems is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main objects of this proposal are: 1. Developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of BSD and their generalizations by Bloch and Kato (BK). 2. Exploiting the techniques involved in BSD in order to prove new results in other areas, including the Sato-Tate conjecture, the congruence number problem and applications to Fuchsian codes. 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 through the works of: (1) Fields medallist Bhargava (Princeton), (2) Skinner (Princeton) and Urban (Columbia), and (3) the principal investigator of this proposal and collaborators. This third approach allows to prove BSD for many new number fields under the assumption that L(E/K,s) does not vanish at s=1. Our new ideas have been highly influential, as it has inspired the subsequent work of mathematicians including Lauder (Oxford), Dasgupta (UCSC), Harris (Columbia), Venkatesh (Stanford), Prasanna (Princeton), Liu (MIT), Greenberg (Calgary), Longo (Padova), Lei (McGill), Loeffler (Warwick), Zerbes (UC London), Kings (Regensburg), Skinner (Princeton). In spite of that, BSD is still far from being solved and still keeps many secrets. The main lines of research in which we plan to achieve important new contributions are: I. BSD over totally real number fields: We plan to prove new 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. II. BSD in rank r=2: Most of the literature applies when the rank is r=0 or 1. We expect to prove p-adic versions of the theorems of Gross- Zagier and Kolyvagin in rank 2. III. Darmon's ICM conjecture on Stark-Heegner points: We plan to solve this fascinating problem on the global rationality of Stark-Heegner points by recasting it in terms of triple p-adic L-functions. IV. Heegner points and p-adic L-functions: We plan to design algorithms for computing Heegner points in new scenarios and prove p-adic Gross-Zagier formulas relating their logarithms to p-adic L-functions. V. Applications of BSD to the Sato-Tate conjecture: We plan to work on this question for abelian varieties of arbitrary dimension, exploring connections with automorphic forms and the crucial role of BSD in the convergence of the error terms. VI. Applications of BSD to the Congruence Number Problem (CNP): Exploiting the background gained in project A on BSD for Hilbert modular forms, we plan to solve CNP for non-rectangle triangles with sides over totally real number fields. VII. Applications of Shimura curves to Fuchsian codes: We plan to exploit our methods for BSD (Shimura curves, uniformization, Heegner points) to the development of transmission schemes for additive white Gaussian noisy (AWGN).
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia
Subprograma Estatal de Generación de Conocimiento
Excelencia: Proyectos I+D
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco