In Pure Mathematics, Number Theory is probably the branch with more applications, and one where computational techniques play a more relevant role. It is well known the ubiquity of Arithmetic in Cryptography and Coding Theory. But the interaction between Number Theory and Computer Science is bidirectional and provides significant benefits to both fields. The Birch and Swinnerton-Dyer conjecture, one of the Millenium Problems, emerged 50 years ago from an extensive numerical experimentation done with a EDSAC 2, and is one of the clearest examples of the contribution of computers to high-level abstract mathematics. In the other direction, from basic arithmetic algorithms to the more abstract arithmetic structures they have affected the development of information technology, both in terms of hardware and software. Our research group has been characterized from its origin by a practical approach to the problems we have dealt with, and by an intensive use of computers as a tool for experimentation and support for their resolution. The elaboration of tables was one of our first contributions, but along our careers we have developed a number of algorithms and computational strategies to solve relevant problems we have addressed. The use of modular symbols, the arithmetic of numbers fields or the determination of equations for curves are good examples. Many techniques developed in our papers have been subsequently applied by other researchers in various fields. The main topics of our work will be number fields, modular curves and elliptic curves. We shall consider the computation of normal bases, the determination of sub-extensions, automorphism groups, rational points ... with a computational approach, with the goal of providing algorithmic solutions or at least propose effective strategies for its resolution. In addition to publishing the results in international journals of high level, we plan the implementation of the designed algorithms, either as free open source software, or as libraries to attach to the standard software packages in our area (SAGE, Magma, Pari). Beyond the strictly scientific objectives, the project will allow us to consolidate internationally in the field of Computational Number Theory, and become a specialized group in computational techniques in Number Theory and its applications, in the style of the well-established Number Theory groups in Bordeaux or Berlin.
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de I+D+i Orientada a los Retos de la Sociedad
Retos de Investigación: Proyectos de I+D+i
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco