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Computational Methods in Number Theory

Total activity: 1
Type of activity
Competitive project
Funding entity
Funding entity code
67.761,31 €
Start date
End date
Hopf-Galois theory, arithmetic geometry, computational number theory, cuerpos de números, curvas modulares, geometría aritmética, modular curves, number fields, teoría de Hopf-Galois, teoría de números computacional
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.
Adm. Estat
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Call year
Funcding program
Programa Estatal de I+D+i Orientada a los Retos de la Sociedad
Funding call
Retos de Investigación: Proyectos de I+D+i
Grant institution
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco


Scientific and technological production

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