Black Box Galois Representations

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\det\rho$, whether or not $\rho$ is residually reducible, and further information about the size of the isogeny graph of $\rho$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $\rho$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees' report

Computational aspects of Galois representations

This thesis contains a series of studies about 2-adic integral Galois representations unramified outside a finite set of primes. There are two main focuses of research: the study of 2-adic integral Galois representations and the study on how to compare two 2-adic integral Galois representations. Firstly, when studying a representation, we develop methods to determine whether the residual image is reducible or irreducible: in the irreducible case the residual image is completely determined. On the other hand, when the residual image is reducible we are able to make a choice of a stable lattice to completely determine the residual image. Lastly, from the choice of lattice, we are able to extend our methods to determine whether the representation is trivial modulo 2k+1 assuming that is trivial modulo 2k. Secondly, when comparing two 2-adic integral Galois representations, we are able to determine whether the representations are isogenous that is, after conjugation if necessary, their residual representations are the same. In some cases, this process follows the approach given in [11] by Ron Livné. Finally, the idea behind these studies was the notion of what we call a "Black Box representation", i.e., a system that will provide the characteristic polynomial of the representation for any prime not in the set of primes

On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.Comment: arXiv admin note: text overlap with arXiv:1901.0538

Paz en Colombia : perspectivas, desafíos, opciones

Este libro pretende ser un aporte para analizar las diversas dimensiones sociales, políticas, económicas, culturales, educativas y éticas de los retos que precedieron al acuerdo de paz y que se desplegarán a partir de ahora. Se publica a pocos días del plebiscito del 2 de octubre de 2016, en el que la sociedad colombiana tendrá en sus manos la posibilidad de refrendar este importantísimo acuerdo, abriendo y multiplicando los desafíos y oportunidades que nos permitieron llegar hasta aquí. La lectura de cada una de las contribuciones de este volumen permite no sólo entender que es urgente y necesario votar por el SÍ a los acuerdos de paz, sino también asumir la inmensa complejidad de una coyuntura que necesitará de muchísima imaginación, compromiso y participación democrática. ¡Viva la Paz! ¡Viva Colombia