Searching for simultaneous arithmetic progressions on elliptic curves

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.Junta de AndalucíaMinisterio de Ciencia y TecnologíaFondo Europeo de Desarrollo Regiona

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.Junta de AndalucíaMinisterio de Educación y CienciaFondo Europeo de Desarrollo Regiona

A complete diophantine characterization of the rational torsion of an elliptic curve

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non–)existence of integral solutions of a system of diophantine equations

Restauración del Tramo Conexión Quart de Poblet - Manises del Parque Fluvial del Turia (Valencia)

[ES] El presente proyecto se sitúa en la zona de ribera del río Turia que pasa por la localidad de Quart de Poblet situado en la provincia de Valencia y pretende ser continuación del parque fluvial del Turia ya ejecutado a nivel de dicha población. Dicho parque fluvial, declarado Parque Natural del Turia con los DECRETOS 43/2007 y 42/2007, por el que se aprueba el Parque Natural del Turia y el Plan de Ordenación de los Recursos Naturales del Turia respectivamente,recorre el cauce de dicho río que va desde la población de Pedralva a Quart de Poblet, y muestra un paisaje dinámico y variado con una notable diversidad de hábitats y especies de fauna y flora. El río Turia-Guadalaviar y su ribera, de gran interés ecológico, bosques mediterráneos de pinar y matorral y la huerta de origen medieval son ambientes representativos de esta área natural. La zona a restaurar alberga aproximadamente 3 hectáreas de superficie y es continua a la zona ya restaurada de la ribera a su paso por dicha localidad. En las últimas décadas, esta zona de ribera se ha ido degradando perdiendo su uso y función, para convertirse en un vertedero en el que las plantas invasoras están acabando con las autóctonas. La degradación que ha sufrido requiere de una actuación para convertirlo en un paseo y zona lúdico-recreativa que, en primer lugar, acerque la población al río pero que a la vez permita retornar en la medida de lo posible a sus condiciones naturales. Una restauración o recuperación del espacio atendiendo a la pureza del término resulta difícil ya que esto supone conseguir la reduplicación más o menos exacta de las condiciones que existían antes de producirse el proceso de degradación, lo cual no sería viable dado: el alto grado de deterioro, el elevado coste y la presión humana de una población con un peso demográfico importante. Por tanto, se va a realizar una restauración con una tipología de rehabilitación en la que se pretende un uso recreativo distinto del original, pero en el que habrá un alto grado de reformación dado que se persigue que el área vuelva a ser habitable por la flora y la fauna silvestres propias del lugar. Así se iniciará la recuperación de la ribera propiamente dicha y se facilita su conexión y armonización con dos municipios Quart de Poblet y Manises con grandes necesidades de espacios verdes y recreativos. Como prioridades se establecen: la recuperación de una zona degradada, el dotar a la población de zonas verdes, eliminar barreras (accesibilidad) y revalorizar el patrimonio medioambiental. Para respetar en la medida de lo posible las condiciones naturales no se dotara al espacio de iluminación urbana, que podría afectar el fotoperiodo de las especies vegetales, ni de riego artificial, dado que se persigue la naturalidad y perfecta adaptación de estas a la vez que el subsuelo de ribera tiene abundante humedad para el desarrollo natural de las mismas.[EN] This project is located in the area of the riverbank of Turia river passing through the town of Quart de Poblet located in the province of Valencia and which objective is to be a continuation of the Turia River Park already executed at the level of that population. Such river park, declared Natural Park Turia with the Decrees 43/2007 and 42/2007, by which they are approved the Natural Park of Turia and Plan of Management of Natural Resources of Turia respectively, go all over the course of the river from the town of Pedralva until Quart de Poblet, showing a dynamic and varied landscape with a remarkable diversity of habitats and species of fauna and flora. Guadalaviar-Turia river and its banks, of great ecological interest, the Mediterranean forests of pine and scrubland and the medieval orchard are representative environments of this natural area. The area to restore is approximately 3 hectares large and is contiguous to the area already restored of the banks on its way through the town. In the last decades, the riverbank has been deteriorated losing its use and function, becoming a dumping ground where invasive plants are destroying native ones. The degradation of the bank requires an intervention to make it a promenade that, first of all, brings the population back to the river and at the same time allows as much as possible to recover their original natural conditions. The restoration or recovery of this space is quite difficult because this means getting more or less exact reduplication of the conditions that existed before the degradation process occurred, which would not be viable given: the high degree of deterioration, the high cost and the human pressure from a population with an important demographic weight. Therefore, it will be made a rehabilitation in which it is intended to get a different recreational use from original, but where there will be a high degree of reformation because it is pursued that the area will be habitable again by the wild flora and fauna of the place. So this recovery will begin in the riverbank itself and it will be facilitated the connection between two municipalities Quart de Poblet and Manises with the bank. The priorities established are: recovering a degraded zone, providing the population of green areas, removing barriers (accessibility) and enhance the environmental heritage. To respect as far as possible the natural conditions no space for urban lighting, which could affect the photoperiod of plants, or artificial irrigation endowing, since the naturalness and perfect adaptation of these at once pursued that subsoil moisture bank has abundant natural development thereof.Tornero Carrascosa, I. (2016). Restauración del Tramo Conexión Quart de Poblet - Manises del Parque Fluvial del Turia (Valencia). http://hdl.handle.net/10251/73334.TFG

Galois Theory, discriminants and torsion subgroups of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve

Computing the rational torsion of an elliptic curve using Tate normal form

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n (4 ≤ n ≤ 10, or n = 12) lie in a oneparameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here a extremely down–to–earth algorithm using the existence of such a family.Junta de Andalucí

Galois Theory, discriminants and torsion subgroups of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.Comment: New version, some typos fixed and the proof of the lemma in the Appendix has been expande