arXiv subjects

Jose M. Tornero

Publications and source records attributed to Jose M. Tornero.

13 recordsLinked to original sources

arXiv2015

Torsion of rational elliptic curves over quadratic fields II

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the present paper, we go further in this study and compute, under this assumption and for every such G, all the possible situations where G\neq H. The result is optimal, as we also display examples for every situation we state as possible. As a consequence, the maximum number of quadratic number fields K such that E(Q)_tors\neq E(K)_tors is easily obtained.

math.NT↗

arXiv2015

Torsion of rational elliptic curves over cubic fields

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.

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arXiv2014

Torsion of rational elliptic curves over quadratic fields

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a quadratic number field.

math.NT↗

arXiv2010

On the ubiquity of trivial torsion on elliptic curves

The purpose of this paper is to give a "down--to--earth" proof of the well--known fact that a randomly chosen elliptic curve over the rationals is most likely to have trivial torsion.

math.NT↗

arXiv2009

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.

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arXiv2007

On the discriminant of elliptic curves with non-trivial torsion

For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve.

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arXiv2007

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.

math.NT↗

arXiv2007

Some geometric aspects of Puiseux surfaces

The following problem is treated: Characterizing the tangent cone and the equimultiple locus of a Puiseux surface (that is, an algebroid embedded surface admitting an equation whose roots are Puiseux power series), using a set of exponents appearing in a root of an equation. The aim is knowing to which extent the well--known results for the quasi--ordinary case can be extended to this much wider family.

math.AG↗

arXiv2007

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.

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arXiv2007

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.

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arXiv2007

Equimultiple locus of embedded algebroid surfaces and blowing--up in characteristic zero

The smooth equimultiple locus of embedded algebroid surfaces appears naturally in many resolution process, both classical and modern. In this paper we explore how it changes by blowing--up.

math.AG↗

arXiv2007

Equimultiple locus of embedded algebroid surfaces and blowing--up in arbitrary characteristic

This paper extends previous results of the authors, concerning the behaviour of the equimultiple locus of algebroid surfaces under blowing--up, to arbitrary characteristic.

math.AG↗

arXiv2002

On the singular braid monoid of an orientable surface

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.

math.GT↗