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Enrique Gonzalez-Jimenez

  1. On the ubiquity of trivial torsion on elliptic curves.

    Чт, 02/04/2010 - 16:01 — SomNambul
    Authors: Enrique Gonzalez-Jimenez, Jose M. Tornero
    Subjects: Number Theory
    Abstract

    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.

  2. On the modularity level of modular abelian varieties over number fields.

    Ср, 02/03/2010 - 16:01 — SomNambul
    Authors: Enrique Gonzalez-Jimenez, Xavier Guitart
    Subjects: Number Theory
    Abstract

    Let f be a weight two newform for Gamma_1(N) without complex multiplication.
    In this article we study the conductor of the absolutely simple factors B of
    the variety A_f over certain number fields L. The strategy we follow is to
    compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's
    formula for the conductor of the restriction of scalars. In this way we obtain
    an expression for the local exponents of the conductor N_L(B). Under some
    hypothesis it is possible to give global formulas relating this conductor with
    N.

  3. Five squares in arithmetic progression over quadratic fields.

    Чт, 09/10/2009 - 10:12 — SomNambul
    Authors: Enrique Gonzalez-Jimenez, Xavier Xarles
    Subjects: Number Theory
    Abstract

    We give several criteria to show over which quadratic number fields
    $\bQ(\sqrt{D})$ there should exists a non-constant arithmetic progressions of
    five squares. This is done by translating the problem to determining when some
    genus five curves C_D defined over Q have rational points, and then using a
    Mordell-Weil sieve argument among others.

  4. Three cubes in arithmetic progression over quadratic fields.

    Ср, 09/02/2009 - 12:02 — SomNambul
    Authors: Enrique Gonzalez-Jimenez
    Subjects: Number Theory
    Abstract

    We study the problem of the existence of arithmetic progressions of three
    cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer.
    For this purpose, we give a characterization in terms of Q(sqrt(D))-rational
    points on the elliptic curve E:y^2=x^3-27. We compute the torsion subgroup of
    the Mordell-Weil group of this elliptic curve over Q(sqrt(D)) and we give
    partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last
    task will be translated to compute if the rank of the quadratic D-twist of the
    modular curve X_0(36) is zero or not.

  5. Three cubes in arithmetic progression over quadratic fields.

    Ср, 09/02/2009 - 12:02 — SomNambul
    Authors: Enrique Gonzalez-Jimenez
    Subjects: Number Theory
    Abstract

    We study the problem of the existence of arithmetic progressions of three
    cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer.
    For this purpose, we give a characterization in terms of Q(sqrt(D))-rational
    points on the elliptic curve E:y^2=x^3-27. We compute the torsion subgroup of
    the Mordell-Weil group of this elliptic curve over Q(sqrt(D)) and we give
    partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last
    task will be translated to compute if the rank of the quadratic D-twist of the
    modular curve X_0(36) is zero or not.

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