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  • Modular Abelian Varieties Over Number Fields

     Guitart Morales, Xavier; Quer Bosor, Jordi
    Canadian journal of mathematics. Journal canadien de mathématiques
    Date of publication: 2014-02-01
    Journal article

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    The main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over Q for congruence subgroups of the form Gamma(1) (N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.; We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

  • Almost totally complex points on elliptic curves

     Guitart Morales, Xavier; Rotger Cerdà, Victor; Zhao, Yu
    Transactions of the American Mathematical Society
    Date of publication: 2014-05
    Journal article

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    Let F=F0 be a quadratic extension of totally real number elds, and let E be an elliptic curve over F which is isogenous to its Galois conjugate over F 0 . A quadratic extension M=F is said to be almost totally complex (ATC) if all archimedean places of F but one extend to a complex place of M . The main goal of this note is to provide a new construction of a supply of Darmon-like points on E , which are conjecturally dened over certain ring class elds of M . These points are constructed by means of an extension of Darmon's ATR method to higher dimensional modular abelian varieties, from which they inherit the following features: they are algebraic provided Darmon's conjectures on ATR points hold true, and they are explicitly computable, as we illustrate with a detailed example that provides numerical evidence for the validity of our conjectures.

  • Computation of ATR Darmon points on nongeometrically modular elliptic curves

     Guitart Morales, Xavier; Masdeu, Marc
    Experimental mathematics
    Date of publication: 2013-03
    Journal article

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    ATR points were introduced by Darmon as a conjectural con- struction of algebraic points on certain elliptic curves for which the Heegner-point method is not in general available. So far, the only numerical evidence, provided by Darmon–Logan and G ̈ artner,concernedcurvesarisingasquotientsofShimuracurves. In those special cases, the ATR points can be obtained from the already existing Heegner points, thanks to results from Zhang and Darmon–Rotger–Zhao. In this paper, we compute for the first time an algebraic ATR point on a curve that is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon’s construction works beyond geometric modularity. To this pur- pose, we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic base field be norm-Euclidean and accelerating the numerical integration of Hilbert modular forms.

  • Continued fractions in 2-stage Euclidean quadratic fields

     Guitart Morales, Xavier; Masdeu, Marc
    Mathematics of computation
    Date of publication: 2013
    Journal article

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  • Abelian varieties with many endomorphisms and their absolutely simple factors

     Guitart Morales, Xavier
    Revista matemática iberoamericana
    Date of publication: 2012
    Journal article

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    We characterize the abelian varieties arising as absolutely simple factors of GL2-type varieties over a number field k. In order to obtain this result, we study a wider class of abelian varieties: the k-varieties A/k satisfying that End0 k(A) is a maximal subfield of End0 ¯k (A). We call them Ribet–Pyle varieties over k. We see that every Ribet–Pyle variety over k is isogenous over ¯k to a power of an abelian k-variety and, conversely, that every abelian k-variety occurs as the absolutely simple factor of some Ribet–Pyle variety over k. We deduce from this correspondence a precise description of the absolutely simple factors of the varieties over k of GL2-type.

  • Fields of definition of building blocks with quaternionic multiplication

     Guitart Morales, Xavier
    Acta arithmetica
    Date of publication: 2012
    Journal article

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  • Remarks on strongly modular jacobian surfaces

     Guitart Morales, Xavier; Quer Bosor, Jordi
    Journal de théorie des nombres de Bordeaux
    Date of publication: 2011
    Journal article

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  • Varietats abelianes sobre Q i formes modulars

     Guitart Morales, Xavier
    Date of publication: 2010-03
    Book chapter

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  • Arithmetic properties of abelian varieties under Galois conjugation

     Guitart Morales, Xavier
    Defense's date: 2010-06-11
    Department of Applied Mathematics II, Universitat Politècnica de Catalunya
    Theses

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  • ARITMÉTICA DE VARIEDADES ALGEBRAICAS: TEORIA Y COMPUTACIÓN

     Fite Naya, Francesc; Guitart Morales, Xavier; Rio Doval, Anna; Rotger Cerdà, Victor; Quer Bosor, Jordi; Vela Del Olmo, Maria Montserrat; Castellà, Francesc; Gómez Molleda, M. Ángeles; Masdeu, Marc; Lario Loyo, Joan Carles
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    On the modularity level of modular abelian varieties over number fields  Open access

     Gonzalez Jimenez, Enrique; Guitart Morales, Xavier
    Journal of number theory
    Date of publication: 2010-07
    Journal article

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    Let f be a weight two newform for Γ1(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 ResL/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 NL (B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree, we find that NL (B) belongs to Z and NL (B)f dim B L = N dim B, where fL is the conductor of L

    Electronic version of an article published as "Journal of number theory", vol. 130, no 7, p. 1560-1570. DOI no 10.1016/j.jnt.2010.03.003.

  • Parametrization of Abelian K-surfaces with quaternionic multiplication

     Guitart Morales, Xavier; Molina Blanco, Santiago
    Comptes rendus de l'Académie des sciences. Série 1, Mathématique
    Date of publication: 2009-12
    Journal article

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    We prove that the Abelian K-surfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin–Lehner involutions.

  • GRUP DE RECERCA EN TEORIA DE NOMBRES DE LA UPC

     Rotger Cerdà, Victor; Vela Del Olmo, Maria Montserrat; Jimenez Urroz, Jorge; Rio Doval, Anna; Lario Loyo, Joan Carles; Tramuns Figueras, Eulalia; Gonzalez Rovira, Josep; Fernandez Gonzalez, Julio; Fite Naya, Francesc; Guitart Morales, Xavier; Molina Blanco, Santiago; Guardia Rubies, Jordi; Quer Bosor, Jordi
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    Tipus de reduccions de corbes  Open access

     Guitart Morales, Xavier
    Date of publication: 2008-01-31
    Book chapter

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    Publicat originalment dins la col·lecció "Notes del Seminari de Teoria de Nombres (UB-UAB-UPC)". Disponible a: