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 Department of Applied Mathematics II
 School
 Terrassa School of Industrial and Aeronautical Engineering (ETSEIAT)
 xavier.guitartestudiant.upc.edu
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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: 20140201
Journal article
Read the abstract View Share Reference managersThe main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the Lfunction L(B/K; s) is a product of Lfunctions of nonCM 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: 201405
Journal article
Read the abstract View Share Reference managersLet 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 Darmonlike 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: 201303
Journal article
Read the abstract View Share Reference managersATR points were introduced by Darmon as a conjectural con struction of algebraic points on certain elliptic curves for which the Heegnerpoint 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 normEuclidean and accelerating the numerical integration of Hilbert modular forms. 
Continued fractions in 2stage 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
Read the abstract View Share Reference managersWe characterize the abelian varieties arising as absolutely simple factors of GL2type varieties over a number field k. In order to obtain this result, we study a wider class of abelian varieties: the kvarieties 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 kvariety and, conversely, that every abelian kvariety 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 GL2type. 
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: 201003
Book chapter
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Arithmetic properties of abelian varieties under Galois conjugation
Guitart Morales, Xavier
Defense's date: 20100611
Department of Applied Mathematics II, Universitat Politècnica de Catalunya
Theses
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Monogràfic sobre treballs de Kenneth Ribet
Date of publication: 201003
Book
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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
Gonzalez Jimenez, Enrique; Guitart Morales, Xavier
Journal of number theory
Date of publication: 201007
Journal article
Read the abstract Access to the full text Share Reference managersLet 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. 15601570. DOI no 10.1016/j.jnt.2010.03.003. 
Parametrization of Abelian Ksurfaces with quaternionic multiplication
Guitart Morales, Xavier; Molina Blanco, Santiago
Comptes rendus de l'Académie des sciences. Série 1, Mathématique
Date of publication: 200912
Journal article
Read the abstract View Share Reference managersWe prove that the Abelian Ksurfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the Krational 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
Guitart Morales, Xavier
Date of publication: 20080131
Book chapter
Read the abstract Access to the full text Share Reference managersPublicat originalment dins la col·lecció "Notes del Seminari de Teoria de Nombres (UBUABUPC)". Disponible a:
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