arXiv subjects
Publications and source records attributed to Jordi Quer.
We investigate the Brauer class of the endomorphism algebra of the motive attached to a non-CM form. The ramification of the algebra is shown in many cases to be controlled by the normalized slopes of the form.
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 zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups $Γ_1(N)$. 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, we show how to find nontrivial examples of strongly modular varieties by twisting, and prove the existence of strongly modular surfaces satisfying certain properties.
This document contains additional information for the paper: J. Quer. Fields of definition of building blocks, Math. Comp. 78 (2009), no. 265, pp. 537--554. It contains a description of the implementation in Magma of the algorithms and computational techniques described in that paper, and also several tables with data that were elaborated using this implementation.
Let M_2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P=[C] is defined over k. Mestre solved the converse problem for curves with Aut(C) isomorphic to C_2. Given a moduli point defined over k, Mestre finds an obstruction to the existence of a corresponding curve defined over k, that is an element in Br_2(k) not always trivial. In this paper we prove that for all the other possibilities of Aut(C), every moduli point defined over k is represented by a curve defined over k. We also give an explicit construction of such a curve in terms of the coordinates of the moduli point.
In this paper we classify curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12 over an arbitrary field k (of characteristic different from 2 in the D_8 case and from 2 and 3 in the D_{12} case) up to k-isomorphism. As an application of the classification of curves of genus 2 obtained, we get precise arithmetic information on their elliptic quotients and on their jacobians. Over the field k=Q, we show that the elliptic quotients of the curves with automorphisms D_8 and D_{12} are precisely the Q-curves of degrees 2 and 3, respectively, and we determine which curves have jacobians of GL_2-type.