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Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

Published online by Cambridge University Press:  20 November 2018

Victor Rotger  and
Carlos de Vera-Piquero
Victor Rotger
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain. e-mail: victor.rotger@upc.edu. e-mail: carlos.de.vera@upc.edu
Carlos de Vera-Piquero
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain. e-mail: victor.rotger@upc.edu. e-mail: carlos.de.vera@upc.edu
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Abstract

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The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$ in cases where the moduli problem is not fine and points in $X\left( K \right)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that one may still attach a Galois representation of $\text{Gal}\left( \bar{K},\,K \right)$ with values in the quotient group $\text{GL}\left( {{T}_{\ell }}\left( A \right) \right)/\,\text{Aut}\left( A \right)$ to a point $P\,=\,\left[ A \right]\,\in \,X\left( K \right)$ represented by an abelian variety $A/\bar{K}$, provided $\text{Aut}\left( A \right)$ lies in the centre of $\text{GL}\left( {{T}_{\ell }}\left( A \right) \right)$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over $\mathbb{Q}$.

Keywords

11G1814G3514G05Shimura curvesrational pointsGalois representationsHasse principleBrauer–Manin obstruction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 5 , 01 October 2014 , pp. 1167 - 1200
Copyright
Copyright © Canadian Mathematical Society 2014

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