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

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 $\mathbb{Q}$ .