Rational points on Shimura curves and Galois representations

This thesis explores one of the essential arithmetical and diophantine properties of Shimura curves and their Atkin-Lehner quotients: the existence of rational points on these families of curves over both number fields and their completions. Due to their moduli interpretation, Shimura curves (and modular curves) are of great arithmetic significance. The research line started by the work of Mazur on rational points on modular curves, leading to the classification of rational torsion subgroups of elliptic curves over Q, has been intensively and successfuly explored by many authors, and the general philosophy is that rational points on modular and Shimura curves over number fields should correspond only to CM-points, except for a few exceptional cases. Aiming to provide more evidence in support of this philosophy, in this thesis we propose new approaches for studying the lack of rational points over number fields on Shimura curves and their Atkin-Lehner quotients. Furthermore, we also wish to show that these curves provide a wealth of counterexamples to the Hasse principle, hence they can be used to test cohomological obstructions to this local-global principle, as for example the Brauer-Manin obstruction. The thesis is divided into two parts. The first of them is devoted to study the arithmetic and the geometry of the cyclic Galois coverings of Shimura curves introduced by Jordan. On the one hand, we determine the group of modular automorphisms of the Shimura curves arising from these coverings, showing in particular that Atkin-Lehner involutions can be lifted through them. As a consequence, we can produce cyclic étale coverings of Atkin-Lehner quotients of Shimura curves, which can be used to study the (non-)existence of rational points on these curves by applying descent techniques. Further, we characterise the existence of local points at bad reduction primes on both the intermediate curves of Jordan's coverings and their quotients by Atkin-Lehner involutions. This part of the thesis exploits the adèlic formalism of Shimura curves, as well as the padic uniformisation theory of Cerednik and Drinfeld, generalising previous work of Jordan-Livné and Ogg. In the second part of the thesis, we propose and investigate a method for proving the non-existence of rational points over a number field K on a coarse moduli space X of abelian varieties with additional structure, with special interest in cases where the moduli problem is not fine and K-rational points may