Rational points on Shimura curves and Galois representations

• Autores: Carlos De Vera Piquero
• Directores de la Tesis: Victor Rotger Cerdà (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2014
• Idioma: catalán
• Tribunal Calificador de la Tesis: Jordi Quer Bosor (presid.) , Marc Masdeu Sabaté (secret.) , Matteo Longo (voc.)
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • 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 not be represented by abelian varieties admitting a model over K (which is the generic situation if the abelian varieties being classified have even dimension). The original inspiration dates back to the works of Mazur and Jordan, in which the authors study the existence K-rational points on modular and Shimura curves, respectively, through the Galois representations attached to the elliptic curves and abelian surfaces parametrised by them. However, they need to assume these varieties to be defined over K (their field of moduli), a hypothesis which need not be correlated to the non-existence of K-rational points on the moduli space. To overcome this, we attach Galois representations to K-rational points on X rather than to the abelian varieties classified by them (what we call "Galois representations over fields of moduli"), inspired by the work of Ellenberg and Skinner on the modularity of Q-curves. We exemplify our method, combined with the cyclic coverings studied in the first part of the thesis, in the case where X is either a Shimura curve or an Atkin-Lehner quotient of it and K is an imaginary quadratic field or the field of rational numbers, respectively. In both cases, we produce new counterexamples to the Hasse principle. And moreover, in the first case we prove that these counterexamples are accounted for by the Brauer-Manin obstruction. The results of this second part complement previous work of Parent-Yafaev, Gillibert or Clark, for example.