## Number theory seminar in memory of F. Momose

### Galois representations attached to points on Shimura curves.

#### Presenters

Carlos de Vera Piquero

#### Abstract

In this talk I will explain a method to study rational points over
a number field $K$ on a coarse moduli space $X$ of abelian varieties with endomorphism structure, especially in the case where the moduli problem is not fine
and points in $X(K)$ may not be represented by an abelian variety admitting a model rational over $K$. The main idea, inspired on the work of Ellenberg and
Skinner on the modularity of $\mathbb{Q}$-curves, is that we can attach certain Galois representations to points in $X(K)$ rather than to the abelian varieties representing them.
This method can be applied to extend some results of Jordan (1986) and
Skorobogatov (2005) on the non-existence of rational points on Shimura curves
over imaginary quadratic fields, obtaining new counterexamples to the Hasse
principle that are accounted for by the Brauer-Manin obstruction. It can also
be applied to produce examples of Atkin-Lehner quotients of Shimura curves
violating the Hasse principle over $\mathbb{Q}$. This is joint work with V. Rotger.

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