Arithmetic Aspects of Hilbert Modular Forms and Varieties

Abstract

Hilbert modular forms and varieties are the natural generalization of elliptic modular forms and curves, when the ground field of rational numbers is replaced by a totally real number field. The aim of these notes is to present the basics of their arithmetic theory and to describe some of the recent results in the area. A special emphasis will be put on the following two subjects: images of Galois representations associated to Hilbert modular forms and cohomology of Hilbert modular varieties with integral coefficients.