At the heart of the Langlands program lies the reciprocity conjecture, which encompasses quadratic reciprocity as well as the correspondence between modular forms and representations of the absolute Galois group of $Q$. The latter can be realized geometrically through quotients of the upper half plane, making essential use of their structure as algebraic curves.

For a general connected reductive group, torsion classes in the cohomology of the associated locally symmetric space generalize modular forms. In this talk, I will describe some of the ideas behind the recent work of Scholze, who constructs Galois representations for torsion classes occurring in the cohomology of higher-dimensional locally symmetric spaces, such as arithmetic hyperbolic 3-manifolds. In general, these spaces lack the structure of algebraic varieties. I will focus first on explaining how to get back to the algebraic world and then give a flavor of the beautiful $p$-adic geometry underlying the construction.