October 31, 2016
The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over $\mathbb Q$ are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is meant to connect number theory with harmonic analysis and representation theory. In this talk, I will survey some exciting recent progress in establishing new reciprocity laws, namely how to construct Galois representations attached to torsion classes which occur in the cohomology of arithmetic hyperbolic $3$-manifolds. This was done by Peter Scholze using $p$-adic geometry and his theory of perfectoid spaces. I will then outline some ideas for better understanding these Galois representations which also crucially rely on $p$-adic geometry.