Seminar

At the heart of the Langlands program lies the reciprocity conjecture, which can be thought of as a non-abelian generalization of class field theory. An example is the correspondence between modular forms and representations of the absolute Galois group of Q. This can be realized geometrically in the cohomology of modular curves, making essential use of their structure as algebraic curves.  

In this talk, I will describe some techniques involved in the recent work of Harris-Lan-Taylor-Thorne and Scholze, who construct Galois representations associated to systems of Hecke eigenvalues occurring in the cohomology of locally symmetric spaces for GL_n. These are real manifolds which generalize modular curves, but lack the structure of algebraic varieties. I will then focus on a very specific property of these Galois representations: the image of complex conjugation, which can be identified by combining Hodge theory with p-adic interpolation techniques. Finally, I will mention some open problems.