Jack Thorne - The Ramanujan conjecture for Bianchi modular forms of weight 2
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2018-05-23

Let  K  be  an  imaginary  quadratic  field.  Conjecturally,  one  should  be  able  to  associate  to  any  cusp  form  on GL_n(A_K) which is cohomological (for the trivial coefficient system) a Galois representation. This can be achieved using our understanding of the classification of automorphic representations of the quasi-
split unitary group U(n, n), which relies upon the stabilization of the twisted trace formula for GL_n.
A detailed understanding of the local properties of these Galois representations opens up the possibility of proving automorphy lifting theorems. I will describe work in progress of a 10 author collaboration that proves such  theorems,  using  as  a  starting  point  very  important  vanishing  theorems  for  the  cohomology  of  non-compact Shimura varieties which are work in progress of Caraiani--Scholze.
A particular consequence is the Ramanujan conjecture for cohomological cusp forms in the case n = 2. (The 10 authors are Allen, F. Calegari, Caraiani, Gee, Helm, Le Hung, J. Newton, Scholze, Taylor, and myself.)