# Abstract Gebhard Böckle

**Gebhard Böckle ** will speak on

** Potential automorphy for unramified \hat G-valued l-adic representations over global function fields**

Abstract: Let G be a split reductive group over a finite field F_q and let K be a global function field with constant field F_q. By fundamental work of Vincent Lafforgue any cuspidal automorphic representation of G(A_K) gives rise to a compatible system of Galois representation of Gal(K^sep/K) valued in the dual group \hat G of G. In joint work with M. Harris, C. Khare and J. Thorne, we investigate the question of when a \hat G-valued continuous l-adic representation of Gal(K^sep/K) is potentially automorphic, i.e. arises potentially from V. Lafforgue's construction. After an introduction and the statement of our potential modularity result, I will try to explain some of the building blocks that go into the result, and how they allow to deduce it.