Wojciech J. Gajda

Abstract:

Let G be a split reductive group over a finite field F_q and let K be a global function field over F_q with adele ring A_K. 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 G^ of G. Suppose conversely that we are given an ell-adic representation rho of Gal(K^{sep}/K) into G^. If rho is potentially unramified and has Zariski dense image, then in joint work with C. Khare and J. Thorne we show that rho is potentially automorphic, i.e., there is a finite extension K' of K, such that the restriction of rho to Gal(K^{sep}/K') arises from V. Lafforgue's construction over K'. The talk aims at explaining some of the building blocks that go into the proof of the result and at suggesting some potential applications.