... m-dimensional Galoisrepresentation with m>n, has an analytic continuation and satisfies a func-tional equation.More precisely, let X be a curve over a finite field, and K the correspondingglobal ... [7].Let, in addition,Qk,exbe the stack classifying the same data (M,κ1, ,κk)as above, but where we allow the last map, i.e., κk, to vanish. In particular,Qkis an open substack in Qk,ex.We ... that areused in the argument proving Theorem 2.14.1. The conjecture1.1. We will first recall the formulation of the Vanishing Conjecture, asit was stated in [7]. Let Bunnbe the moduli stack...