A coarse geometric expansion of a variant of Arthur’s truncated traces and some applications

Let F be a global function field with constant $\mathbb{F}_q$. G reductive group over We establish variant of Arthur's truncated kernel for and its Lie algebra which generalizes original construction. coarse geometric expansion our truncation. As applications, we consider some existence uniqueness problems cuspidal automorphic representations the functions projective line $\mathbb{P}^1_{\mathbb{F}_q}$ two points ramifications.