A Scholzian Approach to the Local Langlands Correspondence for $$\mathrm{GL}_n$$ over function fields

Author/​Artist:
Li, Daniel [Browse]
Format:
Senior thesis
Language:
English
Let $$F$$ is a local field of characteristic $$p$$. Inspired by work of Scholze, we construct a map $$\pi\mapsto\sigma(\pi)$$ from irreducible smooth representations of $$\mathrm{GL}_n(F)$$ to $$n$$-dimensional Weil representations of $$F$$. We prove that this map uniquely satisfies a purely local compatibility condition on traces of a test function $$f_{\tau,h}$$, and we also prove that this map is compatible with parabolically inducing tensor products. It is expected that $$\pi\mapsto\sigma(\pi)$$ equals the local Langlands correspondence for $$\mathrm{GL}_n$$ over $$F$$, up to Frobenius semisimplification.