Nearby cycles of parahoric shtukas, and a fundamental lemma for base change

TL;DR

This paper computes Frobenius-Hecke trace formulas for nearby cycles of shtukas at parahoric places and uses this to prove a base change fundamental lemma for parahoric Hecke algebras, extending Ngo's work to positive characteristic.

Contribution

It provides a geometric proof of the base change fundamental lemma for parahoric Hecke algebras in positive characteristic, generalizing Ngo's spherical case.

Findings

Computed trace of Frobenius with Hecke operators on nearby cycles

Proved a base change fundamental lemma for parahoric Hecke algebras

Extended Ngo's theorem to positive characteristic for GL_n

Abstract

Using the Langlands-Kottwitz paradigm, we compute the trace of Frobenius composed with Hecke operators on the cohomology of nearby cycles, at places of parahoric reduction, of perverse sheaves on certain moduli stacks of shtukas. Following an argument of Ngo, we then use this to give a geometric proof of a base change fundamental lemma for parahoric Hecke algebras for GL_n over local function fields. This generalizes a theorem of Ngo, who proved the base change fundamental lemma for spherical Hecke algebras for GL_n over local function fields, and extends to positive characteristic (for GL_n) a fundamental lemma originally introduced and proved by Haines for p-adic local fields.