$\ell$-adic Poisson Formula and Endoscopy for $p$-Adic Reductive Groups

TL;DR

This paper establishes an $ ext{ell}$-adic Poisson formula for reductive $p$-adic groups, linking trace identities, orbital integrals, endoscopy, and special values of motivic $L$-functions.

Contribution

It introduces an $ ext{ell}$-adic Poisson formula for $p$-adic groups and connects orbital integrals with motivic $L$-function values.

Findings

Proves an $ ext{ell}$-adic Poisson formula for reductive $p$-adic groups.

Relates orbital integrals to special values of motivic $L$-functions.

Reduces trace identities to endoscopy group orbital integrals.

Abstract

For two distinguished prime $\ell$ and $p$, we prove a $\ell$-adic version of the Poisson formula for reductive $p$-adic groups. In order to do this we write an identity for the trace of regular representation and orbital integrals. Next we reduce them to orbital integrals for endoscopy groups and look at this as the special value of $L$-function at $s=0$. And finally show that it is equal to the special value of motivic $L$-function at $s=0$.