Endoscopic Decomposition of Elliptic Fargues–Scholze \(L\)-Packets

The main goal of this note is to show that the local $L$ $L$ -packets of Fargues–Scholze [FS21], corresponding to elliptic $L$ $L$ -parameters, admit an endoscopic decomposition. Our argument is strongly motivated by a beautiful paper of Chenji Fu [Fu25], where the stable case is proven.

To put our results in a more general context, we also construct a general endoscopic decomposition over ℂ based on results of Arthur, and a generalization of this decomposition over arbitrary algebraically closed fields of characteristic zero based on a recent work [KSV25].