Application of Ngo's fundamental lemma

Question

I'm studying Arthur-Selberg trace formula and trying to find various applications of it, from classical ones (Eichler-Selberg formular and the dimension of space of modular forms) to modern ones (Jacquet-Langlands correspondence). I found that the Fundamental lemma is really important theorem since it gives a way to compare geometric side of two different groups. However, I can't find an easy(?) example that uses Fundamental lemmma directly and prove something usuful. As I know, we don't need Fundamental lemma to prove Jacquet-Langlands correspondence (at least for $\mathrm{GL}_{2}$.) Is there any good applications?

Answer 1

The fundamental lemma is a local statement that you want to prove at almost all places to (help) establish a geometric matching of trace formulas. For the Jacquet-Langlands correspondence, the fundamental lemma is trivial, because the groups are isomorphic at almost all places. An "easy" case where the fundamental lemma is nontrivial is, for instance, Langlands proof of base change for GL(2) (though this is not an easy read).

There are various fundamental lemmae for various trace formula comparisons. Ngo's case is important for Arthur's work on endoscopic classification for classical groups.