Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

Ngô carries out the same decomposition-and-support analysis for the endoscopic group $H$, whose parameter space $A_H$ maps naturally into $A$ via the embedding of $H$'s root data into $G$'s. Deligne's purity theorem, from his proof of the Weil conjectures, bounds the Frobenius weights appearing in the cohomology of both fibrations, and combined with the support theorem this forces the point counts of the $G$-fiber and the transported $H$-fiber over generic $a$ to coincide exactly, not merely up to a bounded error (Ngô 2010, §7–8; Hales 2011, §6).

By the product formula for masses — the global count factors as a product of local affine Springer fiber counts, one per place of the curve, almost all trivially equal because $\gamma$ is unramified there — this global equality forces the individual local terms to match at every remaining place, precisely the $\kappa$-weighted orbital integral identity conjectured by Langlands and Shelstad, now established for Lie algebras of any reductive group over a local field of positive characteristic.

This is the climax of Ngô's paper: a global, geometric statement matching cohomology of two families of varieties has been translated back into the local, arithmetic statement from Step 1.