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

Step 8 of 9: From Lie algebras in positive characteristic to the full lemma

In plain words

Steps 3 to 7 proved the fundamental lemma only for Lie algebras and only over power-series fields. Two more translations, both already flagged in Step 2, are needed: moving from power series to $p$ -adic numbers, and moving from Lie algebras back to groups.

Both translations were understood in principle before Ngô's paper appeared, so once the geometric argument closed the hardest case, the rest followed by combining already-available machinery, like the last pieces of a puzzle snapping into place.

\mathrm{FL}(\mathfrak{g}, \mathbb{F}_q((t))) \ \Longrightarrow \ \mathrm{FL}(\mathfrak{g}, F) \ \Longrightarrow \ \mathrm{FL}(G, F) \text{ and its } \kappa\text{-weighted version}

Detailed analysis

The transfer from equal characteristic ( $\mathbb{F}_q((t))$ ) to mixed characteristic ( $p$ -adic fields) for the Lie algebra statement is supplied by the Cluckers–Hales–Loeser motivic transfer principle (announced 2005, published 2011), a model-theoretic argument based on the Ax–Kochen–Ershov transfer principle showing that certain identities between $p$ -adic integrals depend only on the residue characteristic being large enough, and can therefore be checked in the more tractable function-field setting Ngô already solved.

The transfer from Lie algebras back to groups is Waldspurger's reduction from Step 2, run in the other direction: since Ngô proved the Lie-algebra fundamental lemma unconditionally, Waldspurger's equivalence immediately yields the fundamental lemma for the group $G$ itself, and further work of Waldspurger extends it to the $\kappa$ -weighted, twisted-endoscopic version needed in Arthur's trace formula.

Putting Steps 2 through 8 together completes a proof of the Langlands–Shelstad fundamental lemma, in its most general form, for every reductive group over every local field.

Terms in this step

Ax–Kochen–Ershov transfer principle: A 1965 result in model theory stating that many first-order statements about $p$ -adic fields and about power-series fields $\mathbb{F}_p((t))$ of the same characteristic $p$ become equivalent once $p$ is large enough, letting one transfer proofs between the two worlds.

Knowledge used in this step

Lie groups and Lie algebras