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

Step 4 of 9: Globalize with the Hitchin fibration over a curve

In plain words

Instead of studying the fixed-lattice picture at one place at a time, imagine spreading the same data continuously over an entire algebraic curve $X$ , where each point behaves like a separate local field. The moduli space of Higgs bundles on $X$ packages, all at once, a whole family of affine-Springer-fiber-like spaces, one for every point $a$ of a parameter space $A$ .

The payoff is that counting points on a single global fiber over a well-chosen $a$ automatically computes the combination of local counts at every place at once, turning a purely local question into a single geometry question.

M_{Higgs}(G, X) \ \twoheadrightarrow \ A, \quad \text{fibre over } a \in A \ \leftrightarrow \ \{\text{affine Springer fibers at } \gamma_a\}

Detailed analysis

Ngô's key idea is global: the moduli space of $G$ -Higgs bundles on a projective curve $X$ carries a Hitchin map $f: M \to A$ to an affine space $A$ (the Hitchin fibration, built from the characteristic polynomial of the Higgs field), and by the Weil conjectures, counting points on the fiber over $a \in A$ via the Grothendieck–Lefschetz trace formula equals a product of local orbital integrals at the corresponding elements $\gamma_a$ , taken over every place of $X$ at once (Ngô 2010, §4; Hales 2011, §5).

Because the local field $\mathbb{F}_q((t))$ from Step 3 is literally the completed local ring at a point of such a curve, the Hitchin fiber's cohomology decomposes, place by place, into exactly the affine Springer fibers already described — a 'product formula for masses' relating the global count to the local ones.

So the fundamental lemma, a purely local statement, becomes a question about the cohomology of a single, highly structured global family of algebraic varieties over $\mathbb{F}_q$ .

Terms in this step

Hitchin fibration: A map $f: M \to A$ from the moduli space $M$ of $G$ -Higgs bundles on an algebraic curve $X$ to an affine space $A$ , via characteristic polynomials of the Higgs field; introduced by Nigel Hitchin in 1987 for a different question in gauge theory.
Higgs bundle: A pair of a vector bundle on a curve $X$ together with a twisted endomorphism (the Higgs field); its characteristic polynomial gives the point of $A$ under the Hitchin map.

Knowledge used in this step

Algebraic geometry Arithmetic geometry