Global topology of the Hitchin system - GEOM

Heuristics from TMS to FL
let Γ be the finite group scheme π0(ˇPell) over A 0 ell let κ ∈ ˆΓ ∗
(recall ˇPa is the Prym variety of Ca → C)
(Ngô 2008) proves the fundamental lemma in the Langlands
program by proving a cohomological statement for the elliptic
part of the Hitchin fibration, which in the SLn-case is a sheaf
version of H ∗ p( ˜ˇMell)κ ∼ = H ∗+r
p+r (Mell(Hκ))st
there is a natural map f : Γ = JacC [n] ∼ = Z 2g
n → Γ
❀ the action of Γ on H ∗ ( ˇMell) filters through the action of Γ
❀ κ ∈ ˆΓ defines κf ∈ ˆΓ, and H ∗ κf ( ˇMell) ∼ = H ∗ κ( ˇMell)
the refined TMS restricted to ˇMell ❀
H∗ ˇMell)κf p( ∼ ∗+F (γ)
= Hp+r ( ˇMγ/Γ) where w(κf ) = γ
ˇMγ can be identified with Hγ-endoscopy Hitchin systems
❀ TMS|ell ⇔ FL for SLn
❀ Ngô’s proof of FL ❀ TMS|ell, which when n is prime can
be extended to TMS for the whole ˇM
if we allow the Higgs field to have values in D such that
deg(D) > 2g − 2, then we can deduce TMS for every n from
FL from (Chaudouard-Laumon 2009) 39 / 39