Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
The Fundamental Lemma (after Ngô)
Torsten Wedhorn
1. März 2011
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Adeles
Recall the ring of adeles of Q
Y
AQ := { ((xp ), x∞ ) ∈
Qp × R ; almost all xp ∈ Zp }.
p prime
More generally for F ⊃ Q finite extension:
rest
Y
AF :=
Fv .
v place of F
Then: AF locally compact topological F -algebra.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Adelic Lie groups
For G reductive algebraic group over F (e.g., GLn , Sp2n ):
rest
Y
G (AF ) =
G (Fv )
v place of F
This is a locally compact unimodular topological group with G (F )
as discrete subgroup.
For (over-)simplicity: Assume that G (F )\G (AF ) is compact.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Spectral decomposition
G (AF ) (more precisely its “group algebra” Cc∞ (G (AF ))) acts on
the Hilbert space L2 (G (F )\G (AF )) by right translation via
compact operators.
Obtain Hilbert space direct sum of irreducible unitary
representations (“spectral decomposition”):
M
mπ π
L2 (G (F )\G (AF )) =
π
Automorphic representations are representations of G (AF )
occuring in the spectral decomposition.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
(Arthur-)Selberg trace formula
Selberg trace formula
For ϕ ∈ Cc∞ (G (AF )) one has
X
X
vol(Gγ (F )\Gγ (AF ))Oγ (ϕ) =
mπ tr(π(ϕ)),
π
γ∈G (F )/conj
where Gγ is the centralizer of γ in G and
Z
Oγ (ϕ) :=
ϕ(g −1 γg ) dg
Gγ (AF )\G (Af )
the “orbital integral”.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Langlands functoriality
Upshot (without the assumption “G (F )\G (AF ) compact”):
Automorphic representations are linked to orbital integrals.
Langlands correspondence predicts a relation between automorphic
representations of G (AF ) and representations of Gal(F̄ /F ) (in the
dual group of G ).
More generally: Conjectural (!) Langlands functoriality
{relations between Galois representations (or L-groups)}
?
↔ {relations between automorphic representations}
↔ {relations between orbital integrals}
Fundamental Lemma: comparison of (stable) orbital integrals for
different groups
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Reduction to a local question
Orbital integrals for G (AF ) are products of orbital integrals for
G (Fv ) for all places v of F : Can work over local fields.
D. Shelstad (1982): Fundamental Lemma holds for archimedean
fields.
T. Hales (1995), J.-L. Waldspurger (1997): It suffices to show
relation of orbital integrals for all but finitely many
non-archimedean places of F .
J.-L. Waldspurger (2006): It suffices to show a simplified version
for Lie algebras over local fields of characteristic p.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Notation
Fq finite field of characteristic p,
F = Fq ((t)), OF = Fq [[t]] its ring of integers,
G reductive group over F .
Set
g := Lie(G ),
endowed with the adjoint action by G .
For γ ∈ g(F ) let Gγ ⊂ G be its centralizer.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Assumptions
(a) G is unramified, i.e., G (and g) is already defined over Fq .
(b) p does not divide order of the absolute Weyl group of G .
Example:
(1) G = GLn,F and p > n, G = Sp2n,F and p > max(2, n),
(2) G = Un,E /F and p > n, where E = Fq2 ((t)) ⊃ F quadratic
unramified extension, hermitian form given by


1
. 
J :=  . .
∈ Mn (F ).
1
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Endoscopic groups
To simplify: Assume that Cent(G ) is a torus
(e.g., G = GLn , G adjoint, but not G = SLn for n > 1).
Let G ∨ /C be the dual group of G and T ∨ ⊂ G ∨ its maximal torus.
Definition: Unramified endoscopic groups of G are unramified
reductive groups H over F of the following form:
(1) Choose κ ∈ T ∨ .
(2) Let H ∨ := Gκ∨ (centralizer of κ in G ∨ ).
(3) Let H be the dual group of H ∨ .
In general: Endoscopic groups are certain forms of the dual group
of (Gγ∨ )0 depending also on an automorphism of π0 (Gκ∨ ).
Remark: T ∨ maximal torus of H ∨ and κ ∈ (T ∨ )Gal(F̄ /F ) .
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Examples for endoscopic groups
Example I: G = GLn
P
Then G ∨ = GLn and H ∨ ∼
ni = n.
= GLn1 × · · · × GLnr with
Therefore: Endoscopic groupsPof GLn are groups of the form
H∼
ni = n.
= GLn1 × · · · × GLnr with
Example II: G = Sp2n
Then G ∨ = SO2n+1 (J). Choose κ := diag(1, −1, . . . , −1). Then
(Gκ∨ )0 = SO2n (J).
Therefore: SO2n (J) = SO2n (J)∨ endoscopic group of Sp2n .
Note: No non-trivial homomorphisms SO2n (J) → Sp2n (n ≥ 3).
Example III: G = SL2
Unramified Endoscopic groups: SL2 , GL1 , U1,E /F (E ⊃ F
unramified).
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Transfer to endoscopic groups
Fact: H unramified endoscopic group of G . Then every maximal
torus TH of H is isomorphic to a maximal torus TG of G and
WH := NormH (TH )/TH ⊂ WG := NormG (TG )/TG
Set tH := Lie(TH ) and tG := Lie(TG ). Obtain transfer map
ν : tH /WH → tG /WG .
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Characteristic polynomials for GLn
Let d ⊂ gln be the Cartan subalgebra of diagonal matrices,
W ∼
= Sn its Weyl group, permuting the entries of d.
Sending g ∈ gln to its unordered tuple of eigenvalues (with
multiplicity) yields a morphism (“characteristic polynomial”)
χGLn : gln → d/W .
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Characteristic polynomials for G
For general G with Lie algebra g have a similar morphism (of
algebraic varieties)
χG : g → tG /WG ,
with tG Cartan subalgebra of g.
B. Kostant: There is a canonical section
εG : tG /WG → g
of χG (“companion matrix”).
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Transfer for endoscopic groups
Obtain for every endoscopic group H a diagram (with h = Lie(H)):
gS
hK
χH
εH
cH := tH /WH
Torsten Wedhorn
χG
ν
εH
/ cG := tG /WG
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Regular semisimple elements
Call a ∈ tG regular semisimple if its stabilizer in WG is trivial. Let
crs
G ⊂ cG = tG /WG
be the set of WG -orbits of regular semisimple elements (Zariski
open in cG ).
−1 rs
grs := χG
(cG ) = { γ ∈ g ; Gγ is torus }.
Example: a ∈ d regular semisimple iff all entries are different,
γ ∈ glrs
n iff all eigenvalues (in some algebraic closure) are different.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Stable conjugacy
Definition: Call γ, γ 0 ∈ grs (F ) stably conjugate if the following
equivalent conditions are satisfied:
(i) χG (γ) = χG (γ 0 ).
(ii) γ and γ 0 are conjugate by an element of G (F̄ ).
Example: For G = GLn : γ, γ 0 ∈ glrs
n (F ) stably conjugate if and
only if they are conjugate by an element of G (F ).
0 1 , 0 −1 conjugate under SL (F ) if and only if
Example: −1
2
0
1 0
−1 is a sum of two squares in F .
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Conjugacy and stable conjugacy
rs
Notation: Fix a ∈ crs
G (F ). Set γ0 := ε(a) ∈ g (F ).
Then: T := Gγ0 is maximal torus of G .
The map
γ 7→ inv(γ, γ0 ) := { g ∈ G ; Ad(g )(γ) = γ0 } ∈ H 1 (F , T )
yields a bijection
G (F )-conjugacy
↔ Ker(H 1 (F , T ) → H 1 (F , G )) := Aγ0 .
classes in χ−1
(a)
G
Right hand side is finite abelian group.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Tate-Nakayama duality
A finite group. Set AD := Hom(A, C× ) (Pontryagin dual of A).
Tate-Nakayama duality: Obtain homomorphisms
(T ∨ )Gal(F̄ /F ) → π0 ((T ∨ )Gal(F̄ /F ) )
∼
= H 1 (F , T )D
→ AD
γ0 ,
κ 7→ κ̄.
Thus κ ∈ (T ∨ )Gal(F̄ /F ) yields a character on
G (F )-conjugacy
classes in χ−1
G (a)
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Stable orbital integrals
For ϕ ∈ Cc∞ (g(F )) define the κ-orbital integral
X
Oaκ (ϕ) :=
κ̄(inv(γ, γ0 ))Oγ (ϕ),
γ∈χ−1
G (a)/conj
with
Z
ϕ(g −1 γg )dg /dgγ .
Oγ (ϕ) :=
Gγ (F )\G (F )
For κ = 1 we obtain the stable orbital integral
X
SOa (ϕ) := Oa1 (ϕ) =
Oγ (ϕ).
γ∈χ−1
G (a)/conj
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Fundamental lemma
Theorem (Ngô 2008): Let H be an unramified endoscopic group
of G given by κ ∈ (T ∨ )Gal(F̄ /F ) .
rs
Let aH ∈ crs
H (F ) such that aG := ν(aH ) ∈ cG (F ). Then
OaκG (1g(OF ) ) = q r (aH ) SOaH (1h(OF ) ),
where r (aH ) is a certain (computable) integer.
Note: By Fourier inversion orbital integrals are linear combinations
of κ-orbital integrals.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
The fundamental lemma is combinatorial I
Example: G = GLn,F , V := F n , γ : V → V linear endomorphism
with pairwise distinct eigenvalues (i.e., γ ∈ grs ).
Then Gγ ∼
= E1× × · · · × Er× , where Ei ⊂ F finite extension.
An OF -submodule Λ of V is called lattice if it generates V as
F -vector space. Note:
G (F )/G (OF ) = {lattices in V }.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
The fundamental lemma is combinatorial II
Mγ := { Λ lattice of V ; γ(Λ) ⊂ Λ }
= { g ∈ G (F )/G (OF ) ; ad(g )−1 (γ) ∈ g(OF ) }.
Then the orbit set Mγ /Gγ (F ) is finite and
X
SOγ (1g(OF ) ) =
x∈Mγ /Gγ (F )
1
,
vol(Gγ (F )x )
where Gγ (F )x is the stabilizer of x in Gγ (F ) (a compact subgroup).
Have similar description for κ-orbital integrals and for other
classical groups.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Express stable orbital integrals geometrically
Fix γ ∈ g(F )rs .
D. Kazhdan, G. Lusztig (1988): Exists Fq -scheme Mγ (“affine
Springer fiber”) such that
Mγ (Fq ) = { g ∈ G (F )/G (OF ) ; ad(g )−1 (γ) ∈ g(OF ) }.
It carries an action by a group scheme Pγ such that the action of
Gγ (F ) factors through Gγ (F ) Pγ (Fq ).
Thus the quotient stacks calculates stable orbital integrals:
#[Mγ /Pγ ](Fq ) = SOγ (1g(OF ) ).
There is also a variant for κ-orbital integrals.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Grothendieck-Lefschetz machine
Note: Mγ and Pγ are finite-dimensional but only locally of finite
type.
Via Grothendieck-Lefschetz (identifying C ∼
= Q̄` ):
Oγκ (1g(OF ) ) = (#Pγ0 (Fq ))−1 tr(Frobq , H ∗ (Mγ , Q̄` )κ ).
Fact: Mγ and Pγ depend essentially only on a := χG (γ).
Notation: Ma , Pa .
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Globalization
Fix X smooth projective geometrically connected curve over Fq of
genus g ≥ 2. L a line bundle on X .
Think: OF = ObX ,v for some v ∈ X .
Write Mv ,a , Pv ,a instead of Ma , Pa .
Example (SL2 ): Higgs bundles for SL2 : pairs (V , φ) with
(a) V is a vectorVbundle of rank 2 on X with trivialized
determinant 2 V = OX (“SL2 -bundle”),
(b) φ : V → V ⊗ L such that tr(φ) = 0 ∈ H 0 (X , L ).
Let M be the moduli space of all Higgs bundles (an Artin stack).
Set A := H 0 (X , L ⊗2 ) and define the “characteristic polynomial”:
f : M → A, (V , φ) 7→ det(φ).
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Original construction
Original construction (Hitchin, 1987): L = Ω1X canonical bundle
of 1-forms.
Then (Serre duality): M is the cotangent bundle over the moduli
space of principal G -bundles over its stable locus. Thus it carries a
symplectic structure.
The association (V , φ) 7→ det(φ) ∈ H 0 (X , (Ω1X )⊗2 ) defines a
family of d = dim H 0 (X , (Ω1X )⊗2 ) = 12 dim M Poisson-commuting
algebraically independent functions. In other words, M is an
algebraic completely integrable system.
For arbitrary L : Loose symplectic structure.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Hitchin fibration
Generalization to arbitary G :
Assume L sufficiently ample (L = (L 0 )⊗2 with deg L 0 > g ).
Globalization of the characteristic polynomial (+ twisting by L ):
Obtain “Hitchin fibration” f : M → A and a Picard stack
π : P → A acting on M such that for all a ∈ A:
Y
[Ma /Pa ] =
[Mv ,a /Pv ,a ],
v ∈X
where Ma := f −1 (a) and Pa := π −1 (a).
Advantage: Hitchin fibers behave much better in families then
affine Springer fibers.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Elliptic part
There exists an open Aell of A, where M → A is proper and
P → A is of finite type.
Main technical part of Ngô’s proof:
Relate p H ∗ (f∗ell Q̄` ) for G and its endoscopic groups.
For this Ngô proves a deep theorem in the theory of perverse
sheaves:
p H ∗ (f ell Q̄ ) is a direct sum of simple perverse sheaves
`
∗
(Decomposition theorem). Ngô determines their supports.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Variants
There exist several variants of the fundamental lemma:
• Transfer conjecture (“Fundamental Lemma” for arbitrary
locally constant functions on G (F ) with compact support
(instead of 1G (OF ) )): Follows from Fundamental lemma
(Waldspurger).
• Non-standard Fundamental Lemma (relating stable orbital
integrals for semisimple groups with isogeneous root systems):
Proved by Ngô on the way.
• Twisted fundamental lemma: Follows from Fundamental
lemma and the Non-standard Fundamental Lemma
(Waldspurger).
• Weighted fundamental lemma: Proved recently by Chaudouard
and Laumon, extending Ngô’s techniques.
Torsten Wedhorn
The Fundamental Lemma (after Ngô)
Motivation: Trace Formulas
Statement of the Fundamental Lemma
On the proof
Applications
Applications
The Fundamental Lemma has been used to prove the following
major theorems in number theory:
• Arthur’s forthcoming classification of automorphic
representations of classical groups.
• Sato-Tate conjecture for elliptic curves over totally real fields
(Barnet-Lamb, Geraghty, Harris, Taylor).
• A (weak version) of the global Langlands correspondence for
certain number fields (Shin).
• Cohomology of non-compact Shimura varieties and their Galois
representations (Morel).
• Iwasawa’s main conjecture for GL2 (Skinner, Urban)
• The Birch, Swinnerton-Dyer conjecture for a positive fraction
of elliptic curves over Q (Bhargava, Shankar).
Torsten Wedhorn
The Fundamental Lemma (after Ngô)