p-adic periods and local Langlands correspondences
Laurent Fargues
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
`
´
I ∀s ∈ S,
Fs , Fil(F ⊗ OS )s defines a Hodge structure of weight
|{z}
|
{z
}
R-v.s.
of dim. 2
(1, 0), (0, 1)
C-line in Fs ⊗R C
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
`
´
I ∀s ∈ S,
Fs , Fil(F ⊗ OS )s defines a Hodge structure of weight
|{z}
|
{z
}
R-v.s.
of dim. 2
C-line in Fs ⊗R C
(1, 0), (0, 1) → Fil ∩ Fil = (0) (Hodge condition)
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
`
´
I ∀s ∈ S,
Fs , Fil(F ⊗ OS )s defines a Hodge structure of weight
|{z}
|
{z
}
R-v.s.
of dim. 2
I
C-line in Fs ⊗R C
(1, 0), (0, 1) → Fil ∩ Fil = (0) (Hodge condition)
∼
ρ : R2 −−→ F rigidification (kill the Betti monodromy)
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
`
´
I ∀s ∈ S,
Fs , Fil(F ⊗ OS )s defines a Hodge structure of weight
|{z}
|
{z
}
R-v.s.
of dim. 2
I
C-line in Fs ⊗R C
(1, 0), (0, 1) → Fil ∩ Fil = (0) (Hodge condition)
∼
ρ : R2 −−→ F rigidification (kill the Betti monodromy)
GL2 (R)
via the action on ρ
"
X
The archimedean case
X = moduli space of rigidifed rank 2 variations of Hodge structures of weight
(1, 0), (0, 1)
S = smooth complex analytic space
X (S) = {(F, Fil(F ⊗R OS ), ρ)}/ ∼ where :
I F local system of R-vector spaces of rank 2 on S
I Fil(F ⊗R OS ) locally direct factor sub bundle of rank 1
`
´
I ∀s ∈ S,
Fs , Fil(F ⊗ OS )s defines a Hodge structure of weight
|{z}
|
{z
}
R-v.s.
of dim. 2
I
C-line in Fs ⊗R C
(1, 0), (0, 1) → Fil ∩ Fil = (0) (Hodge condition)
∼
ρ : R2 −−→ F rigidification (kill the Betti monodromy)
"
GL2 (R)
X
via the action on ρ
Period morphism :
∨
X
,→
(F, Fil, ρ)
7−→
X = P1C
`
´
ρ−1
Fil ⊂ F ⊗ OS −−→ OS2
Hodge structure
7−→
Hodge filtration
∼
The archimedean case
The period morphism
∨
X ,→ X = P1C
is an GL2 (R)-equivariant embedding.
The archimedean case
The period morphism
∨
X ,→ X = P1C
is an GL2 (R)-equivariant embedding.
Image described by Hodge condition :
X = H± = C \ R ⊂ P1C
The archimedean case
The period morphism
∨
X ,→ X = P1C
is an GL2 (R)-equivariant embedding.
Image described by Hodge condition :
X = H± = C \ R ⊂ P1C
Uniformisation :
Y (S) = {Elliptic curves /S}/ ∼
(groupı̈d of...)
The archimedean case
The period morphism
∨
X ,→ X = P1C
is an GL2 (R)-equivariant embedding.
Image described by Hodge condition :
X = H± = C \ R ⊂ P1C
Uniformisation :
Y (S) = {Elliptic curves /S}/ ∼
(groupı̈d of...)
˘
∼
e (S) = Elliptic curves π : E → S + ρ : Z2 −−
Y
→
R 1 π∗ Z
| {z }
relative Betti coho
e has an action of GL2 (Z)
Via the action of GL2 (Z) on Z2 , Y
¯
/∼
The archimedean case
The period morphism
∨
X ,→ X = P1C
is an GL2 (R)-equivariant embedding.
Image described by Hodge condition :
X = H± = C \ R ⊂ P1C
Uniformisation :
Y (S) = {Elliptic curves /S}/ ∼
(groupı̈d of...)
˘
∼
e (S) = Elliptic curves π : E → S + ρ : Z2 −−
Y
→
R 1 π∗ Z
| {z }
relative Betti coho
e has an action of GL2 (Z)
Via the action of GL2 (Z) on Z2 , Y
Hodge structure on R 1 π∗ R + rigidification ρ induces
∼
e −−
Y
→X
Forget the rigidication : GL2 (Z)-invariant morphism
e −→ Y
Y
Induces
ˆ
˜ ˆ
˜
e = GL2 (Z)\H±
Y ' GL2 (Z)\Y
¯
/∼
In general...
Same picture for more general hermitian symmetric spaces/ Shimura varieties...
Hodge filtration embedding
∨
X ,→ X
∨
where X = hermitian symmetric space, X = compact dual.
Unformisation of Shimura variety :
`
´ ∼
G (Q)\ X × G (Af )/K −−→ ShK (C)
Rapoport-Zink spaces (Serre-Tate, Lubin-Tate, Drinfeld, Rapoport-Zink)
H = a p-divisible group over Fp
For example H = A[p ∞ ] where A = abelian variety over Fp , or a direct factor
of it
Rapoport-Zink spaces (Serre-Tate, Lubin-Tate, Drinfeld, Rapoport-Zink)
H = a p-divisible group over Fp
For example H = A[p ∞ ] where A = abelian variety over Fp , or a direct factor
of it
c : W (Fp )-schemes where p is locally nilpotent
M
−→
Sets
S
7−→
(H, ρ)/ ∼
I
H = p-divisible group/S
I
rigidification
ρ : H ×Fp (S mod p) −→ H ×S (S mod p)
is a quasi-isogeny (virtual isogeny)
Rapoport-Zink spaces
Theorem (Rapoport-Zink)
I
c is representable by a formal scheme locally formally of finite type over
M
Spf(W (Fp )).
→ locally of the form
Spf(W (Fp )Jx1 , . . . , xn Khy1 , . . . , ym i/Ideal
´
Rapoport-Zink spaces
Theorem (Rapoport-Zink)
I
I
c is representable by a formal scheme locally formally of finite type over
M
Spf(W (Fp )).
cred (locally of finite type over Fp ) has
The reduced special fiber M
projective irreducible components.
→ locally of the form
Spf(W (Fp )Jx1 , . . . , xn Khy1 , . . . , ym i/Ideal
´
Rapoport-Zink spaces
Theorem (Rapoport-Zink)
I
I
c is representable by a formal scheme locally formally of finite type over
M
Spf(W (Fp )).
cred (locally of finite type over Fp ) has
The reduced special fiber M
projective irreducible components.
→ locally of the form
Spf(W (Fp )Jx1 , . . . , xn Khy1 , . . . , ym i/Ideal
´
r
Y
` ´×
J = End H Q '
GLmi (Dλi )
p
i=1
where (λi )i are the Dieudonné-Manin slopes wt. multiplicities (mi )i
Dλ := division algebra with invariant λ̄ ∈ Q/Z
c via its action on the rigidification ρ.
Then J acts on M
Rapoport-Zink tower
can = generic fiber as a Berkovich analytic space/W (Fp )Q
M
W
J
Rapoport-Zink tower
can = generic fiber as a Berkovich analytic space/W (Fp )Q
M
W
J
n := ht(H).
`
can
π1 M
´
"
Tp (univ .deformation) = rank n étale Zp -local system
can
M
Rapoport-Zink tower
can = generic fiber as a Berkovich analytic space/W (Fp )Q
M
W
J
n := ht(H).
`
can
π1 M
´
"
Tp (univ .deformation) = rank n étale Zp -local system
can
M
→ GLn (Qp )-equivariant tower of étale coverings
J
(MK )K ⊂GLn (Zp )
GLn (Zp )
|
GLn (Qp )
can = MGL (Z )
M
n p
GLn (Zp )-pro Galois covering + action of GLn (Qp ) (Hecke correspondences)
Period mapping (Katz, Gross-Hopkins, Rapoport-Zink)
D(H)Qp = rational Dieudonné module of H
1
(Hcris
)
Period mapping (Katz, Gross-Hopkins, Rapoport-Zink)
D(H)Qp = rational Dieudonné module of H
1
(Hcris
)
D = convergent isocrystal associated to the Dieudonné crystal of the universal
deformation = locally free OM
can -module of finite rank + Gauss-Manin
connection ∇ on D
Fil D = Hodge filtration
Period mapping (Katz, Gross-Hopkins, Rapoport-Zink)
1
(Hcris
)
D(H)Qp = rational Dieudonné module of H
D = convergent isocrystal associated to the Dieudonné crystal of the universal
deformation = locally free OM
can -module of finite rank + Gauss-Manin
connection ∇ on D
Fil D = Hodge filtration
Rigidification ρ induces rigidification
∼
ρ∗ : D(H)Qp ⊗W (Fp )Q OM
−→ D
can −
p
D(H)Qp = D∇=0
Then Hodge filtration induces a period morphism
J
can −−−→ F
π:M
where F = an analytic Grassmanian associated to D(H)Qp
Period mapping
I
can → F is quasi-étale (Grothendieck-Messing deformation theory,
π:M
analog of Kodaira-Spencer def. theory)
Period mapping
I
I
can → F is quasi-étale (Grothendieck-Messing deformation theory,
π:M
analog of Kodaira-Spencer def. theory)
can ) = ∂(F) = ∅ ⇒ π is étale
But ∂(M
Period mapping
I
can → F is quasi-étale (Grothendieck-Messing deformation theory,
π:M
analog of Kodaira-Spencer def. theory)
can ) = ∂(F) = ∅ ⇒ π is étale
But ∂(M
I
The morphism π is Hecke invariant
I
g
/ MGLn (Z)∩gGLn (Zp )g −1
∼
NNN
p
p
NNN
p
p
N&
xppp
can
can WWW
g
M
g
g M
WWWW π
π ggggg
WWWW
ggg
WWWW
WWW+ sggggggg
F
MGLn (Z)∩gGLn (Zp )g −1
and the fibers of π are the Hecke orbits ' GLn (Qp )/GLn (Zp )
Period mapping
I
can → F is quasi-étale (Grothendieck-Messing deformation theory,
π:M
analog of Kodaira-Spencer def. theory)
can ) = ∂(F) = ∅ ⇒ π is étale
But ∂(M
I
The morphism π is Hecke invariant
I
g
/ MGLn (Z)∩gGLn (Zp )g −1
∼
NNN
p
p
NNN
p
p
N&
xppp
can
can WWW
g
M
g
g M
WWWW π
π ggggg
WWWW
ggg
WWWW
WWW+ sggggggg
F
MGLn (Z)∩gGLn (Zp )g −1
and the fibers of π are the Hecke orbits ' GLn (Qp )/GLn (Zp )
I
Set F a = Im(π) = open subset of F (admissible locus)
Theorem (de Jong)
can −→ F a is an étale covering in the sens of de Jong and the Zp -local
π:M
system given by Tp (univ .def .) descends to a Qp -local system in the sens of de
Jong corresponding to Vp (univ .def .) on F a
Theorem (de Jong)
can −→ F a is an étale covering in the sens of de Jong and the Zp -local
π:M
system given by Tp (univ .def .) descends to a Qp -local system in the sens of de
Jong corresponding to Vp (univ .def .) on F a
∼
can , after fixing a basis Znp −−
→ if x =geo. point of M
→ Tp (Hxuniv )
can , x)
π1dJ (M
π1dJ (F a , π(x))
/ π1 (M
can , x)
/ GLn (Z
 _ p)
/ GLn (Qp )
En résumé : « M∞ := lim MK »
←−
K
O
M∞
GLn (Zp )
GLn (Qp )
can
M
rigidification of p-adic étale coho.
π
Fa
/
rigidification of p-adic cristalline coho.
\
J
|
GLn (Qp )
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
There is a natural open subset, the weakly amissible open subset, F wa , defined
by Fontaine’s weakly admissibility condition associated to the filtered isocrystal
F a ⊂ F wa ⊂ F
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
There is a natural open subset, the weakly amissible open subset, F wa , defined
by Fontaine’s weakly admissibility condition associated to the filtered isocrystal
F a ⊂ F wa ⊂ F
I
By Fontaine (and his disciples...) the classical points of F a and F wa
coincide. But in general F a ( F wa (counterexample by Rapoport and
Zink, other one by Hartl).
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
There is a natural open subset, the weakly amissible open subset, F wa , defined
by Fontaine’s weakly admissibility condition associated to the filtered isocrystal
F a ⊂ F wa ⊂ F
I
I
By Fontaine (and his disciples...) the classical points of F a and F wa
coincide. But in general F a ( F wa (counterexample by Rapoport and
Zink, other one by Hartl).
In general Faltings has given a p-adic Hodge theoretic description of F a
proving a necessary condition given by Fontaine is sufficient to be
admissible, but it is quite abstract and difficult to compute...
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
There is a natural open subset, the weakly amissible open subset, F wa , defined
by Fontaine’s weakly admissibility condition associated to the filtered isocrystal
F a ⊂ F wa ⊂ F
I
I
I
By Fontaine (and his disciples...) the classical points of F a and F wa
coincide. But in general F a ( F wa (counterexample by Rapoport and
Zink, other one by Hartl).
In general Faltings has given a p-adic Hodge theoretic description of F a
proving a necessary condition given by Fontaine is sufficient to be
admissible, but it is quite abstract and difficult to compute...
F wa has an interpretation in terms of semi-stability in the
Harder-Narasimhan sens of a filtered isocrystal and even semi-stability in
the sens of Mumford (Totaro).
Fontaine’s condition
What is the image of the period mapping F a ⊂ F ?
In the archimedean setting : given by Hodge condition → easily compute the
∨
open subset X ⊂ X
There is a natural open subset, the weakly amissible open subset, F wa , defined
by Fontaine’s weakly admissibility condition associated to the filtered isocrystal
F a ⊂ F wa ⊂ F
I
I
I
I
By Fontaine (and his disciples...) the classical points of F a and F wa
coincide. But in general F a ( F wa (counterexample by Rapoport and
Zink, other one by Hartl).
In general Faltings has given a p-adic Hodge theoretic description of F a
proving a necessary condition given by Fontaine is sufficient to be
admissible, but it is quite abstract and difficult to compute...
F wa has an interpretation in terms of semi-stability in the
Harder-Narasimhan sens of a filtered isocrystal and even semi-stability in
the sens of Mumford (Totaro).
F a has an interpretation in terms of semi-stability of a vector bundle on a
« curve lying on F »(joint work with J.M. Fontaine)
p-adic uniformization (Serre-Tate, Cherednik-Drinfeld, Varshavsky,
Rapoport-Zink)
Theorem (Rapoport-Zink)
The spaces MK uniformize p-adically pieces of the p-adic analytic spaces
associated to some Shimura varieties.
p-adic uniformization (Serre-Tate, Cherednik-Drinfeld, Varshavsky,
Rapoport-Zink)
Theorem (Rapoport-Zink)
The spaces MK uniformize p-adically pieces of the p-adic analytic spaces
associated to some Shimura varieties.
I
In general there is no p-adic uniformization of the whole Shimura variety.
Only case : Drinfeld case.
p-adic uniformization (Serre-Tate, Cherednik-Drinfeld, Varshavsky,
Rapoport-Zink)
Theorem (Rapoport-Zink)
The spaces MK uniformize p-adically pieces of the p-adic analytic spaces
associated to some Shimura varieties.
I
In general there is no p-adic uniformization of the whole Shimura variety.
Only case : Drinfeld case.
I
In the non-basic case (when J is not an inner form of G : non-isoclinic
case) R.Z. theorem is non-satisfactory. It has been strenghtened by
Harris-Taylor and in general by Mantovan (Igusa varieties theory).
Local Langlands correspondences
One can define R.Z. tower of spaces for more general reductive groups than
GLn/Qp (not all).
G = reductive group over Qp
J = inner form of a Levi subgroup of the quasi-split inner form of G
(MK )K ⊂G (Zp )
W
J
|
G (Qp )
Local Langlands correspondences
One can define R.Z. tower of spaces for more general reductive groups than
GLn/Qp (not all).
G = reductive group over Qp
J = inner form of a Levi subgroup of the quasi-split inner form of G
(MK )K ⊂G (Zp )
W
|
G (Qp )
J
Consider
lim
−→
K ⊂G (Zp )
ˆ p , Q` ) |
Hc• (MK ⊗C
J×G (Qp )×WE
WE = Weil group of the reflex field, [E : Qp ] < +∞
Action of J × G (Qp ) is smooth (Berkovich for the action of J)
Local Langlands correspondences
One can define R.Z. tower of spaces for more general reductive groups than
GLn/Qp (not all).
G = reductive group over Qp
J = inner form of a Levi subgroup of the quasi-split inner form of G
(MK )K ⊂G (Zp )
W
|
G (Qp )
J
Consider
lim
−→
K ⊂G (Zp )
ˆ p , Q` ) |
Hc• (MK ⊗C
J×G (Qp )×WE
WE = Weil group of the reflex field, [E : Qp ] < +∞
Action of J × G (Qp ) is smooth (Berkovich for the action of J)
General conjecture (vague form) : this representations realize local Langlands
correspondences between rep. of G (Qp ), J and WE
Results : the Lubin-Tate case
Lubin-Tate case : deform a 1-dimensional formal p-divisible group/Fp .
×
G = GLn/F , J = D1/n
where [F : Qp ] < +∞, D1/n = div. alg. wt. invariant 1/n
over F
a n−1 π
can =
M
B̊
−−→ Pn−1
Z
In this case π is surjective (Lafaille, Gross-Hopkins) : F a = F wa = F = Pn−1 .
Results : the Lubin-Tate case
Lubin-Tate case : deform a 1-dimensional formal p-divisible group/Fp .
×
G = GLn/F , J = D1/n
where [F : Qp ] < +∞, D1/n = div. alg. wt. invariant 1/n
over F
a n−1 π
can =
M
B̊
−−→ Pn−1
Z
In this case π is surjective (Lafaille, Gross-Hopkins) : F a = F wa = F = Pn−1 .
×
Rep(D1/n
) → Rep(GLn (F )) : Jacquet Langlands correspondence
Rep(GLn (F )) ↔ Rep(WF ) : local Langlands correspondence (in fact we want
to use those cohomology spaces to prove it exists...)
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
I
Berkovich developped his theory of étale cohomology of analytic
spaces/formal vanishing cycles...
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
I
Berkovich developped his theory of étale cohomology of analytic
spaces/formal vanishing cycles...
I
Boyer proved Carayol’s conjecture for the supercuspidal part in the equal
characteristic case (Fq ((π))) (the local Langlands corr. in the equal char.
case was allready knwon by Laumon-Rapoport-Stuhler)
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
I
Berkovich developped his theory of étale cohomology of analytic
spaces/formal vanishing cycles...
I
Boyer proved Carayol’s conjecture for the supercuspidal part in the equal
characteristic case (Fq ((π))) (the local Langlands corr. in the equal char.
case was allready knwon by Laumon-Rapoport-Stuhler)
I
Harris and Taylor proved the local Langlands conjecture in the p-adic case,
using those cohomology spaces and computed
Pcompletly the alternate sum
of the cohomology of the Lubin-Tate tower, i (−1)i [Hci ], as virtual
representations in terms of the local Langlands correspondences
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
I
Berkovich developped his theory of étale cohomology of analytic
spaces/formal vanishing cycles...
I
Boyer proved Carayol’s conjecture for the supercuspidal part in the equal
characteristic case (Fq ((π))) (the local Langlands corr. in the equal char.
case was allready knwon by Laumon-Rapoport-Stuhler)
I
Harris and Taylor proved the local Langlands conjecture in the p-adic case,
using those cohomology spaces and computed
Pcompletly the alternate sum
of the cohomology of the Lubin-Tate tower, i (−1)i [Hci ], as virtual
representations in terms of the local Langlands correspondences
I
Boyer was able to separate the terms in the alternate sum and computed
each individual cohomology spaces in any degree without taking an
alternate sum (uses in particular an amelioration of Berkovich’s invariance
under formal completion of vanishing cycles (F.))
Results : the Lubin-Tate case
I
Deligne and Carayol proved the result for GL2 . Carayol gave a general
conjecture for the supercuspidal part in the GLn case (depending on a
good étale cohomology theory for rigid analytic spaces...)
I
Berkovich developped his theory of étale cohomology of analytic
spaces/formal vanishing cycles...
I
Boyer proved Carayol’s conjecture for the supercuspidal part in the equal
characteristic case (Fq ((π))) (the local Langlands corr. in the equal char.
case was allready knwon by Laumon-Rapoport-Stuhler)
I
Harris and Taylor proved the local Langlands conjecture in the p-adic case,
using those cohomology spaces and computed
Pcompletly the alternate sum
of the cohomology of the Lubin-Tate tower, i (−1)i [Hci ], as virtual
representations in terms of the local Langlands correspondences
I
Boyer was able to separate the terms in the alternate sum and computed
each individual cohomology spaces in any degree without taking an
alternate sum (uses in particular an amelioration of Berkovich’s invariance
under formal completion of vanishing cycles (F.))
I
Dat computed the smooth-equivariant cohomology complex of the
Lubin-Tate tower, using the result by Boyer
Results : the Drinfled case
×
In this case switch the G and J of the Lubin-Tate case : G = D1/n
and
G = GLn/F
a
π
can '
Ω −−→ Pn−1
M
Z
where
Ω = Pn−1 \
[
H
H∈P̌n−1 (F )
can , the period
and F wa = F a = Ω : on each connected component of M
morphism is an iso. onto its image.
Results : the Drinfled case
×
In this case switch the G and J of the Lubin-Tate case : G = D1/n
and
G = GLn/F
a
π
can '
Ω −−→ Pn−1
M
Z
where
Ω = Pn−1 \
[
H
H∈P̌n−1 (F )
can , the period
and F wa = F a = Ω : on each connected component of M
morphism is an iso. onto its image.
Theorem (Faltings, F.)
The cohomology of the Drinfeld tower is isomorphic to the one of the
Lubin-Tate tower after switching G and J...
Remark : in fact this theorem is used by Boyer and Dat in the preceding
theorems about the L.T. tower.
Other cases
Theorem (F.)
G = GLn/F with F |Qp unramified and J = Qp -points of an inner form of G
equal either to the trivial inner form or an anisotropic inner form. Then the
supercuspidal part of the cohomology of the R.Z. tower realizes local Langlands
correspondences.
Same type of result for a p-adic unitary group in three variables
Other cases
Theorem (F.)
G = GLn/F with F |Qp unramified and J = Qp -points of an inner form of G
equal either to the trivial inner form or an anisotropic inner form. Then the
supercuspidal part of the cohomology of the R.Z. tower realizes local Langlands
correspondences.
Same type of result for a p-adic unitary group in three variables
Restrictions on the groups : lack of results in harmonic analysis at that time,
not on the geometric part of the proof.
Since this theorem (2001), the technics of harmonic analysis have evolved and
it is quite probable one can use my technic of proof to prove the supercuspidal
part of any basic (J is an inner form of G ) R.Z. space (linear group, symplectic
groups, unitary groups) realizes local Langlands correspondences.