On the geometry and cohomology of some simple Shimura varieties

224 Pages·01.02 MB·English

by  Michael Harris, Richard Taylor

On the geometry and cohomology of some
simple Shimura varieties
Michael Harris 1 Richard Taylor 2
Department of Mathematics, Department of Mathematics,
University of Paris 7, Harvard University,
Paris, Cambridge,
France. MA 02138,
U.S.A.
August 30, 1999
1Institut de Math´ematiques de Jussieu, U.M.R. 7586 du CNRS
2Partially supported by NSF Grant DMS-9702885 and by the Miller Institute at
the University of California at Berkeley
Introduction
This paper has twin aims. On the one hand we prove the local Langlands
conjecture for GL over a p-adic field. On the other hand in many cases we
n
are able to identify the action of the decomposition group at a prime of bad
reduction on the l-adic cohomology of the “simple” Shimura varieties studied
by Kottwitz in [Ko4]. These two problems go hand in hand.
The local Langlands conjecture is one of those hydra like conjectures which
seems to grow as it gets proved. However the generally accepted formulation
seems to be the following (see [He2]). Let K be a finite extension of Qp. Fix a
non-trivial additive character ψ : K C×. We will denote the absolute value
→
on K which takes uniformisers to the reciprocal of the number of elements in
the residue field by . We will let W denote its Weil group. Recall that
K K
| |
local class field theory gives us a canonical isomorphism
Art : K Wab.
K × → K
(NormalisedsothatgeometricFrobeniuselementscorrespondtouniformisers.)
The local Langlands conjecture provides some sort of description of the whole
of W in the same spirit.
K
WewillletIrr(GL (K))denotethesetofisomorphismclassesorirreducible
n
admissible representations of GLn(K) over C (or what comes to the same
thing: irreducible smooth representations). If [π ] Irr(GL (K)) and [π ]
1 ∈ n1 2 ∈
Irr(GL (K)) then there is an L-factor L(π π ,s) and an epsilon factor
n2 1 × 2
(cid:143)(π π ,s,ψ) associated to the pair π , π (see for instance [JPSS]).
1 2 1 2
×
On the other hand let WDRep (W ) denote the set of isomorphism classes
n K
of n-dimensional Frobenius semi-simple Weil-Deligne representations of the
Weil group, WK, of K over C. By a Frobenius semi-simple Weil-Deligne
representation of WK over C we mean a pair (r,N) where r is a semi-simple
representation of W on a finite dimensional complex vector space , V, which
K
is trivial on an open subgroup and an element N End (V) such that
∈ C
r(σ)Nr(σ) 1 = Art 1(σ) N
− | K− |K
for all σ W . Again if [(r,N)] WDRep (W ) then there is an L-factor
∈ K ∈ n1 K
L((r,N),s) and an epsilon factor (cid:143)((r,N),s,ψ) associated to (r,N) (see for
instance [Tat2] and section 12 of this paper for the precise normalisations we
are using).
By a local Langlands correspondence for K we shall mean a collection of
bijections
rec : Irr(GL (K)) WDRep (W )
K n −→ n K
for every n 1 satisfying the following properties.
≥
1
1. If π Irr(GL (K)) then rec (π) = π Art 1.
∈ 1 K ◦ K−
2. If [π ] Irr(GL (K)) and [π ] Irr(GL (K)) then
1 ∈ n1 2 ∈ n2
L(π π ,s) = L(rec (π ) rec (π ),s)
1 2 K 1 K 2
× ⊗
and
(cid:143)(π π ,s,ψ) = (cid:143)(rec (π ) rec (π ),s,ψ).
1 2 K 1 K 2
× ⊗
3. If [π] Irr(GL (K)) and χ Irr(GL (K)) then rec (π (χ det)) =
n 1 K
∈ ∈ ⊗ ◦
rec (π) rec (χ).
K K
⊗
4. If [π] Irr(GL (K)) and π has central character χ then detrec (π) =
n K
∈
rec (χ).
K
5. If [π] Irr(GL (K)) then rec (π ) = rec (π) (where denotes con-
n K ∨ K ∨
∈ ∨
tragredient).
Henniart showed (see [He5]) that there is at most one set of bijections rec
K
with these properties. The commonest formulation of the local Langlands
conjecture for GL is the following theorem.
n
Theorem A A local Langlands correspondence rec exists for any finite ex-
K
tension K/Qp.
However it seems to us that one would really like more than this simple
existence theorem. On the one hand it would be very useful if one had some
sort of explicit description of this map rec . Our methods shed no light on
K
this. One might well hope that the methods of Bushnell, Henniart and Kutzko
will lead to an explicit version of this theorem. On the other hand one would
also like to know that the local reciprocity map rec is compatible with global
K
reciprocity maps whenever the global map is known to exist. Our methods
do not resolve this latter question but they do shed considerable light on it.
For instance in the cases considered by Clozel in [Cl1] we settle this question
affirmatively up to semisimplification (in particular we do not identify the two
N’s).
Maybe a remark on the history of this problem is in order. The exis-
tence of rec with the desired properties follows from local class
K|Irr(GL1(K))
field theory (due originally to Hasse [Has]), but this preceded the general con-
jecture. The key generalisation to n > 1 is due to Langlands (see [Lan]), who
formulated some much more wide ranging, if less precise, conjectures. The
formulation in the form described here, with its emphasis on epsilon factors
2
of pairs, seems to be due to Henniart (see [He2]). Henniart’s formulation has
the advantage that there is a most one such correspondence, but as remarked
above it limits somewhat the scope of Langlands’ original desirata. The exis-
tence of rec with the desired properties was established by Kutzko
K|Irr(GL2(K))
([Ku]), following earlier partial work by a number of people. The existence of
rec with almost all the desired properties was established by Hen-
K|Irr(GL3(K))
niart ([He1]). In particular, his correspondence had enough of these properties
to characterise it uniquely. Both the work of Kutzko and Henniart relied on
a detailed classification of all elements of Irr(GL (K)). These methods have
n
since been pushed much further, but to date have not provided a construction
of rec which demonstrably has the desired properties on Irr(GL (K)) for any
K n
n > 3. In the case of completions of functions fields of transcendence degree 1
over finite fields, the corresponding theorem was proved by Laumon, Rapoport
and Stuhler ([LRS]).
We will let Cusp(GL (K)) denote the subset of Irr(GL (K)) consisting
n n
of equivalence classes of supercuspidal representations. Let Rep (W ) denote
n K
the subset of WDRep (W ) consisting of equivalence classes of pairs (r,N)
n K
with N = 0. Also let Irr (W ) denote the subset of Rep (W ) consisting of
n K n K
equivalence classes of pairs (r,0) with r irreducible. It follows from important
work of Zelevinsky [Z] that it suffices to construct bijections
rec : Cusp(GL (K)) Irr (W )
K n n K
−→
with the properties listed above (see [He2].) In a key breakthrough, Henniart
[He4] showed that there did exist bijections
rec : Cusp(GL (K)) Irr (W ),
K n n K
−→
which preserved conductors and were compatible with twists by unramified
characters. He was however unable to show that these bijections had enough
of the other desired properties to characterise them uniquely. The usefulness
of this result is that it allows one to use counting arguments, for instance
any injection Cusp(GL (K)) (cid:44) Irr (W ) satisfying the desired properties
n n K
→
must be a bijection. (This result is usually referred to as the numerical local
Langlands theorem.)
We will give a natural construction of a map
rec : Cusp(GL (K)) Irr (W ),
K n n K
−→
which we will show is compatible with the association of l-adic representa-
tions to many automorphic forms on certain unitary groups. Using this global
3
compatibility and some instances of non-Galois automorphic induction dis-
covered by one of us (M.H., see [Har2]) we will see that there is a subset
Cusp(GL (K)) Cusp(GL (K)) such that
n (cid:48) n
⊂
recK : Cusp(GLn(K))(cid:48) ∼ Irrn(WK)
−→
and such that rec has all the desired properties. The subset
K|Cusp(GLn(K))(cid:48)
Cusp(GL (K)) may be described as those elements of Cusp(GL (K)) which
n (cid:48) n
become unramified after some series of cyclic base changes. Appealing to
Henniart’s numerical local Langlands theorem ([He4]) we can conclude that
Cusp(GL (K)) = Cusp(GL (K)) and so deduce theorem A.
n n (cid:48)
One of us (M.H. see [Har1]) had previously given a different construction
of a map
rec : Cusp(GL (K)) Irr (W ).
K(cid:48) n −→ n K
In some cases he was able to show its compatibility with the association of l-
adic representationsto certain classes of automorphic forms on unitary groups.
As a result he could deduce the local Langlands conjecture only for p > n (see
[BHK] and [Har2]). A posteriori we can show that rec = rec . Since the
(cid:48)K K
distribution of a preliminary version of this paper, but before the distribution
of the final version, Henniart [He6] has given a much simpler proof of theorem
A by making much cleverer use of the non-Galois automorphic induction of
[Har2] and of his own numerical local Langlands theorem [He4]. He does not
needthe a prioriconstruction of a maprec compatiblewith someinstances of
K
theglobalcorrespondence, andthusheisabletoby-passallthemainresultsin
this paper. For the reader interested only in theorem A his is clearly the better
proof. None the less we believe the results of this paper are still important as
they establish many instances of compatibility between the global and local
correspondences.
Let us now explain our construction of maps
rec : Cusp(GL (K)) Rep (W ).
K n −→ n K
To this end choose a prime l = p and fix an isomorphism C = Qac. Let k
(cid:54) ∼ l
denote the residue field of K. For any g 1 there is, up to isomorphism,
≥
a unique one-dimensional formal -module Σ /kac of -height g. Then
K K,g K
O O
EndOK(ΣK,g) ⊗Z Q ∼= DK,g, the division algebra with centre K and Hasse
invariant 1/g. Drinfeld showed that the functor which associates to any Ar-
tinian local -algebra A with residue field kac the set of isomorphism classes
K
O
of deformations of Σ to A is prorepresented by a complete noetherian lo-
K,g
cal algebra R with residue field kac. (In fact he showed that R is
K K,g K,g
O
4
a formal power series ring in g 1 variables over the ring of integers of the
−
completion of the maximal unramified extension of K.) We will let Σ de-
K,g
note the universal deformation of Σ over R . (In the case g = 1 one just
K,g K,g
obtains the base change to the ring of integers of the completion of th(cid:101)e maxi-
mal unramified extension of K of any Lubin-Tate formal module over K.)
K
O
Drinfeld further showed that for any integer m 0 there is a finite flat R -
K,g
≥
algebra R over which Σ has a universal Drinfeld level pm-structure. We
K,g,m K,g
will consider the direct limit over m of the formal vanishing cycle sheaves of
SpfRK,g,m with coefficients(cid:101)in Qalc. This gives a collection {ΨiK,l,g} of infinite-
dimensional Qac vector spaces with natural admissible actions of the subgroup
l
of GL (K) D W consisting of elements (γ,δ,σ) such that
g × K×,g × K
detδ detγ 1 Art 1σ = 1.
| || |− | −K |
For any irreducible representation ρ of D we set
K×,g
Ψi (ρ) = Hom (ρ,Ψi ).
K,l,g OD×K,g K,l,g
This becomes an admissible GL (K) W -module. In the case g = 1 we
g K
×
have Ψi = (0) for i > 0, while it follows from the theory of Lubin-Tate
K,l,1
formal groups (see [LT]) that Ψ0K,l,1(ρ) = Qlac with an action of K× ×WK via
ρ 1 (ρ Art 1) (see section 3.4 of [Car3]).
− × ◦ K−
To describe Ψi (ρ) in greater generality we must recall that Deligne,
K,l,g
Kazhdan and Vigneras (see [DKV]) and Rogawski ([Rog2]) have given a bijec-
tion between irreducible representations of D and (quasi-)square integrable
K×,g
irreducible admissible representations of GL (K) characterised by a natural
g
character identity (see appendix IV). This generalises work of Jacquet and
Langlands in the case g = 2 so we will denote the correspondence ρ JL(ρ).
(cid:55)→
Carayol essentially conjectured ([Car3]) that if JL(ρ) is supercuspidal then
ΨgK−,l1,g(ρ) ∼= JL(ρ)∨ ×recK(JL(ρ)⊗|det|(1−g)/2).
We do not quite prove this (though it may be possible by our methods to
do so). However motivated by Carayol’s conjecture our first main theorem
is the following. To state it let [Ψ (ρ)] denote the virtual representation
K,g
(−1)g−1 gi=−01(−1)i[ΨiK,l,g(ρ)].
(cid:80)
Theorem B If π is an irreducible supercuspidal representation of GL (K)
g
then there is a (true) representation
rl(π) : WK → GLg(Qalc),
5
such that in the Grothendieck group
[Ψ (JL(π) ] = [π r (π)].
K,l,g ∨ l
⊗
In the case n = 1 Lubin-Tate theory allows one to identify
r (π) = π 1 Art 1.
l − ◦ −K
We use this theorem to define
rec : Cusp(GL (K)) Rep (W )
K g −→ g K
by the formula
rec (π) = r (π ( det)(1 g)/2).
K l ∨ −
⊗ | |◦
It will also be convenient for us to extend r to all irreducible admissible
l
representations of GL (K) as follows. If π is an irreducible admissible repre-
g
sentation of GL (K), then we can find positive integers g ,...,g which sum to
g 1 t
g and irreducible supercuspidal representations π of GL (K) such that π is
i gi
a subquotient of n-Ind(π ... π ), where we are using the usual normalised
1 t
× ×
induction (see appendix I). Then we set
t
rl(π) = rl(πi)⊗|Art−K1|(gi−g)/2.
i=1
(cid:77)
This is well defined and
rec (π) = (r (π ( det)(1 g)/2),N),
K l ∨ −
⊗ | |◦
for some N.
Our second key result is that r is compatible with many instances of the
l
global Langlands correspondence. The following theorem strengthens a theo-
rem of Clozel [Cl1] (in which he only identifies [R(Π) ] for all but finitely
|WFy
many places y, and specifically for none of the bad places).
Theorem C Suppose that L is a CM field and that Π is a cuspidal automor-
phic representation of GLg(AL) satisfying the following conditions:
Π = Πc,
∨ ∼
•
Π has the same infinitesimal character as some algebraic representation
• ∞
over C of the restriction of scalars from L to Q of GLg,
6
and for some finite placexofL the representationΠ is squareintegrable.
x
•
Then there is a non-zero integer a(Π) and a continuous representation R(Π)
of Gal(Lac/L) over Qac such that for any finite place y of L not dividing l we
l
have
[R(Π) ] = a(Π)[r (Π )].
|WLy l y
In the case n = 2 and K = Qp and F+ = Q both theorems B and C were
essentially proved by Deligne in his beautiful letter [De2]. (The argument was
completed by Brylinski [Bry].) Carayol [Car2] generalised Deligne’s method
to essentially prove both theorems B and C in the general n = 2 case. We will
simplygeneraliseDeligne’sapproachton > 2. ThecombinationoftheoremsB
andC, Henniart’snumericallocalLanglandstheorem[He4]andthenon-Galois
automorphic induction of [Har2] suffice to prove theorem A.
Both theorems B and C follow without great difficulty from an analysis
of the bad reduction of certain Shimura varieties. We will next explain this
analysis. Unfortunately we must first establish some notation. Let E denote
an imaginary quadratic field in which p splits: p = uuc. Let F+ denote a
totally real field of degree d and set F = EF+. Fix a place w of F above u.
Let B be a division algebra with centre F such that
the opposite algebra Bop is isomorphic to B E;
E,c
• ⊗
B is split at w;
•
at any place x of F which is not split over F+, B is split;
x
•
at any place x of F which is split over F+ either B is split or B is a
x x
•
division algebra,
if n is even then 1+dn/2 is congruent modulo 2 to the number of places
•
of F+ above which B is ramified.
Let n denote [B : F]1/2. We can pick a positive involution of the second kind
on B (i.e. = c and tr (xx ) > 0 for all nonzero x B). If β B = 1
∗ ∗|F B/Q ∗ ∈ ∈ ∗ −
then we will let
G denote the algebraic group with G(Q) the subgroup of elements x
• ∈
(Bop)× so that x∗βx = ν(x)β for some ν(x) Q×,
∈
ν : G Gm the corresponding character,
• →
G the kernel of ν,
1
•
7
and ( , ) the pairing on B defined by
•
(x,y) = (tr tr )(xβy ).
F/Q ◦ B/F ∗
We can and will choose β such that
G is quasi-split at all rational primes x which do not split in E
•
and G1(R) ∼= U(n 1,1) U(n)[F+:Q]−1.
• − ×
If U G(A∞) is an open compact subgroup we will consider the following
⊂
moduli problem. If S is a connected F-scheme and s is a closed geometric
point of S then we consider equivalence classes of quadruples (A,λ,i,η) where
A is an abelian scheme of dimension [F+ : Q]n2;
•
λ : A A is a polarisation;
∨
• →
• i : B (cid:44)→ End(A)⊗Z Q such that λ◦i(b) = i(b∗)∨ ◦λ for all b ∈ B and
such that
tr(b ) = (c tr tr )(nb)+tr (b) (c tr )(b)
Lie(A) F/E B/F B/F B/F
| ◦ ◦ − ◦
for all b B;
∈
• η is a π1(S,s)-invariant U-orbit of isomorphisms of B ⊗Q A∞-modules
η : V ⊗Q A∞ → VAs which take the standard pairing ( , ) on V to a
scalar multiple of the λ-Weil pairing on VA .
s
We consider two such quadruples equivalent if the abelian varieties are isoge-
nous in a way that preserves the rest of the structure (but only need preserve
the polarisation up to Q× multiples). The set of equivalence classes is canon-
ically independent of the choice of s. If U is sufficiently small this moduli
problem is represented by a smooth proper scheme of finite type X /F.
U
If ξ is a representation of the algebraic group G over Qac then we can define
l
a lisse Qalc sheaf Lξ on XU. Then we will consider the Qalc-vector spaces
Hi(X, ) = limHi (X Fac, ).
Lξ U et U × Lξ
→
This is naturally an admissible G(A∞) Gal(Fac/F)-module. In fact we can
×
write
Hi(X, ) = π Ri(π),
Lξ ⊗ ξ
π
(cid:77)
8
where π runs over irreducible admissible representations of G(A∞) and Rξi(π)
is a finite dimensional continuous representation of Gal(Qac/Q). We will focus
on the virtual representation
[R (π)] = ( 1)n 1 ( 1)i[Ri(π)].
ξ − − − ξ
i
(cid:88)
Kottwitz (see [Ko4]) determined tr[R (π)](Frob ) in terms of ρ for all but
ξ x
finitely many places x of F. He thus completely determined the virtual rep-
resentation [R (π)]. We will extend Kottwitz’s description to all places x l of
ξ
(cid:54) |
F (see corollary 11.12).
We have an isomorphism
G(Qp) ∼= Eu×c × (Bxop)×
xu
(cid:89)|
and hence a decomposition
G(A∞) ∼= G(A∞,p)×Eu×c × (Bxop)×.
xu
(cid:89)|
Thus we may decompose an irreducible admissible representation π of G(A∞)
as
π = πp π π .
∼ p,0 x
⊗ ⊗
xu
(cid:79)|
For h = 1,...,n we will let Pop denote the parabolic subgroup of GL (F )
h n w
consisting of block lower triangular matrices with an (n h) (n h)-block
− × −
in the top left and an h h block in the bottom right. It has Levi com-
×
ponent GL (F ) GL (F ). Let Nop denote its unipotent radical. Sup-
n−h w × h w h
pose that ρ is an irreducible representation of D . We will let ϕ
F×w,n h JL(ρ)∨ ∈
C (GL (F )) denote a pseudo-coefficient for JL−(ρ) (so that ϕ is
∞ n h w ∨ JL(ρ)
− ∨
compactly supported mod centre, and for any irreducible tempered admissi-
ble representation α of GL (F ) with the same central character as JL(ρ)
n h w ∨
−
we have trα(ϕ ) = vol(D /F ) if α = JL(ρ) and = 0 otherwise).
JL(ρ)∨ F×w,n h w× ∼ ∨
Then we define a homomorphism −
n-red(h) : Groth(GL (F )) Groth(GL (F ))
ρ n w −→ h w
as a composite
Groth(GL (F )) Groth(GL (F ) GL (F )) Groth(GL (F )),
n w n h w h w h w
→ − × →
where
9