Automorphic vector bundles with global sections on $G$-${\tt Zip}^{\mathcal Z}$-schemes

0.26 MB·

by  Wushi Goldring

AUTOMORPHIC VECTOR BUNDLES WITH GLOBAL
SECTIONS ON G-ZipZ-SCHEMES
7
1
0 WUSHIGOLDRING,JEAN-STEFANKOSKIVIRTA
2

b Abstract. A general conjecture is stated on the set of automorphic vector
e bundlesadmittingnonzeroglobalsectionsonschemesendowedwithasmooth,
F surjectivemorphismtoastackofG-zipsofHodgetype;suchschemesinclude
Hodge-type Shimura varieties with hyperspecial level. We prove our conjec-
3 ture forgroups oftype A1; thisincludesallHilbert-Blumenthal varieties. An
1
exampleusingflagspacesisgiventoshowthatourconjecture canfailforzip

datanotofHodgetype.
]
T
N
.
h Introduction
t
a Thispapercontributestoourgeneralprogramlaunchedin[7]. Ourfocushereis
m
akeyexampleofthegeometry-by-groupsphilosophywhichemanatesfromstacksof
[ G-Zips, the Ekedahl-Oort (EO) stratification of Shimura varieties and their Hasse

2 invariants. See our sequel papers to [7] and joint work with B. Stroh and Y.
v Brunebarbe for further developments in our program[8, 6, 9, 3].
3 Let X be the special fiber of a Hodge-type Shimura variety at a place of good
3 reduction, attached to a Shimura datum (G,X). Write G := G ×F , where
3 Zp p
G isareductiveZ -modelofG . ByZhang[19],itisknownthatthereexistsa
0 Zp p Qp
0 smoothmorphismofstacksX →G-ZipZ,whereG-ZipZ isthecorrespondingstack
. of G-zips (see §1.1.1). In this setting, an encapsulation of the geometry-by-groups
1
0 philosophy is the following:
7
Question A. To what extent is the global geometry of the Shimura variety X
1
captured by the group-engendered stack G-ZipZ ?
:
v
i InitiallyitmayappeartothereaderthattheglobalgeometryofX ismuchricher
X than the information contained in the finite stack G-ZipZ, so that the answer to
r QuestionAshouldbe“minimal”. However,thepreviouspapers[14,13,7,8]already
a
deduced nontrivial information about the global geometry of X from a study of
group-theoreticalHasse invariantsonG-ZipZ (andthe closely relatedstacks ofzip
flags). Forexample,weshowedbythismethodthattheEkedahl-Oortstratification
of X is uniformly principally pure [7, Cor. 3.1.3] and that all Ekedahl-Oort strata
of the minimal compactification Xmin are affine (loc. cit., Th. 3.3.1).
Motivatedby aconjecture thatF.Diamondcommunicatedtous [4], this note is
concernedwithanotherexamplewhereafacetofglobalgeometryofX isunderstood
purelyintermsofG-ZipZ: thequestionofwhichautomorphicvectorbundlesadmit
W.G.DepartmentofMathematics,StockholmUniversity,StockholmSE-10691,Swe-
den
J.-S. K. Department of Mathematics, South Kensington Campus, Imperial College
London, London SW7 2AZ, UK
E-mail address: [email protected], [email protected].
1
2 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES
globalsectionsonX. OurforthcomingjointworkwithStrohandBrunebarbe[9,3]
will study the closely related question of which automorphic vector bundles are
ample on X and on the associated partial flag spaces defined in section 4.1.
OurmethodsrevealthatitisinfactmorenaturaltoreplacetheShimuravariety
X by an arbitrary scheme and study the more general:
Question B. Let X be an arbitrary scheme, endowed with a smooth surjective
morphism ζ :X →G-ZipZ. To what extent is the global geometry of X controlled
by the stack G-ZipZ?
The passagefromQuestionAconcerningShimuravarietiestothe muchbroader
contextofQuestionBisofcentralimportancetoourgeneralprogram. Thisbroad-
ening of perspective will be dissected elsewhere. For a more precise question re-
garding global sections of automorphic vector bundles, see Question C in §2.2.
The main object of study in this note is a collection of ‘automorphic’ vector
bundles on the stack G-ZipZ. Denote by L the Levi subgroup of G given by Z
(see §1.1.1) and choose a Borel pair (B,T) appropriately adapted to Z. A B∩L-
dominant character λ ∈ X∗(T) gives rise to a vector bundle V(λ) on the stack
G-ZipZ (see §1.2.1). Put V (λ) := ζ∗(V(λ)). When X is the special fiber of
X
a Hodge-type Shimura variety, the V (λ) recover the usual automorphic vector
X
bundles on X.
Let C (resp. C ) denote the set of λ ∈ X∗(T) such that V(λ) (resp. V (λ))
X X
admits nonzero global sections. We are interested in the saturated cones hCi and
hC igeneratedbythem. TheinclusionhCi⊂hC iholdsingeneral. Intheveinof
X X
QuestionB,weconjecture thathCi=hC iwheneverthe zipdatum Z is ofHodge
X
type (see [8, Def. 1.4.3]), the map ζ is smooth and surjective on each connected
component of X, and global sections of the structure sheaves of strata closures in
X are locally constant (see Assumption 2.2.2 and Conjecture 2.2.5). In particular,
our conjecture includes the case when X is a Shimura variety of Hodge-type with
hyperspecial level at p. Note that the cone hCi can be determined explicitly in
group-theoreticalterms.
Asanexample,weexaminethecasewhenGisoftypeA (i.e.,G isisomorphic
1 Fp
to SLn , see §3.1). This case applies to Hilbert modular varieties. Even though
2,Fp
G is not exactly the group attached to Hilbert modular varieties, the difference is
harmless with our methods (see § 2.3). Let Z be the zip datum of G attached to
a Borel subgroup of G. Zip strata are parametrized by subsets of {1,...,n}. For
a subset S, denote by X the corresponding closed stratum and by X∗ its Zariski
S S
closure (see §2.1).
Theorem (Theorem 3.2.3). Assume the following:
(1) For each connected component X◦ ⊂ X, the map ζ : X◦ → G-ZipZ is
smooth and surjective.
(2) For every S ⊂{1,...,n}, every h∈H0(X∗,O ) is locally constant.
S X
Then for any G-admissible subset S ⊂{1,...,n}, the equality hC i=hC i holds.
X,S S
In particular, one has hC i=hCi.
X
For the definition of G-admissible, see Definition 3.2.1. The cone hCi is given
explicitly in Corollary 3.2.4. Diamond shared with us his conjecture that when X
is a Hilbert modular variety, the cone hC i is equal to that spanned by Goren’s
X
partial Hasse invariants [10]. As Diamond later informed us, a similar question
AUTOMORPHIC BUNDLES ON ZIP-SCHEMES 3
had been raised by Andreatta-Goren [1, Question 15.8]. Inspired by Diamond’s
Conjecture, the Theorem above offers a reinterpretation and generalization to the
setting of G-Zips and to our framework of Questions A and B.
After we announced the results of this paper and communicated them to Dia-
mond, we received the preprint [5], which among other results provides a different
proof of the second statement of Theorem 3.2.3 when X is the special fiber of a
Hilbertmodularvarietyataplaceofgoodreduction(Cor.1.3ofloc. cit.). Itseems
tousthattheapproachofDiamond-KassaeiusesspecialpropertiesofHilbertmod-
ular varieties (e.g., the results of Tian-Xiao that in the Hilbert case EO strata are
P1-bundles over quaternionic Shimura varieties).
By contrast, our methods only use the map ζ and its basic properties, which
hold for all Hodge-type Shimura varieties and even more general G-ZipZ-schemes.
For example, in forthcoming work, we show that our methods can be used to
generalizeTh.3.2.3tothegroupsGSp(4)andGU(2,1)(forsplitp),therebyproving
Conj. 2.2.5 in these cases.
We now give an overview of the paper. In §1, we recall the theory of G-zips
and our previous results on Hasse invariants. In §2, we define a family of cones
inside X∗(T) and formulate our conjectural generalization of Th. 3.2.3 to zip data
of Hodge type, see Conj. 2.2.5. The proof of Th. 3.2.3 is the subject of §3. In §4,
weexhibit the flagspaceofGSp(6) asanexample ofhowConj.2.2.5canfailwhen
the zip datum is not of Hodge type.
Acknowledgments
WethankFredDiamondforsharinghisconjecturewithusandforhelpfulcorre-
spondence. We are indebted to Benoît Stroh for our collaborationon relatedques-
tions concerning ampleness of automorphic bundles. We are grateful to Torsten
Wedhorn for helpful comments on an earlier version of this paper.
W.G.thankstheUniversityofZurichforprovidingexcellentworkingconditions
and the opportunity to present some of the results of this paper during a visit in
the fall of 2016.
1. Review of previous results
1.1. Review of zip data.
1.1.1. Zip data and the stack of G-zips. We fix an algebraic closure k of F . Let
p
G be a reductive group over F , and denote by ϕ : G → G be the Frobenius
p
homomorphism. LetZ :=(G,P,L,Q,M,ϕ)beaFrobeniuszipdatum. Recallthat
this means that P,Q are parabolic subgroups of G and L ⊂ P and M ⊂ Q are
k
Levi subgroups, with the property that ϕ(L)=M. We say that Z is a zip datum
of Borel-type if P is a Borel subgroup of G (this implies that Q is also a Borel
subgroup). The zip group is the subgroup of P ×Q defined by
(1.1.1) E :={(x,y)∈P ×Q, ϕ(x)=y}
where x ∈ L and y ∈ M denote the Levi components of the elements x and y
respectively. We letG×GactonGby(a,b)·g :=agb−1,andwegetby restriction
an action of E on G. The stack of G-zips of type Z is isomorphic to the quotient
stack
(1.1.2) G-ZipZ ≃[E\G].
4 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES
For convenience, we assume that there exists a Borel pair (B,T) defined over F
p
such that B ⊂ Q. Then there exists an element z ∈ W such that zB ⊂ P, and
(B,T,z) defines a W-frame for Z (see Definition 1.3.1 in [8]).
1.1.2. Zip stratification. Let Φ ⊂ X∗(T) (resp. Φ ) be the set of T-roots in G
L
(resp. L). Let Φ+ (resp. Φ+) be the system of positive roots given by putting
L
α ∈ Φ+ (resp. α ∈ Φ+) when the α-root group U is contained in B (resp.
L −α
B := B∩L). Write ∆ ⊂ Φ for the set of simple roots. For α ∈ Φ, let s ∈ W
L + α
be the corresponding reflection. Then (W,{s ,α∈∆}) is a Coxeter group and we
α
denote by ℓ:W →N the length function. Let I ⊂∆ (resp. J ⊂∆) be the type of
P (resp. Q). Let W ⊂W be the subgroup generated by the s for α∈I. Let w
I α 0
(resp. w ) be the longest element in W (resp. W ). Denote by IW (resp. WJ)
0,I I
the subset of elements w ∈ W which are minimal in the coset W w (resp. wW ).
I J
The set IW (resp. WJ) is a set of representatives for the quotient W \W (resp.
I
W/W ). Forw ∈W,wefixarepresentativew˙ ∈N (T),suchthat(w w )· =w˙ w˙
J G 1 2 1 2
wheneverℓ(w w )=ℓ(w )+ℓ(w )(thisispossiblebychoosingaChevalleysystem,
1 2 1 2
see [2], Exp. XXIII, §6).
If h ∈ G(k) is an element of the group, we denote by O (h) the E-orbit of h
Z
in G. Similarly, we write o (h):= [E\O (h)] for the corresponding locally closed
Z Z
substack of [E\G]. By Theorem 7.5 in [18], the map
(1.1.3) IW →{E-orbits in G}, w 7→Gw :=O (z˙w˙)
Z
defines a bijection between IW and the set of E-orbits in G. There is a dual
parametrization(§11 in loc. cit.) given by a bijection:
(1.1.4) WJ →{E-orbits in G}, w7→G′w :=O (z˙w˙).
Z
Furthermore, one has the following dimension formulas:
(1.1.5) dim(Gw)=ℓ(w)+dim(P) for all w ∈IW
(1.1.6) dim(G′w)=ℓ(w)+dim(P) for all w ∈WJ.
1.2. Vector bundles and Hasse invariants.
1.2.1. Vector bundles on G-ZipZ. We make the identification X∗(L) = X∗(P) =
X∗(E) via the inclusion L ⊂ P and the first projection E → P. Similarly every
representationofL gives riseto one of E. ThereforeeveryL-module givesa vector
bundle on G-ZipZ.
Associated to a Φ+-dominant character λ ∈ X∗(T), one has an L-equivariant
L
line bundle L onthe flag varietyL/B and consequentlythe L-module H0(λ):=
λ L
H0(L/B ,L ). Denote by V(λ) the corresponding vector bundle on the stack
L λ
G-ZipZ. In case λ∈X∗(T) is not Φ+-dominant, set V(λ)=0.
L
Givena schemeX anda morphismζ :X −→G-ZipZ, denote the vectorbundle
ζ∗(V(λ)) on X by V (λ).
X
1.2.2. Canonical sections on strata. Recallthatforanyw ∈IW,thereexistsn≥1
such that for all λ∈X∗(L), the line bundle V(λ)n admits a nonzero section
(1.2.1) f ∈H0([E\G ],V(λ)n).
w,λ w
which is unique up to a scalar.
In [7, Th. 3.1.2] the authors proved the following: If λ is ample and orbitally
p-close, there exists an N ≥ 1 such that for all w ∈ IW the section fN extends
w,λ
AUTOMORPHIC BUNDLES ON ZIP-SCHEMES 5
to a section over E\G . Furthermore, the non-vanishing locus of this section
w
is [E\G ]. This conclusion holds also when Z is the zip datum attached to a
w
(cid:2) (cid:3)
maximal-type zip datum (G,µ) and λ is of maximal-type (see [8, Def. 1.4.4]. In
particular, this applies to the Hodge bundle in the context of Shimura varieties of
Hodge-type.
2. The conjecture
2.1. Notation. In this section we consider a reductive group G over F and a zip
p
datumZ :=(G,P,L,Q,M,ϕ). We denote byX :=G-ZipZ the associatedstackof
G-zips. We fix a k-scheme X and a morphism of stacks
(2.1.1) ζ :X −→X.
In (1) through (4) below, all locally closed subsets are endowed with the reduced
structure. For all w ∈IW, we define:
(1) X as the locally closed substack [E\G ]⊂X.
w w
(2) X∗ := E\G as the Zariski closure of X .
w w w
(3) X as the preimage ζ−1(X ).
w (cid:2) (cid:3) w
(4) X∗ as the preimage ζ−1(X∗).
w w
2.2. Cones. We now define several subsets of X∗(T). In the following, w denotes
an element of IW:
(2.2.1) C :={λ∈X∗(T), H0(X,V(λ))6=0}
(2.2.2) C :={λ∈X∗(T), H0(X∗,V(λ))6=0}
w w
(2.2.3) C :={λ∈X∗(T), H0(X,V (λ))6=0}
X X
(2.2.4) C :={λ∈X∗(T), H0(X∗,V (λ))6=0}
X,w w X
Definition 2.2.1. Let M be a finitely generated free abelian group of rank r.
(a) A cone in M is a subset C ⊂M satisfying x,y ∈C ⇒x+y ∈C .
0 0 0
(b) We say that C is saturated if nx∈C ⇒x∈C for all n≥1 and x∈M.
0 0 0
(c) For a subset C of a free abelian group of finite-type M, we define
0
(2.2.5) hC i:={x∈M,∃n≥1,nx∈C }.
0 0
If C is a cone, then hC i is a saturatedcone. One sees immediately that C and
0 0
C is a cone of X∗(T) for all w ∈ IW. If X is irreducible, then C is a cone.
w w X,w
We will make the following assumption:
Assumption 2.2.2.
(a) The zip datum Z is of Hodge-type ([8, Def. 1.4.3]).
(b) For all connected components X◦ ⊂ X, the map ζ : X◦ → X is smooth and
surjective.
(c) For all w ∈W, every section h∈H0(X∗,O ) is locally constant.
w X
Remark 2.2.3. Assumption 2.2.2 implies the following:
(a) X, X are smooth.
w
(b) X∗ is the Zariski closure of X .
w w
(c) X∗ and X are equi-dimensional of dimension dim(X)+ℓ(w)−ℓ(w ).
w w 0
(d) The decomposition X = X is a stratification of X, in the sense that
w∈IW w
the Zariski closure of a stratum is a union of strata.
F
6 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES
(e) The inclusions C ⊂C and hC i⊂hC i hold.
w X,w w X,w
In this setting, a particular instance of Question B is:
Question C. For which w ∈IW does one have hC i=hC i?
w X,w
Remark 2.2.4.
(a) The inclusion C ⊂ C is usually proper. For example, if X is the special
w X,w
fiber of the modular curve, then H0(X,ωn)6=0⇐⇒n=(p−1)m,m≥0.
(b) We conjecture that, when X is a Hilbert modular threefold at a totally inert
prime and w ∈IW has length two, then hC i6=hC i.
w X,w
We submit the following:
Conjecture 2.2.5. IfAssumption2.2.2 is satisfied, theequalityhCi=hC iholds.
X
We will give in section 4.2 an example of a pair (X,ζ) satisfying Assumptions
2.2.2(b)and2.2.2(c)butnot2.2.2(a)forwhichhCi=6 hC i. Thefollowingsimple
X
proposition serves to illustrate some of the concepts introduced above and will be
used later.
Proposition 2.2.6. Assume Z is a Borel-type zip datum (i.e P is a Borel sub-
group). If ℓ(w)=1, the equality hC i=hC i holds.
w X,w
Proof. Let λ ∈ C and assume λ ∈/ hC i. Since ℓ(w) = 1, we have −λ ∈ hC i.
X,w w w
So there exists m ≥ 1 such that −mλ ∈ C and mλ ∈ C . Hence there exist
w X,w
nonzero sections h∈H0(X∗,V(−mλ)) and f ∈H0(X∗,V (mλ)).
w w X
Since X∗ is reduced, there exists an irreducible component X′ ⊂ X where
w w w
f| 6= 0. Since ζ is smooth, ζ∗(h) is non-zero on X′ . So ζ∗(h)f ∈ H0(X′ ,O )
Xw′ w w X
is nonzero too, therefore must be constant by Asssumption 2.2.2(c). Thus h is
nowhere vanishing on X and V(mλ)| ≃O ; this contradicts λ∈/ hC i. (cid:3)
w Xw Xw w
2.3. Changing the group. Note that X∗(G) ⊂ C for all w ∈ IW. Let G′ be
w
the derived group of G. Intersecting with G′, the zip datum Z yields naturally a
zip datum Z′ for G′. The map ι : G′ → G induces a homeomorphism G′-ZipZ →
G-ZipZ. Consider the fiber product
X′ ζ′ // G′-ZipZ′
ιX ι
(cid:15)(cid:15) (cid:15)(cid:15)
X ζ // G-ZipZ.
DenotebyC′,C′ andC′ ,C′ (forw ∈IW)thecones(2.2.1)to(2.2.4)relatively
X w X′,w
to the map ζ′. Let T′ and T be maximal tori of G′ and G respectively, satisfying
T ∩G′ = T′. Write ι∗ : X∗(T) → X∗(T′) for the restriction map. Note that,
given λ ,λ ∈ X∗(T), one has V(λ ) ∼= V(λ ) (as line bundles) if and only if
1 2 1 2
ι∗(λ )=ι∗(λ ).
1 2
Itisclearthatι∗C ⊂C′ andι∗C ⊂C′ forallw ∈IW. Theseinclusions
w w X,w X′,w
are strict in general. However by [7, Lemma 6.2.7] the saturated cones are equal:
Lemma 2.3.1. For all w ∈IW, one has hι∗C i=hC′ i and hι∗C i=hC′ i.
w w X,w X′,w
In particular,
hC i=hC i⇐⇒hC′ i=hC′ i.
w X,w w X′,w
AUTOMORPHIC BUNDLES ON ZIP-SCHEMES 7
2.4. Shimura varieties. Let X be the special fiber of a Hodge-type Shimura va-
rietywithhyperspeciallevelatp. LetGbe the correspondingreductivegroupover
F and Z the zip datum of X. By [19], there is a smooth morphism of stacks
p
(2.4.1) ζ :X →G-ZipZ,
Themapζ satisfiesAssumption2.2.2. Condition(a)isclearlysatisfiedbydefini-
tion. Anumberofworkshaverecentlyshownthatζ issurjectiveoneveryconnected
componentX◦ ofX cf.[16,12]. Furthermore,2.2.2(c)followsfromKoecher’sprin-
ciple for strata, proved by Lan-Stroh in [15, Th. 2.5.10]. In particular, Conjecture
2.2.5 applies to Shimura varieties.
3. Example : Groups of type A
1
Inthissection,westudy QuestionC forF -groupsGoftypeA . Asacorollary,
p 1
wededuceresultsaboutHilbertmodularvarieties. Inforthcomingwork,weobtain
similarresults when G=GSp(4)andwhen G=GU(2,1)(and p split), whence we
deduce results about Siegel modular threefolds and Picard modular surfaces.
We prove Conjecture 2.2.5 when G is of type A . If G is F -split or if split over
1 p
the quadratic extension Fp2, we show that hCwi = hCX,wi holds for all w ∈ W,
which gives a complete answer to Question C.
Inthegeneralcase,wedefineasetofadmissiblestrataforwhichonehashC i=
w
hC i. However, we conjecture that not all w satisfy this equality of cones.
X,w
3.1. Notation. Let n≥1 be an integer; let n=n +···+n , be a partition of n
1 r
with n ≥1 for all i=1,...,r. Consider the F -reductive group G defined by
i p
(3.1.1) G:=G ×...×G , G :=Res (SL )
1 r i Fpni/Fp 2,Fpni
Define N = m n for all 1 ≤ m ≤ r and N := 0. Denote again by σ the
m i=1 i 0
permutation of {1,...,n} defined as a product σ =c ···c where c is the n -cycle
1 r i i
P
c =(N (N −1)···(N +1)) for i=1,...,r. There is an isomorphism
i i i i−1
(3.1.2) G ≃SLn
k 2,k
such that the action of σ ∈Gal(k/F ) on G(k)≃SL (k)n is given by
p 2
(3.1.3) σ(x ,...,x ):=(ϕ(x ),ϕ(x ),...,ϕ(x )).
1 n σ(1) σ(2) σ(n)
Let T ⊂SL be the diagonaltorus. We identify X∗(T)=Z by sending m∈Z
2,k
to the character diag(x,x−1) 7→ xm. Define T := T ×...×T ⊂ G and identify
k
similarly X∗(T) = Zn. Let B ⊂ SL be the Borel subgroup of upper-triangular
2,k
matrices,and define B :=B×...×B ⊂G. Deenote by B the opposite Borel. The
−
Weyl group ofeG is W =S ×...×S .
2 2
Let Z be the Boreel-type zip datum (G,B,T,B ,T,eϕ). Denote by X the cor-
−
responding stack of G-zips. Fix a map ζ : X → X satisfying Assumptions 2.2.2.
Define a Zariski open subset U ⊂SL as theeneon-evanieshing locus of the function
2
a b
(3.1.4) h:SL →A1, h: 7→a.
2,k k c d
(cid:18) (cid:19)
anddenotebyZ ⊂SL the zerolocusofthe function h(note thatZ is areduced
2,k
subscheme). We identify the elements of W withsubsets S ⊂{1,...,n}by the map
(3.1.5) W →P({1,...,n}), τ =(τ ,...,τ )7→{i∈{1,...,n}:τ =1}.
1 n i
8 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES
For a subset S ⊂{1,...,n}, write
(3.1.6) S =S ⊔...⊔S , S :=S∩{N +1,...,N }.
1 r i i−1 i
The zip stratum corresponding to a subset S ⊂{1,...,n} is defined by:
n
(3.1.7) G := G
S S,i
i=1
Y
where G := U if i ∈ S and G := Z if i ∈/ S. For a subset S ⊂ {1,...,n},
S,i S,i
denote by X := [E\G ] ⊂ G-ZipZ and X ⊂ X the corresponding locally closed
S S S
subsets, endowed with the reduced structure, and define similarly X∗ and X∗ as
S S
their respective Zariski closures.
WriteC andC fortheconescorrespondingtothezipstratumG ,asdefined
S X,S S
in section 2.2. Denote by e ,...,e ∈ Zn the natural basis of Zn. For any subset
1 n
S ⊂{1,...,n}, we define a Q-basis B =(β ,...,β ) of Qn by:
S 1,S n,S
−e +pe if i∈S,
(3.1.8) β := i σ(i)
i,S
(ei+peσ(i) if i∈/ S.
The cone hC i is the set of characters λ∈X∗(T) such that
S
(3.1.9) λ= a β
i i,Se
i∈S
X
where a ∈N for all i∈S and a ∈Z for all i∈/ S.
i i
3.2. The result. Let d ∈ Z be an integer. For any subset R ⊂ Z consisting of
≥1
d consecutive integers, the map φ : R → Z/dZ, k 7→ k is a bijection. Let Z/dZ
R
actonitself byaddition. Then φ yieldsa naturalactionofZ/dZonthe following
R
objects:
(1) The set R itself.
(2) The powerset P(R).
(3) The set of pairs (S,j) where S ⊂R and j ∈S.
Definition 3.2.1. Let d≥1 be an integer and R⊂Z be a subset consisting of d
consecutive numbers.
(1) A normalized admissible pair of R is a pair (S,x ) such that S ⊂ R is of the
r
form S ={x ,...,x } with x <...<x and x −x odd for all 1≤i≤r−1.
1 r 1 r i+1 i
(2) AnadmissiblepairofRisapair(S,j)thatisintheZ/dZ-orbitofanormalized
admissible pair for R.
(3) A G-admissible pair is a pair (S,j) such that j ∈ S for some 1 ≤ m ≤ r
m
(with the same notations as in (3.1.6)) such that (S ,j) is an admissible pair
m
of {N +1,...,N }.
m−1 m
(4) A subset S ⊂{1,...,n} is G-admissible if the pair (S,j) is G-admissible for all
j ∈S.
Remark 3.2.2.
(1) The subset {i} is G-admissible for all i∈{1,...,n}.
(2) The subset {1,...,n} is G-admissible.
(3) If σ(i)=i for all i∈{1,...,n} (equivalently if r =n), then every subset S
is G-admissible.
AUTOMORPHIC BUNDLES ON ZIP-SCHEMES 9
Theorem 3.2.3. Let ζ : X →X be a map satisfying Assumption 2.2.2. For each
G-admissible subset S ⊂{1,...,n}, the equality hC i=hC i holds.
X,S S
Theorem3.2.3willbeprovedattheendofthepaper,asacorollaryofProposition
3.3.3below. Asanapplication,letF beatotallyrealnumberfieldandassumethat
X is the special fiber of a Hilbert modular variety attached to F. Using Lemma
2.3.1, we may reduce to Theorem 3.2.3. Hence:
Corollary 3.2.4. Conjecture 2.2.5 holds for the special fiber of Hilbert modular
varieties at primes of good reduction. In other words,
n
(3.2.1) hC i=hCi= a (−e +pe ), a ∈N
X i i σ(i) i
( )
i=1
X
3.3. Proof of Theorem 3.2.3. Assume λ∈Qn is a quasi-character expressed in
the basis B . Choose an element j ∈ S, and consider the subset S \{j}, and the
S
corresponding basis B of Qn. We want to decompose λ in the basis B .
S\{j} S\{j}
For this, it suffices to determine the decomposition of the vector β = −e +
j,S j
pe in B . Write s := |S | for i = 1,...r and s := |S| = r s . Let
σ(j) S\{j} i i i=1 i
m∈{1,...,r} such that j ∈S . Let a ,...,a ∈Q be the unique rationalnumbers
m 1 n
P
such that
n
(3.3.1) β = a β .
j,S i i,S\{j}
i=1
X
Oneseesimmediatelythata =0forj ∈/ {N +1,...,N }. For1≤a≤b≤n,
i m−1 m
we define γ(a,b,S)∈{±1} by the formula:
(3.3.2) γ(a,b,S):=(−1)|{x∈S, a≤x≤b}|.
Ford≥1, x=(x ,...,x )∈Qd andy =(y ,...,y )∈Qd,define x∗y asthe vector
1 d 1 d
(x y ,...,x y ). Then we have the following formula:
1 1 d d
(3.3.3)
aNm−1+1 (−1)sm+Nm+j γ(Nm−1+1,j−1,S) 2pj−Nm−1−1
aNm.−1+2 (−1)sm+.Nm+j+1 γ(Nm−1+.2,j−1,S)  2pj−N.m−1−2 
. . . .
. . . .
       
 aj−2   (−1)nm+sm−1   γ(j−2,j−1,S)   2p2 
       
δ aj−1 = (−1)nm+sm ∗ γ(j−1,j−1,S) ∗ 2p 
 aj   1   1  pnm +(−1)sm+nm+1
       
 aj+1   −1   1   2pnm−1 
       
 aj+2   1   γ(j+1,j+1,S)   2pnm−2 
 ..   ..   ..   .. 
 .   .   .   . 
       
 aNm   (−1)Nm+j   γ(j+1,Nm−1,S)   2pj−Nm−1 
       
where δ =pnm +(−1)sm+nm.
Lemma 3.3.1. Let S 6= ∅ be a subset and i ∈ S. Let λ ∈ Zn with coordinates
(x ,...,x ) in B . Then there exists M (depending on S and λ) such that for
1 n S
m≥M, one has
H0(X∗,λ−mβ )=0.
S i,S
10 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES
Proof. We prove the result by induction on s = |S|. Let m ∈ {1,...,r} such that
i ∈ S . By Assumption 2.2.2(c), the result is clear if s = 1, so we assume s > 1.
m
Also, we may assume S = {x ,...,x } with x < ... < x and i = x . Let
m 1 sm 1 sm 1
j ∈S−{i}. By induction, there exists M(λ,j)≥1 such that
(3.3.4) H0(X∗ ,λ−mβ )=0
S−{j} i,S
for m ≥ M(λ,j). Consider the unique (up to scalar) non-zero section h ∈
j
H0(X∗,β ). The vanishing locus of h is X∗ , and the vanishing locus of
S j,S j S−{j}
ζ∗(h ) is X∗ by smoothness of ζ. Multiplication by ζ∗(h ) yields a shortexact
j S−{j} j
sequence of sheaves
0→VX(λ−mβi,S −βj,S)|X∗ →VX(λ−mβi,S)|X∗ →VX(λ−mβi,S)|X∗ →0
S S S−{j}
and a long exact sequence of cohomology:
0→H0(X∗,λ−mβ −β )→H0(X∗,λ−mβ )→H0(X∗ ,λ−mβ )→...
S i,S j,S S i,S S−{j} i,S
Hence for m≥M(λ,j), one has an isomorphism
(3.3.5) H0(X∗,λ−mβ −β )≃H0(X∗,λ−mβ )
S i,S j,S S i,S
Now there exists an integer M(λ−β ,j)≥M(λ,j) such that
j,S
(3.3.6) H0(X∗ ,λ−β −mβ )=0
S−{j} S,j i,S
form≥M(λ−β ,j). Applyingtheexactsequenceaboveforthischaractershows
j,S
that
H0(X∗,λ−mβ −2β )≃H0(X∗,λ−mβ −β )≃H0(X∗,λ−mβ )
S i,S j,S S i,S j,S S i,S
form≥M(λ−β ,j). Continuingthisway,itisclearthatwecanfindM′(λ,j)≥1
j,S
such that for m≥M′(λ,j), there exists λ′ with coordinates (x′,...,x′ ) in B such
1 n S
that x′ <0 and
j
(3.3.7) H0(X∗,λ−mβ )≃H0(X∗,λ′−mβ ).
S i,S S i,S
Henceforlargem,thereexistsµwithcoordinates(y ,...,y )inB suchthaty <0
1 n S j
for all j ∈ S and H0(X∗,λ−mβ ) ≃ H0(X∗,µ). By Assumption 2.2.2 (c), this
S i,S S
space is zero. (cid:3)
Lemma 3.3.2. Let d ≥ 1 be an integer and R ⊂ Z be a subset consisting of
d consecutive numbers. Let (S,j) be a normalized admissible pair for R. Write
S = {α ,...,α } with s = |S| and α < ... < α . Then the integer α +α +s is
1 s 1 s 1 s
odd.
Proof. Since (S,j) is a normalized admissible pair for R, one has j = α and
s
α −α is odd for all i=1,...,s−1. Hence,
i+1 i
s−1
(3.3.8) α −α = (α −α )
s 1 i+1 i
i=1
X
has the same parity as s−1, hence the result. (cid:3)
Proposition 3.3.3. Let (S,j) be a G-admissible pair. Let λ ∈ Zn be a character
with coordinates (x ,...,x ) in B . If x <0, then H0(X∗,λ)=0.
1 n S j S