Table Of ContentManin problems for Shimura varieties of Hodge type
Adrian Vasiu, Binghamton University, July 2007
7 ABSTRACT. Let k be a perfect field of characteristic p > 0. We prove the existence
0 of ascending and descending slope filtrations for Shimura p-divisible objects over k. We
0 ¯
use them to classify rationally these objects over k. Among geometric applications, we
2
mention two. First we formulate Manin problems for Shimura varieties of Hodge type.
l
u
Under two mild conditions (checked for p≥3 in [45]) we solve them. Second we formulate
J
integral Manin problems. We solve them for some Shimura varieties of PEL type.
4
2
] MSC (2000). 11G10, 11G15, 11G18, 11G35, 14F30, 14G15, 14L05, 14K22, and 20G25.
T
N
. KEY WORDS.F-crystals,F-isocrystals,Newtonpolygons,p-divisibleobjectsandgroups,
h
t reductive group schemes, abelian and Shimura varieties, Hodge cycles, and stratifications.
a
m
[
1. Introduction
5
v
0 Let p ∈ N be a prime. Let k be a perfect field of characteristic p. Let W(k) be the
1
ring of Witt vectors with coefficients in k. Let B(k) be the field of fractions of W(k). Let
4
9 σ := σ be the Frobenius automorphism of k, W(k), or B(k). An F-isocrystal over k is
k
0 a pair (M[1],φ), where M is a free W(k)-module of finite rank and φ : M[1]→∼ M[1] is
2 p p p
0 a σ-linear automorphism of M[1]. If we have φ(M) ⊆ M, then the pair (M,φ) is called
/ p
h an F-crystal over k. We denote also by φ the σ-linear automorphism of End(M[1]) that
t p
a
takes x ∈ End(M[1]) to φ◦x◦φ−1 ∈ End(M[1]).
m
p p
The F-isocrystals were introduced by Dieudonn´e in his work on finite, flat, commu-
:
v
tative group schemes of p-power order over k. In [11, Thms. 1 and 2] the F-isocrystals
i
X over k¯ are classified: an F-isocrystal over k¯ is uniquely determined up to isomorphisms
r by its Newton polygon. Manin reobtained this classification (see [28, Ch. 2, Section 4])
a
and brought the topic into the context of abelian varieties as follows (see [28, Ch. 4]).
Let (D,λ ) be a principally quasi-polarized p-divisible group over k of height 2r. Let
D
(M ,φ ,ψ ) be its principally quasi-polarized Dieudonn´e module (see [1, Ch. 4]). The
0 0 0
pair (M ,φ ) is an F-crystal over k, ψ is a perfect alternating form on M , and we have
0 0 0 0
pM ⊆ φ (M ) ⊆ M and ψ (φ (x),φ (y)) = pσ(ψ (x,y)), where x, y ∈ M . The Newton
0 0 0 0 0 0 0 0 0
polygon of (M ,φ ) has the following three properties:
0 0
(*) its slopes belong to the interval [0,1], its starting and ending points are (0,0) and
(2r,r), and the multiplicity of a slope γ is the same as the multiplicity of the slope 1−γ.
The first two properties of (*) are particular cases of a theorem of Mazur (see [19,
Thm. 1.4.1]). The third property of (*) is a consequence of the existence of λ and thus
D
1
of ψ ; in the geometric context of abelian varieties, it appears for the first time in [28].
0
The original Manin problem conjectured that each Newton polygon that satisfies (*), is
the Newton polygon of an abelian variety over k of dimension r. It was first solved in [39]
(see also [32] and [33] for two more recent proofs). In all that follows, expressions of the
form GL(M) and GSp(M ,ψ ) are viewed as reductive group schemes over W(k). Thus
0 0
GL(M)(W(k)) is the group of W(k)-linear automorphisms of M, etc.
Let T be a split, maximal torus of GSp(M ,ψ ) whose fibre over k normalizes
0 0 0
the kernel of the reduction mod p of φ . Let N be the normalizer of T ∩ Sp(M ,ψ )
0 0 0 0 0
in Sp(M ,ψ ). It is easy to see that there exists g ∈ Sp(M ,ψ )(W(k)) such that
0 0 0 0 0
(g φ )(Lie(T ))) = Lie(T ). The three properties of (*) are equivalent to:
0 0 0 0
1.1. Fact. For each g ∈ Sp(M ,ψ )(W(k)), there exists w ∈ N (W(k)) such that the
0 0 0 0
Newton polygon of (M ,gφ ) is the same as the Newton polygon of (M ,w g φ ).
0 0 0 0 0 0
This equivalence can be checked easily by considering the actions of w g φ ’s on the
0 0 0
rank 1 direct summands of M normalized by T . Since many years (see the paragraph
0 0
before Subsection 1.3 below) it was expected that Fact 1.1 also holds if GSp(M ,ψ )
0 0
is replaced by an arbitrary reductive, closed subgroup scheme of GL(M ) related in a
0
natural way to φ . The interest in the resulting problems stems from the study of Shimura
0
varieties. A general study of such problems was started by Kottwitz in [21]. See [8], [31],
[30], and [43, Subsection 2.5] for different types of Shimura varieties. For instance, the
Shimura varieties of PEL type are moduli spaces of polarized abelian varieties endowed
with endomorphisms (see [38] and [7]). Also the Shimura varieties of abelian (resp. of
Hodge) type are moduli spaces of polarized abelian motives (resp. of polarized abelian
varieties) endowed with Hodge cycles (see [7] and [30]). The Shimura varieties of abelian
type are the main testing ground for many parts of the Langlands program (like zeta
functions, local correspondences, etc.). The deep understanding of their zeta functions
depends on the Langlands–Rapoport conjecture (see [26], [27], and [29]) on F -valued
p
points of special fibres of their good integral models in mixed characteristic (0,p). To solve
this conjecture and to aim in extending [36] and [18] to all Shimura varieties, one needs
a good theory of isomorphism classes of F-isocrystals with additional structures that are
crystalline realizations of abelian motives associated naturally to these F -valued points.
p
This paper, [44], and [45] are part of a sequence meant to contribute to such a theory.
The below notion Shimura p-divisibleobject axiomatizesall crystallinerealizationsone can
(or hopes to) associate to points with values in perfect fields of “good” integral models in
mixed characteristic (0,p) of arbitrary (quotients of) Shimura varieties. The main goals
¯
of the paper are to classify rationally such p-divisible objects over k and to generalize the
original Manin problem to contexts related to Shimura varieties of Hodge type.
1.2. The language. In this paper we will use an integral language that is closer in
spirit to the works [25], [14], and [49] which precede [21]. For a, b ∈ Z, b≥a, we define
S(a,b) := {a,a+1,... ,b}.
1.2.1. Definitions. A p-divisible object with a reductive group over k is a triple (M,φ,G),
where M is a free W(k)-module of finite rank, the pair (M[1],φ) is an F-isocrystal over
p
k, and G is a reductive, closed subgroup scheme of GL(M), such that there exists a direct
2
sum decomposition
(1) M = ⊕ F˜i(M)
i∈S(a,b)
for which the following two axioms hold:
(a) we have φ−1(M) = ⊕b p−iF˜i(M) and φ(Lie(G )) = Lie(G );
i=a B(k) B(k)
(b) the cocharacter µ of GL(M) such that β ∈ G (W(k)) acts through µ on F˜i(M)
m
as the multiplication by β−i, factors through G.
Following [34, Section 2] we refer to µ : G → G as a Hodge cocharacter of (M,φ,G)
m
and to (1) as its Hodge decomposition. If G = GL(M), then often we do not mention
G and so implicitly “with a reductive group”. For i ∈ S(a,b) let Fi(M) := ⊕b F˜j(M)
j=i
and let φ : Fi(M) → M be the restriction of p−iφ to Fi(M). Triples of the form
i
(M,(Fi(M)) ,φ) show up in [25], [14], [49], etc. If n ∈ N, then the reduction mod pn
i∈S(a,b)
of(M,(Fi(M)) ,(φ ) )isanobjectoftheabeliancategoryMF (W(k))used
i∈S(a,b) i i∈S(a,b) [a,b]
in [25], [14], and [12]. This and the fact that such a reduction is a natural generalization of
a truncated Barsotti–Tate group of level n over W(k) (for instance, for p≥3 see [12, Thm.
7.1]) justifies our terminology “p-divisible object”. Let b ∈ S(0,b − a) be the smallest
L
number with the property that we have a direct sum decomposition
(2) Lie(G) = ⊕bL F˜i(Lie(G))
i=−bL
such that β ∈ G (W(k)) acts through µ on F˜i(Lie(G)) as the multiplication by β−i.
m
We have b = 0 if and only if µ factors through the center of G. If b = 1, then
L L
µ is called a minuscule cocharacter of G. If bL≤1, then we say (M,φ,G) is a Shimura
p-divisible objectover k. If (a,b) = (0,1), then we say (M,φ,G) is a Shimura F-crystal over
k; they were extensively used in [43, Section 5] and in many previous works on Shimura
varieties of PEL type (see [46], [27], [23], etc.).
For g ∈ G(W(k)) let
C := (M,gφ,G).
g
We have φ−1(M) = (gφ)−1(M) and thus C is also a p-divisible object with a reductive
g
group over k that has µ as a Hodge cocharacter. By the extension of C to a perfect field
g
k that contains k, we mean the triple C ⊗ k := (M ⊗ W(k ),gφ⊗σ ,G ).
1 g k 1 W(k) 1 k1 W(k1)
1.2.2. Definition. Let g, g ∈ G(W(k)). By an inner (resp. by a rational inner)
1
isomorphism between C and C we mean an element h ∈ G(W(k)) that normalizes
g g1
φ−1(M) and (resp. an element h ∈ G(B(k))) such that we have hgφ = g φh.
1
1.2.3. Notations. For g ∈ G(W(k)) let S(g) be the set of Newton polygon slopes of
(M[p1],gφ). Let M[p1] = ⊕γ∈S(g)Mγ(g) be the direct sum decomposition stable under
φ and such that all Newton polygon slopes of (M (g),φ) are γ. If m ∈ N is such that
γ
mγ ∈ Z, then there exists a B(k)-basis for M (g) formed by elements fixed by p−mγφm.
γ
Let γ < γ < ··· < γ be the numbers in S(g) listed increasingly. For γ ∈ S(g) let
1,g 2,g ng,g
Wγ(M,gφ) := M∩(⊕α∈S(g)∩[γ,∞)Mα(g)) and Wγ(M,gφ) := M∩(⊕α∈S(g)∩(−∞,γ]Mα(g)).
3
Due to the axioms 1.2.1 (a) and (b), the Newton quasi-cocharacter of GL(M[1]) defined
p
by gφ (its definition is reviewed in Subsubsection 2.2.1) factors through G (see Claim
B(k)
2.2.2). Let ν be this factorization.
g
1.3. The basic results. In Sections 3 and 4 we mainly study the case when bL≤1 and
we deal with two aspects of the classification of C ’s up to inner isomorphisms. The two
g
aspects are: slope filtrations and ν ’s. We list the basic results.
g
1.3.1. Theorem. Let g ∈ G(W(k)). There exists a unique parabolic subgroup scheme
P+(gφ) of G such that all Newton polygon slopes of the F-isocrystal (Lie(P+(gφ) ),gφ)
G G B(k)
(resp. (Lie(G )/Lie(P+(gφ) ),gφ)) are non-negative (resp. are negative). If b ≤ 1,
B(k) G B(k) L
then there exists a Hodge cocharacter of (M,gφ,G) that factors through P+(gφ).
G
1.3.2. Corollary. Let g ∈ G(W(k)). We assume bL≤1. We have:
(a) There exists a Hodge cocharacter of (M,gφ,G) that normalizes Wγ(M,gφ) for
all γ ∈ S(g).
(b) Up to a rational inner isomorphism, we can assume that we have a direct sum
decomposition M := ⊕γ∈S(g)M∩Mγ(g) and that µ normalizes M∩Mγ(g) for all γ ∈ S(g).
If b ≤ 1, then from Corollary 1.3.2 (a) and axiom 1.2.1 (a) we get that the filtration
L
(3) (Wγng,g(M,gφ),gφ) ⊆ (Wγng−1,g(M,gφ),gφ) ⊆ ··· ⊆ (Wγ1,g(M,gφ),gφ)= (M,gφ)
is a filtration in the category p − M(W(k)) of p-divisible objects over k (the morphisms
being W(k)-linear maps that respect after inverting p the Frobenius endomorphisms). We
refer to it as the descending slope filtration of (M,gφ). Replacing “negative” by “positive”,
we get another parabolic subgroup scheme P−(gφ) of G and the ascending slope filtration
G
(4) (W (M,gφ),gφ) ⊆ (W (M,gφ),gφ)⊆ ··· ⊆ (W (M,gφ),gφ) = (M,gφ)
γ1,g γ2,g γng,g
of(M,gφ)inthecategoryp−M(W(k)). Wecandefine Shimurap-divisibleobjectsoverany
field l of characteristic p and we can always speak about their ascending slope filtrations
over a Cohen ring K(l) of l. Thus we view (4) and its analogue over K(l) as a natural
extension of Grothendieck’s slope filtrations of p-divisible groups over l (see [47]).
We shift to the rational classification of C ’s. For the remaining part of Section 1,
g
we will assume that G is split. Let T be a split, maximal torus of G such that µ factors
through it. Until Section 2 we also assume that φ(Lie(T)) = Lie(T) (in Subsection 2.5 we
check that we can always achieve this by replacing φ with g φ for some g ∈ G(W(k))).
0 0
1.3.3. Theorem. Let g ∈ G(W(k)). Let N be the normalizer of T in G. We assume that
bL≤1 and that G is split. We have:
(a) There exists w ∈ N(W(k)) such that ν and ν are G(B(k))-conjugate.
w g
¯
(b) If k = k, then there exist w ∈ N(W(k)) and h ∈ G(B(k)) such that hwφ = gφh.
Standard properties needed in Sections 3 and 4 are gathered in Section 2. See
Subsections 2.3 and 4.1 for the proofs of Theorem 1.3.1 and Corollary 1.3.2. The proof
4
of Theorem 1.3.3 is in two steps (see Subsection 4.2). The first step works for all b ∈
L
N ∪ {0} and shows two things. First, using the classification of adjoint group schemes
over Z , it shows that there exists w ∈ N(W(k)) such that all Newton polygon slopes
p
¯
of (Lie(G ),wφ) are 0 (see Subsubsection 4.2.2). Second, if k = k and if all Newton
B(k)
polygon slopes of (Lie(G ),gφ) and of (Lie(G ),g φ) are 0, then standard arguments
B(k) B(k) 1
as in [21] and [35] show that there exist rational inner isomorphisms between C and C
g g1
(see Subsections 2.6 and 2.7). The second step is an inductive one. It works only if b ≤ 1
L
(see Subsubsection 4.2.3). The idea is: if (Lie(G ),gφ) has non-zero Newton polygon
B(k)
slopes, then C has standard forms that reduce the situation to a context in which there
g
exists a Levi subgroup scheme L of P+(gφ) such that the triple (M,gφ,L) is a Shimura
G
p-divisible object. The standard forms (see Section 3) are also the essence of Theorem
1.3.1 and Corollary 1.3.2. They are rooted on the simple property 2.4 (c) and on the fact
that each intersection of two parabolic subgroups of G contains a maximal split torus
k
of G . See Subsections 4.3 to 4.7 for examples and complements to Corollary 1.3.2 and
k
Theorem 1.3.3.
The notion rational inner isomorphism is only a variant of the σ-conjugacy notion of
[21]; directconnectionstothe“B(G )-language”of[21]and[35]aremadeinSubsections
B(k)
2.6 and 4.5. The notion inner isomorphism is a natural extension of the classification ideas
of [28]. Parabolic subgroup schemes as P+(gφ) were first used in [42] for (a,b) = (0,1).
G
To our knowledge, Theorem 1.3.1 is a new result. The Newton polygon translation of
Theorem 1.3.3 (a) was indirectly hinted at by the Langlands–Rapoport conjecture. Chai
extrapolated this conjecture and stated rather explicitly that Theorem 1.3.3 (a) ought to
hold (see [6]). Theorem 1.3.1 and versions of Corollary 1.3.2 and Theorem 1.3.3 were first
part of our manuscripts math.NT/0104152 and math.NT/0209410 (only few particular
cases of Theorem 1.3.3 (b) were known before math.NT/0104152 and most of them could
be deduced from [28]). The paper [24] was written after the mentioned two manuscripts;
one can use [24, Theorem 4.3] to recover Theorem 1.3.3 (b). See Corollary 4.4 for an
interpretation of Theorem 1.3.3 (b) in terms of equivalence classes. The last part of
Theorem 1.3.1 and Theorem 1.3.3 (and thus also (3) and (4)) do not hold in general if
bL≥2 (see Example 2.3.4).
1.4. On geometric applications. In Subsection 5.1 we introduce the standard Hodge
situations. They give birth to good moduli spaces in mixed characteristic (0,p) of prin-
cipally polarized abelian varieties endowed with (specializations of) Hodge cycles, that
generalize the moduli spaces of principally polarized abelian varieties endowed with endo-
morphisms used in [46], [23], and [27]. In Subsection 5.2 we formulate Manin problems
for standard Hodge situations. Under two mild assumptions (see Subsubsection 5.2.1) we
solve them: see the Main Theorem 5.2.3. For p≥3, these two mild conditions are par-
ticular cases of [45, Main Thm. 1.2 and Lemma 2.5.2 (a)]. The proof of Main Theorem
relies on Theorem 1.3.3 (b), on Fontaine comparison theory, and on properties of Shimura
varieties and reductive group schemes. In particular, we get a new solution to the original
Manin problem mentioned before Fact 1.1 (cf. Subsubsection 5.2.4 (a)). In Subsection 5.3
we follow [35, Thm. 3.8] and introduce rational stratifications of the special fibres of the
mentioned good moduli spaces. In Subsection 5.4 we formulate ıintegral Manin problems
for standard Hodge situations. Theorem 5.4.2 solves them in many cases that pertain to
5
Shimura varieties of PEL type; the simplest example implies that each principally quasi-
¯
polarized p-divisible group over k of height 2r is the one of a principally polarized abelian
¯
variety over k of dimension r (cf. Example 5.4.3 (a)). These integral problems are nat-
ural extrapolations of the “combination” between the Manin problems and a conjecture
of Milne (see [43, Conjecture 5.6.6] and see [45] for refinements of it). Section 5 prepares
the background for a future work in which we will extend [46] to the context of many
standard Hodge situations. This extension will be a major step toward the complete proof
of the Langlands–Rapoport conjecture for Shimura varieties of abelian type with respect
to primes p ≥ 5.
2. Preliminaries
See Subsection 2.1 for our conventions and notations. In Subsections 2.2 and 2.3 we
include complements on Newton polygons. We deal with two aspects: Newton (quasi-)
cocharacters and parabolic subgroup schemes that correspond to either non-negative or
to non-positive Newton polygon slopes. Our approach to Newton quasi-cocharacters is
slightly different from the standard one that uses the pro-torus of character group Q (see
[35] and [34]). In Subsections 2.4 to 2.7 we list different properties of C ’s.
g
2.1. Notations and conventions. Reductive group schemes have connected fibres. If
Spec(R) is an affine scheme and if H is a reductive group scheme over R, let Hder, Z(H),
Hab, and Had be the derived group scheme, the center, the maximal abelian quotient, and
respectively the adjoint group scheme of H. We have H/Z(H) = Had and H/Hder = Hab.
Let Z0(H) be the maximal torus of Z(H). Let Lie(F) be the Lie algebra over R of
a smooth, closed subgroup scheme F of H. If R = W(k), then H(W(k)) is called a
hyperspecial subgroup of H(B(k)) (see [41]). Let R → R be a homomorphism. If it is
0
finite and flat, let Res H be the affine group scheme over Spec(R ) that is the Weil
R/R0 0
restriction of scalars of H (see [4, Subsection 7.6]). In general, the pull back of an R -
0
scheme X or X (resp. X with ∗ an index) to R is denoted by X (resp. X ). Let E¯
R0 ∗ R ∗R
be an algebraic closure of a field E.
For an R-module N let N∗ := Hom (N,R). Let N⊗s ⊗ N∗⊗t, with s,t ∈ N∪{0},
R R
be the tensor product of s-copies of N with t-copies of N∗ taken in this order. Let
T(N) := ⊕ N⊗s ⊗ N∗⊗t.
s,t∈N∪{0} R
A family of tensors of T(N) is denoted in the form (v ) , with J as the set of indices.
α α∈J
We emphasize that we use the same notation for two tensors or bilinear forms obtained one
from another by an extension of scalars. Let N be another R-module. Each isomorphism
1
f : N →∼ N extends naturally to an isomorphism T(N)→∼ T(N ) and therefore we speak
1 1
about f taking v to some specific element of T(N ). A bilinear form on N is called perfect
α 1
if it induces an isomorphism N →∼ N∗. We view GL(N) as a reductive group scheme over
R. If f and f are two Z-endomorphisms of N, then f f := f ◦f .
1 2 1 2 1 2
Until Section 5, whenever we consider a p-divisible object with a reductive group
(M,φ,G) over k, the following notations a, b, S(a,b), µ, b , C ’s, F˜i(M) and Fi(M)
L g
with i ∈ S(a,b), and F˜i(Lie(G)) with i ∈ S(−b ,b ) will be as in Subsubsection 1.2.1.
L L
6
Often we do not mention “over k”. Let P be the parabolic subgroup scheme of G that
normalizes Fi(M) for all i ∈ S(a,b). From Subsection 2.3 onward, γ ∈ S(g), M (g),
γ
Wγ(M,gφ), W (M,gφ), and ν , will be as in Subsections 1.2.3 and 2.2.3. If all Newton
γ g
polygon slopes of (Lie(G ),φ) are 0, then following [21, Subsection 5.1] we say (M,φ,G)
B(k)
is basic. We denote also by φ the σ-linear automorphism of M∗[1] that takes x ∈ M∗[1]
p p
to σxφ−1 ∈ M∗[1]. Thus φ acts on T(M[1]) in the natural tensor way. The identification
p p
End(M) = M⊗ M∗ is compatible withthe two φ actions (defined here and before Fact
W(k)
1.1). If (a,b) = (0,1), then we refer to (M,F1(M),φ,G) as a Shimura filtered F-crystal.
2.2. Quasi-cocharacters. Let H be a reductive group scheme over a connected scheme
S. Let χ(H) be the set of cocharacters of H. Let Λ(H) := χ(H) × Q \ {0}. Let R(H)
be the smallest equivalence relation on Λ(H) that has the following property: two pairs
(µ ,r ), (µ ,r ) ∈ Λ(H) are in relation R(H) if there exists n ∈ N such that nr , nr ∈ Z,
1 1 2 2 1 2
g.c.d.(nr ,nr ) = 1, and µnr1 = µnr2. Let Ξ(H) be the set of equivalence classes of R(H).
1 2 1 2
An element of Ξ(H) is called a quasi-cocharacter of H. Identifying χ(H) with the subset
χ(H)×{1} of Λ(H), we view naturally χ(H) as a subset of Ξ(H). If H is a split torus, then
χ(H) = X (H) has a natural structure of a free Z-module and we can identify naturally
∗
Ξ(H) = X (H) ⊗ Q. The group H(S) acts on Ξ(H) via its inner conjugation action
∗ Z
on χ(H). If f : H → H is a homomorphism of reductive group schemes over S, then
1
f (µ ,r ) := (f ◦ µ ,r ) ∈ Λ(H ). The resulting map f : Λ(H) → Λ(H ) is compatible
∗ 1 1 1 1 1 ∗ 1
with the R(H) and R(H ) relations; the quotient map f : Ξ(H) → Ξ(H ) is compatible
1 ∗ 1
with the H(S)- and H (S)-actions. We say † ∈ Ξ(H ) factors through H if † ∈ f (Ξ(H)).
1 1 ∗
2.2.1. The slope context. Let C = (M,φ,G) be a p-divisible object with a reductive
1M
group over k. Let S(1 ) be as Subsubsection 1.2.3. For γ ∈ S(1 ) we write γ = aγ,
M M bγ
where a ∈ Z, b ∈ N, and g.c.d.(a ,b ) = 1. Let d := l.c.m.(b |γ ∈ S(1 )). Each
γ γ γ γ γ M
¯
F-isocrystal over k is a direct sum of simple F-isocrystals which have only one Newton
¯
polygon slope, cf. Dieudonn´e’s classification of F-isocrystals over k. We have a direct sum
decomposition
(5) (M ⊗W(k) B(k¯),φ⊗σk¯) = ⊕γ∈S(1M)(M¯γ(1M),φ⊗σk¯)
¯ ¯
of F-isocrystals over k such that (M (1 ),φ ⊗ σ ) has only one Newton polygon slope
γ M k¯
γ. Let e¯ ∈ End(M) ⊗ B(k¯) be the semisimple element that acts on M¯ (1 ) as
W(k) γ M
the multiplication by daγ. The Galois group Gal(k¯/k) acts on B(k¯) having B(k) as its
bγ
fixed field and acts on T(M ⊗ B(k¯)) having T(M[1]) as its set of fixed elements.
W(k) p
¯ ¯
For each automorphism τ ∈ Aut(B(k)/B(k)) defined by an element of Gal(k/k), φ ⊗σ
k¯
and 1 ⊗ τ commute. Thus e¯ is fixed by Gal(k¯/k) i.e., e¯ ∈ End(M[1]) =
M⊗W(k)B(k¯) p
¯
End(M)⊗ B(k). Thus (5)isthetensorizationwithB(k)ofadirect sum decomposition
W(k)
(M[p1],φ) = ⊕γ∈S(1M)(Mγ(1M),φ).
Let ν˜ ∈ χ(GL(M[1])) be such that ν˜ (p) acts on M (1 ) as the multiplication
1M p 1M γ M
daγ
by p bγ . Let ν1M := [(ν˜1M, d1)] ∈ Ξ(GL(M[p1])). By abuse of language, we say that ν1M(p)
acts on M (1 ) as the multiplication by pγ. As the decompositions (5) are compatible
γ M
7
with morphisms and tensor products of F-isocrystals, ν˜ and ν factor through the
1M 1M
subgroup of GL(M[1]) that fixes all tensors of T(M[1]) fixed by φ.
p p
2.2.2. Claim. Both ν˜ and ν factor through G .
1M 1M B(k)
Proof: Let σ := φ◦ µ(p). We have σ (M) = φ(⊕b p−1F˜i(M)) = φ(φ−1(M)) = M, cf.
0 0 i=1
axiom 1.2.1 (a). Thus σ is a σ-linear automorphism of M. As σ normalizes Lie(G )
0 0 B(k)
(cf. axioms 1.2.1 (a) and (b)), it also normalizes Lie(G) = Lie(G )∩End(M). Let
B(k)
M := {x ∈ M|σ (x) = x}
Zp 0
and g := {x ∈ Lie(G)|σ (x) = x}. To prove the Claim we can assume k = k¯. As k = k¯,
Zp 0
we have M = M ⊗ W(k) and Lie(G) = g ⊗ W(k) ⊆ End(M )⊗ W(k).
Z Z Z Z Z Z
p p p p p p
In this paragraph we check that there exists a unique reductive subgroup G of
Q
p
GL(M [1]) whose Lie algebra is g [1]. The uniqueness part is implied by [3, Ch. II,
Zp p Zp p
Subsection 7.1]. To check the existence part, we consider commutative Q -algebras A such
p
that there exists a reductive, closed subgroup scheme G of GL(M ⊗ A) whose Lie
A Zp Zp
algebra is g ⊗ A. For instance, A can be B(k) itself and thus we can assume A is
Z Z
p p
a finitely generated Q -subalgebra of B(k) (cf. [10, Vol. III, Exp. XIX, Rm. 2.9]). By
p
replacing A with A/J, where J is a maximal ideal of A, we can assume A is a finite field
extension of Q . Even more, we can assume that A is a finite Galois extension of Q . As
p p
Lie(G ) = g ⊗ A, from [3, Ch. II, Subsection 7.1] we get that the natural action of the
A Zp Zp
Galois group Gal(A/Q ) on g ⊗ A is defined naturally by an action of Gal(A/Q ) on
p Zp Zp p
the subgroup G of GL(M ⊗ A). This last action is free. As G is an affine scheme,
A Zp Zp A
the quotient G of G by Gal(A/Q ) exists (cf. [4, Ch. 6, 6.1, Thm. 5]) and it is a
Qp A p
subgroup of GL(M [1] whose Lie algebra is g [1].
Zp p Zp p
Our notations match i.e., G is the pull back of G to B(k) (cf. [3, Ch. II,
B(k) Qp
Subsection 7.1]). Let G be the Zariski closure of G in GL(M ); its pull back to W(k)
Z Q Z
p p p
is G. Thus G is a reductive, closed subgroup scheme of GL(M ); its Lie algebra is g .
Z Z Z
p p p
Let (t ) be a family of tensors of T(M ) such that G is the subgroup of
α α∈J Zp Qp
GL(M [1]) that fixes t for all α ∈ J, cf. [9, Prop. 3.1 (c)]. Each t is fixed by both σ
Zp p α α 0
and µ(p). Thus we have φ(t ) = t for all α ∈ J. Therefore ν˜ fixes each t , cf. end of
α α 1M α
Subsubsection 2.2.1. Thus both ν˜ and ν factor through G . (cid:3)
1M 1M B(k)
2.2.3. Definition. By the Newton cocharacter (resp. quasi-cocharacter) of C we mean
1M
the factorization of ν˜ (resp. of ν ) through G . Similarly, for g ∈ G(W(k)) let
1M 1M B(k)
ν ∈ Ξ(G ) be the Newton quasi-cocharacter of C .
g B(k) g
2.3. Sign parabolic subgroup schemes. Let LS(1 ) be the set of Newton poly-
M
gon slopes of (Lie(G ),φ). The composite of ν with the homomorphisms G →
B(k) 1M B(k)
GL(M[1]) → GL(End(M)[1]), is the Newton quasi-cocharacter of (End(M)[1],φ). Thus
p p p
the Newton polygon slope decomposition
(Lie(GB(k)),φ) = ⊕γ∈LS(1M)(Lγ,φγ)
is such that ν (p) acts via inner conjugation on L as the multiplication by pγ. As
1M γ
Im(ν˜ ) isa split torus of G , the centralizer C (φ) of ν˜ in G isa reductive group
1M B(k) G 1M B(k)
of the same rank as G (see [10, Vol. III, Exp. XIX, 2.8]). We have Lie(C (φ)) = L .
B(k) G 0
8
2.3.1. Lemma. There exists a unique parabolic subgroup scheme P+(φ) of G such that
G
we have Lie(PG+(φ)B(k)) = ⊕γ∈LS(1M)∩[0,∞)Lγ. The group CG(φ) is a Levi subgroup of
PG+(φ)B(k) and L>0 := ⊕γ∈LS(1M)∩(0,∞)Lγ is the nilpotent radical of Lie(PG+(φ)B(k)).
Proof: Byreplacingk withafiniteGaloisextensionofk, wecanassumeGissplit. Thusthe
groupC (φ)isalsosplit. LetLie(G ) = Lie(T (φ)) g betherootdecomposition
G B(k) G Lα∈Φ α
withrespecttoasplit,maximaltorusT (φ)ofC (φ). HereΦisarootsystemofcharacters
G G
of T (φ) whose irreducible factors are indexed by the simple factors of Gad . Let γ ,
G B(k) 1
γ ∈ LS(1 ). If γ +γ ∈/ LS(1 ), let L := 0. As φ is a σ-linear Lie automorphism
2 M 1 2 M γ1+γ2
of Lie(G ), we have [L ,L ] ⊆ L . Thus the set Φ := {α ∈ Φ|g ⊆ L ⊕L }
B(k) γ1 γ2 γ1+γ2 + α 0 >0
is a closed subset of Φ. Let Φ be a simple factor of Φ. The intersection Φ ∩ Φ is
0 0 +
a closed subset of Φ . As ν˜ factors through T (φ), each L is a direct sum of some
0 1M G γ
g ’s and we have g ⊆ L if and only if g ⊆ L . Thus the intersection Φ ∩Φ is a
α α γ −α −γ 0 +
parabolic subset of Φ in the sense of [5, Ch. VI, 7, Def. 4] i.e., it is closed and we have
0
Φ = −(Φ ∩Φ )∪Φ ∩Φ . There exists a unique parabolic subgroup P+(φ) of G
0 0 + 0 + G B(k) B(k)
whose Lie algebra is L ⊕L , cf. [10, Vol. III, Exp. XXVI, Prop. 1.4]. Obviously L
0 >0 >0
is a nilpotent ideal of L ⊕L and thus also of the nilpotent radical n of L ⊕L . As
0 >0 0 >0
C (φ) is reductive and Lie(C (φ)) = L , we have n∩L = 0. Thus n = L .
G G 0 0 >0
Therefore Lie(P+(φ) ) = L ⊕n = Lie(C (φ))⊕n and thus C (φ) is a Levi sub-
G B(k) 0 G G
group of P+(φ) . As the W(k)-scheme that parametrizes parabolic subgroup schemes
G B(k)
of G is projective (cf. [10, Vol. III, Exp. XXVI, Cor. 3.5]), the Zariski closure P+(φ) of
G
P+(φ) in G is a parabolic subgroup scheme of G. (cid:3)
G B(k)
A similar argument shows that there exists a unique parabolic subgroup scheme
PG−(φ) of G such that we have Lie(PG−(φ)B(k)) = ⊕γ∈LS(1M)∩(−∞,0]Lγ. The group CG(φ)
is also a Levi subgroup of P−(φ) . Replacing the role of ν˜ with the one of µ, we
G B(k) 1M
similarly get that the parabolic subgroup scheme P of Subsubsection 1.2.1 exists and is
uniquely determined by the equality Lie(P) = ⊕bL F˜i(Lie(G)). The next Corollary is only
i=0
a variant of [21, Subsection 5.2].
2.3.2. Corollary. The following three statements are equivalent:
(a) P+(φ) = G (or P−(φ) = G);
G G
(b) C is basic;
1M
(c) ν˜ (or ν ) factors through Z0(G ).
1M 1M B(k)
Proof: EachstatementisequivalenttothestatementthatL = Lie(C (φ))isLie(G ).(cid:3)
0 G B(k)
2.3.3. Definition. We call P+(φ) (resp. P−(φ)) the non-negative (resp. non-positive)
G G
parabolic subgroup scheme of C . We call C (φ) the Levi subgroup of C .
1M G 1M
¯
2.3.4. Example. Suppose that k = k, (a,b) = (0,2), rk (M) = 4, G = GL(M),
W(k)
rk (F˜2(M)) = rk (F˜0(M)) = 2, and there exists a W(k)-basis {e ,... ,e } for M
W(k) W(k) 1 4
formed by elements of F˜0(M)∪F˜2(M) and such that φ(e ) = pnie , where n = n = 0
i i 1 2
and n = n = 2. Let T be the maximal torus of GL(M) that normalizes W(k)e for all
3 4 i
i ∈ S(1,4). Let g ∈ G(W(k)) be such that S(g) = {1, 3}, cf. [16, Thm. 2]. For each
2 2
element w ∈ G(W(k)) that normalizes T, the F-crystal (M,wφ) over k has at least one
9
Newton polygon slope which is an integral (to check this one can assume that w permutes
the set {e ,... ,e }). Thus ν and ν are not G(B(k))-conjugate i.e., Theorem 1.3.3 does
1 4 g w
not hold in this case. The parabolic subgroup scheme of G that normalizes M ∩M (g) is
3
2
P+(gφ). If there exists a Hodge cocharacter of (M,gφ,G) that factors through P+(gφ),
G G
then (M∩M (g),gφ) is a p-divisible object whose Hodge numbers belong to the set {0,2}
3
2
and this, based on Mazur’s theorem (see [19, Thm. 1.4.1]), contradicts the fact that the
end point of Newton polygon of (M ∩ M (g),gφ) is (2,3). Thus also the last part of
3
2
Theorem 1.3.1 does not hold in this case.
2.4. Lemma. (a) An element h ∈ G(W(k)) normalizes φ−1(M) if and only if we have
(6) h(F˜i(M)) ⊆ ⊕b pmax{0,i−j}F˜j(M) ∀i ∈ S(a,b).
j=a
In particular, each element h ∈ P(W(k)) normalizes φ−1(M).
(b) For h ∈ G(B(k)) let h := φhφ−1 ∈ GL(M)(B(k)). We have h ∈ G(B(k)).
1 1
Also, h normalizes φ−1(M) if and only if h ∈ G(W(k)).
1
(c) We have bL≤1 if and only if each element h ∈ Ker(G(W(k)) → G(k)) normalizes
φ−1(M).
Proof: We prove (a). As h(M) = M, the inclusions of (6) are equivalent to the inclusion
h(φ−1(M)) ⊆ φ−1(M). As h(M) = M and as pbφ−1(M) ⊆ M, by reasons of lengths of
artinian W(k)-modules we get that h(φ−1(M)) ⊆ φ−1(M) if and only if h(φ−1(M)) =
φ−1(M). Thus (6) holds if and only if h normalizes φ−1(M). If h ∈ P(W(k)), then
h(F˜i(M)) ⊆ h(Fi(M)) = Fi(M) = ⊕b F˜j(M); thus (6) holds and therefore h normalizes
j=i
φ−1(M). Thus (a) holds. To prove (b) we can assume k = k¯. Let σ = φ◦µ(p) : M →∼ M
0
be as in the proof of Claim 2.2.2. We have h = σ µ(1)hµ(p)σ−1 ∈ σ G(B(k))σ−1 =
1 0 p 0 0 0
G(B(k)). Thus h ∈ G(W(k)) if and only if h (M) = M. As φ−1(h (M)) = h(φ−1(M)),
1 1 1
we have h (M) = M if and only if h(φ−1(M)) = φ−1(M). Thus (b) holds.
1
We prove (c). We assume that b ≤ 1 and we check that each h ∈ Ker(G(W(k)) →
L
G(k)) normalizes φ−1(M). Based on (b), we only have to show that h ∈ G(W(k)). As
1
bL≤1, F˜−1(Lie(G)) is a commutative Lie subalgebra of Lie(G) that is the Lie algebra
of a connected, closed, smooth, commutative subgroup scheme U of G. As Lie(P) =
1
F˜0(Lie(G))⊕F˜1(Lie(G)), we have a direct sum decomposition Lie(G) = Lie(U )⊕Lie(P)
1
of W(k)-modules (cf. (2)). Thus the product morphism U × P → G is ´etale around
1 W(k)
the point (1 ,1 ) ∈ U (W(k)) × P(W(k)). Therefore we can write h = u p , where
M M 1 h h
p ∈ P(W(k)) and u ∈ U (W(k)) are both congruent to 1 mod p. Thus h = h h ,
h h 1 M 1 2 3
where h := φu φ−1 and h := φp φ−1. From (a) and (b) we get that h ∈ G(W(k)).
2 h 3 h 3
To check that h ∈ G(W(k)) we can assume k = k¯. We have h = σ µ(1)u µ(p)σ−1.
2 2 0 p h 0
As uh ∈ Ker(U1(W(k)) → U1(k)) and bL≤1, we have µ(p1)uhµ(p) ∈ U1(W(k)). Thus
h ∈ σ U (W(k))σ−1 6 G(W(k)). Therefore h = h h ∈ G(W(k)).
2 0 1 0 1 2 3
We assume that all elements h ∈ Ker(G(W(k)) → G(k)) normalize φ−1(M) and
we check that bL ≤ 1. We show that the assumption bL≥2 leads to a contradiction.
Both Lie algebras F˜−bL(Lie(G)) and F˜bL(Lie(G)) are non-trivial and commutative. Let
U be the non-trivial, connected, smooth, commutative, closed subgroup scheme of G
bL
10
