On a Conjecture of Rapoport and Zink

0.32 MB·English

by  Urs Hartl

On a Conjecture of Rapoport and Zink
Urs Hartl ∗
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0 February 8, 2008
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Abstract

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In their book Rapoport and Zink constructed rigid analytic period spaces Fwa for
1
Fontaine’s filtered isocrystals and period morphisms from moduli spaces of p-divisible

] groups to some of these period spaces. They conjectured the existence of an ´etale
T
bijective morphism Fa →Fwa of rigid analytic spaces and of interesting local systems
N
of Q -vector spaces on Fa. For those period spaces possessing period morphisms de
p
h. Jong pointed out that one may take Fa as the image of the period morphism, viewed
t as a morphism of Berkovichspaces, and take the rational Tate module of the universal
a
m p-divisible group as the desired local system on Fa. In this article we construct for
Hodge-Tate weights 0 and 1 an intrinsic Berkovich open subspace of Fwa through
[
which the period morphism factors and which we conjecture to be the image of the

1 period morphism. We present indications supporting our conjecture and we show that
v only in exceptional cases our open subspace equals all of Fwa.
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Mathematics Subject Classification (2000): 11S20, (14G22, 14L05, 14M15)
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5
0
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Introduction
0
/
h
t RapoportandZinkfixanF-isocrystal(D,ϕD)overFpalg andconsiderthepartialflagvariety
a
m F˘ over K0 := W(Fpalg)[1p] parametrizing filtrations of D with fixed Hodge-Tate weights.
: They show that the set of weakly admissible filtrations is a rigid analytic subspace (F˘wa)rig
v b
i of F˘, which is called the period space; see [28, Proposition 1.36]. They conjecture the
X
existence of an´etale morphism(F˘a)rig → (F˘wa)rig of rigid analytic spaces which is bijective
r b b
a on rigid analytic points, and of a local system of Q -vector spaces on (F˘a)rig such that at
p b
every point µ of (F˘a)rig the fiber of the local system is the crystalline Galois representation
b
corresponding by the Colmez-Fontaine Theorem [12] to the filtration on (D,ϕ ) given by
D
the image of µ in (F˘wa)rig. We recall the precise formulation of the conjecture and the
b
definition of (F˘wa)rig in Section 1. De Jong [22] pointed out that to study local systems it
b
is best to work in the category of Berkovich’s K -analytic spaces rather than rigid analytic
0
spaces. For convenience of thereader wereview Berkovich’s definition andtherelation with
rigid analytic spaces in Appendix A. We show that in fact (F˘wa)rig is the rigid analytic
b
space associated with an open K -analytic subspace F˘wa of F˘ (Proposition 1.3).
0 b
If the Hodge-Tate weights all are 0 or 1 we present in this article an open K -analytic
0
subspace F˘a of F˘wa whose associated rigid analytic space (F˘a)rig is a candidate for the
b b b
∗The author acknowledges support of the Deutsche Forschungsgemeinschaft in form of DFG-grant
HA3006/2-1
1
1 THE CONJECTURE OF RAPOPORT AND ZINK 2
space searched for in Rapoport and Zink’s conjecture. The construction of F˘a is as follows.
b
With any analytic point µ of F˘ (these are the ones of which K -analytic spaces consist;
0
see Definition A.1) we associate in Section 4 a ϕ-module M over B† and we let F˘a
µ rig b
†
be the set of those µ for which M is isoclinic of slope zero. The ring B is defined in
µ rig
e
Section 2 and the notion of ϕ-modules is recalled in Section 3. Our construction is inspired
by Berger’s [4] construction which associates with any rigid analytic pointeµ ∈ F˘ (the ones
whose residue field H(µ) is finite over K ) a (ϕ,Γ)-module over the Robba ring. Due to
0
the restriction to rigid analytic points Berger’s approach works even without the above
assumption on the Hodge-Tate weights. However, the techniques we use in Section 5 where
we prove that F˘a is an open K -analytic subspace of F˘ require in an essential way to work
b 0
with K -analytic spaces, which are topological spaces in the classical sense, and not rigid
0
analytic spaces. Namely we show that the complement F˘ rF˘a is the image of a compact
b
set undera continuous map (Theorem 5.2). It is worth noting that this crudemap is in fact
only continuous and not a morphism of K -analytic or rigid analytic spaces. We also show
0
that only in rare cases the inclusion F˘a ⊂ F˘wa is an equality (Remark 5.5). The fact that
b b
this inclusion may be strict was noted in [28] and [22] as a peculiarity. We give a natural
explanation for this phenomenon.
For Hodge-Tate weights 0 and 1 Rapoport and Zink also study a period morphism from
the moduli space Man of p-divisible groups with additional structure to F˘wa. The problem
b
todeterminetheimageofthisperiodmorphismwasalreadymentionedbyGrothendieck[19].
We prove in Section 6 that the period morphism factors through our set F˘a. We conjec-
b
ture that F˘a is its image and, moreover, that (F˘a)rig and the rational Tate module of the
b b
universal p-divisible group over Man are the space and the local system searched for in
the conjecture of Rapoport and Zink. We present our motivation for this conjecture in
Section 7. The fact that the inclusion F˘a ⊂ F˘wa may be strict while it induces a bijec-
b b
tion on rigid analytic points by the Colmez-Fontaine Theorem [12] again demonstrates the
necessity to work with Berkovich’s K -analytic spaces instead of rigid analytic spaces.
0
To end this introduction let us mention that the ideas in this article are inspired by our
analogous theory in equal characteristic [21] where we were able to prove the analogue of
the Rapoport-Zink Conjecture and the surjectivity of the period morphism onto F˘a. In the
b
beginningwe had hoped to beable to extend our approach here also to Hodge-Tate weights
other than 0 and 1, butwe did not succeed. Not even the construction of the ϕ-moduleM
µ
†
over B , let alone the openness result could be established along this line. The reason lies
rig
in the fact that for analytic points µ ∈ F˘ whose residue field H(µ) is not finite over K the
0
ring Be† is not a H(µ)-algebra (like Fontaine’s field B is not a C -algebra). So Berger’s
rig dR p
construction could not be adapted.
e
1 The Conjecture of Rapoport and Zink
In this section we explain the construction of Rapoportand Zink’s p-adic periodspaces and
their conjecture mentioned above. Let us first recall the definition of filtered isocrystals.
We denote by Falg an algebraic closure of the finite field with p elements and by K :=
p 0
W(Falg)[1] the fraction field of the ring of Witt vectors over Falg. Let ϕ = W(Frob ) be
p p p p
the Frobenius lift on K .
0
Definition 1.1. An F-isocrystal over Falg is a finite dimensional K -vector space D
p 0
equipped with a ϕ-linear endomorphism ϕ . If K is a field extension of K and Fil•D
D 0 K
1 THE CONJECTURE OF RAPOPORT AND ZINK 3
is an exhaustive separated decreasing filtration of D := D ⊗ K by K-subspaces we
K K0
say that D = (D,ϕ ,Fil•D ) is a filtered isocrystal over K. The integers h for which
D K
Fil−hD 6= Fil−h+1D are called the Hodge-Tate weights of D. We let t (D) be the
K K N
p-adic valuation of detϕ (with respect to any basis of D) and we let
D
t (D) = i·dim gri (D ).
H K Fil• K
i∈Z
X
The filtered isocrystal D is called weakly admissible1 if
t (D) = t (D) and t (D′) ≤ t (D′) (1.1)
H N H N
for any subobject D′ = D′,ϕ | ,Fil•D′ of D, where D′ is any ϕ -stable K -subspace
D D′ K D 0
of D equipped with the induced filtration Fil•D′ on D′ := D′⊗ K.
(cid:0) (cid:1) K K K0
To construct period spaces let G be a reductive linear algebraic group over Q . Fix a
p
conjugacy class {µ} of cocharacters
µ :G → G
m
defined over subfields of C (the p-adic completion of an algebraic closure of K ). Let E be
p 0
thefield of definition of the conjugacy class. Itis a finiteextension of Q . Twococharacters
p
in this conjugacy class are called equivalent if they induce the same weight filtration on
the category Rep G of finite dimensional Q -rational representations of G. There is a
Qp p
projective variety F over E whose C -valued points are in bijection with the equivalence
p
classes of cocharacters (from the fixed conjugacy class). Namely let V in Rep G be any
Qp
faithful representation of G. With a cocharacter µ defined over a field K one associates the
filtration FiliV := V of V := V ⊗ K given by the weight spaces
µ K j≥i K,j K Qp
V L:= v ∈ V : µ(z)·v = zjv for all z ∈ G (K) .
K,j K m
This defines a closed embe(cid:8)dding of F into a partial flag variety of V (cid:9)
F ⊂−→ Flag(V)⊗ E, (1.2)
Qp
where the points of Flag(V) with value in a Q -algebra R are the filtrations Fi of V ⊗ R
p Qp
by R-submodules which are direct summands such that rk gri is the multiplicity of the
R F•
weight i of the conjugacy class {µ} on V.
Now let b ∈ G(K ). For any V in Rep G one obtains an F-isocrystal
0 Qp
(V ⊗ K , b·ϕ).
Qp 0
A pair (µ,b) with a cocharacter µ : G → G defined over K ⊃ K and an element
m 0
b ∈ G(K ) is called weakly admissible if for some faithful representation V in Rep G the
0 Qp
filtered isocrystal
I(V) := (V ⊗ K , b·ϕ, Fil•V )
Qp 0 µ K
is weakly admissible. In fact this then holds for any V in Rep G; see [28, 1.18]. If (µ,b) is
Qp
weakly admissible and K/K is finite then the filtered isocrystal I(V) is admissible, that is
0
1Thisusedtobetheterminology untilColmez–Fontaine [12]showed thatforK/K finiteweakly admis-
0
sible implies admissible. Since we consider also infinite extensions K/K for which the Colmez–Fontaine
0
Theorem fails we stick to the old terminology.
1 THE CONJECTURE OF RAPOPORT AND ZINK 4
it arises from a crystalline Galois-representation ρ : Gal(Kalg/K) → GL(U) via Fontaine’s
functor
D (U) := (U ⊗ B )Gal(Kalg/K)
cris Qp cris
which by the Colmez-Fontaine theorem [12] is an equivalence of categories from crystalline
representations of Gal(Kalg/K) to weakly admissible filtered isocrystals over K. The as-
signment
Rep G −→ (Q vector spaces), V 7→ U (1.3)
Qp p
defines a fiber functor on Rep G.
Qp
Let E˘ = EK be the completion of the maximal unramified extension of E. In what
0
follows we consider cocharacters µ defined over finite extensions K of E˘. Let F˘rig be the
rigid analytic space over E˘ associated with the variety F˘ = F ⊗ E˘. Rapoport and Zink
E
define the p-adic period space associated with (G,b,{µ}) as
(F˘wa)rig := µ ∈ F˘rig : (µ,b) is weakly admissible .
b
They show that (F˘wa)rig is an(cid:8)admissible open rigid analytic subspa(cid:9)ce of F˘rig; see [28,
b
Proposition 1.36]. Namely they reduce to the case where b is decent with integer s, that is
for some positive integer s and some cocharacter sv :G → G defined over K
m 0
b·ϕ(b)·...·ϕs−1(b) = sv(p).
(In fact sv is the s-fold multiple of the slope cocharacter v; see [28, 1.7].) If b is decent with
integer s then (F˘wa)rig has a natural structure of rigid analytic subspace of (F ⊗ E )rig
b E s
over Es = EW(Fps)[1p] from which it arises by base change to E˘. Rapoport and Zink make
the
Conjecture 1.2. ([28, p. 29]) There exists an ´etale morphism (F˘a)rig → (F˘wa)rig of rigid
b b
analytic spaces over E˘ which is bijective on (rigid analytic) points and there exists a tensor
functor from Rep G to the category Q -Loc of local systems of Q -vector spaces on
Qp p (F˘a)rig p
b
(F˘a)rig with the following property:
b
For any point µ ∈ (F˘wa)rig(K) with K/E˘ finite, the fiber functor which associates
b
with a representation in Rep G the fiber at µ of the corresponding local system
Qp
is isomorphic to the fiber functor (1.3).
(1.4)
The notion of local systems of Q -vector spaces on rigid analytic spaces was studied by
p
deJong [22, 4] whopointed out that this is bestdone working with Berkovich’s E˘-analytic
spaces rather§than rigid analytic spaces. Solet F˘an betheE˘-analytic space associated with
the variety F˘. See Appendix A for the notion of E˘-analytic space and the relation with
rigid analytic spaces.
Proposition 1.3. There exists an open E˘-analytic subspace F˘wa of F˘an whose associated
b
rigid analytic space is the period space (F˘wa)rig.
b
Proof. From its construction in [28, Proposition 1.34] the period space (F˘wa)rig possesses
b
an admissible covering {Xrig} by admissible subsets Xrig ⊂ F˘rig such that F˘rigrXrig is
i i∈N i i
rig rig rig
a finite union SpB of affinoid subdomains and X ⊂ X . The X correspond to
j i,j i i+1 i
S
2 FONTAINE’S RINGS 5
open E˘-analytic subspaces X := F˘an r M(B ) of F˘an. Moreover, it follows from [28,
i j i,j
Proposition 1.34] that theclosureX of X inF˘an is afiniteunionofE˘-affinoid subdomains
i Si
and X¯ ⊂ X . Thus the union F˘wa := X is an open E˘-analytic subspace of F˘an
i i+1 b i∈N i
whose associated rigid analytic space equals (F˘wa)rig. (Use [6, Lemma 1.6.2] to see that
S
the Grothendieck topology induced from F˘wa on {µ ∈ F˘wa : H(µ)/E˘ is finite} coincides
b b
with the Grothendieck topology of (F˘wa)rig.)
b
After recalling some of Fontaine’s rings and some facts on ϕ-modules in the next two
sections we will construct in Sections 4 and 5 for Hodge-Tate weights 0 and 1 an open
E˘-analytic subspace F˘a of F˘wa such that (F˘a)rig is a candidate for the space searched for
b b b
in Conjecture 1.2.
Let us also mention the following fact for local systems on an E˘-analytic space X.
Proposition 1.4. ([22, Theorem 4.2].) For any geometric point x¯ of X there is a natural
Q -linear tensor functor
p
w : Q -Loc → Rep π´et(X,x¯)
x¯ p X Qp 1
(cid:0) (cid:1)
which assigns to a local system V the π´et(X,x¯)-representation V . It is an equivalence if X
1 x¯
is connected.
Here Rep π´et(X,x¯) is the category of continuous Q -linear representations of the
Qp 1 p
´etale fundamental group π´et(X,x¯) of X which was defined by de Jong [22].
(cid:0) (cid:1)1
2 Fontaine’s Rings
We recall from Colmez [11] some of the rings used in p-adic Hodge theory. Let O be a
K
complete valuation ring of rank one which is an extension of Z and let K be its fraction
p
field. Let v be the valuation on O which we assume to be normalized so that v (p) = 1.
p K p
In p-adic Hodge theory it is usually assumed that O is discretely valued with perfect
K
residue field. However, in Section 4 we need the more general situation introduced here.
Let C be the completion of an algebraic closure Kalg of K and let O be the valuation
C
ring of C. For n ∈ N let ε(n) be a primitive pn-th root of unity chosen in such a way that
(ε(n+1))p = ε(n) for all n. We define
E+ := E+(C) := x = (x(n)) : x(n) ∈ O , (x(n+1))p = x(n) .
n∈N0 C
(cid:8) (cid:9)
Feix the eelements ε := (1,ε(1),ε(2),...) and π := ε − 1 of E+. With the multiplication
xy := x(n)y(n) , the addition x+y := lim (x(m+n) +y(m+n))pm , and the
n∈N0 m→∞ n∈N0
valuati(cid:0)on vE(x)(cid:1):= vp(x(0)), E+(C) becomes(cid:0)a complete vaeluation ring of(cid:1)rank one with
algebraically closed fraction field, called E := E(C), of characteristic p. Next we define
e
A+ := A+(C) := W E+(C) and e e
A := A(C) := W(cid:0)E(C) (cid:1)the rings of Witt vectors,
e e e
B+ := B+(C) := A+(cid:0)(C)[1(cid:1)] and
e e e p
B := B(C) := A(C)[1] the fraction field of A(C).
e e e p
e e e e
2 FONTAINE’S RINGS 6
By Witt vector functoriality there is the Frobenius lift ϕ := W(Frob ) on the later
p
rings. For x ∈ E(C) we let [x] ∈ A(C) denote the Teichmu¨ller lift. Set π := [ε]−1. If
x= ∞i≫−∞pi[xi]∈ A(C) then we set wk(x) := min{vE(xi) : i≤ k}. For r > 0 let
e e
A(0,rP] := A(0,r](C) := x ∈A(C) : lim w (x)+ k = +∞ ,
e k r
k→+∞
Be(0,r] := Be(0,r](C) := A(cid:8) (0,r](eC)[1], (cid:9)
p
B† := B†(C) := B(0,r](C).
e e e
0<r
[
e e e
One has A+(C) ⊂ A(0,r](C) for all r > 0. On B(0,r](C) there is a valuation defined for
x= ∞ pi[x ] as
i≫−∞ i
e e e
v(0,r]P(x) := min{wk(x)+ kr : k ∈Z} = min{vE(xi)+ ri : i∈ Z}.
NowonedefinesB]0,r](C)astheFr´echetcompletionofB(0,r](C)withrespecttothefamilyof
semi-valuations v[s,r](x) := min{v(0,s](x),v(0,r](x)} for 0 < s ≤ r. This means in concrete
terms that a seqeuence of elements x ∈ B(0,r](C) ceonverges in B]0,r](C) if and only if
n
lim v[s,r](x −x ) = +∞ for all 0 < s ≤ r. Also if r ≥ s we let B[s,r](C) be the
n→∞ n+1 n
completion of B(0,r](C) with respect to v[s,re]. We view B]0,r](C) asea subring of BI(C) for
any closed subinterval I ⊂(0,r]. If I =]0,r] or I = [s,r] the functions e
e e ∞ e
f : B(0,r](C) −→ E(C) defined by x= pi[f (x)] (2.1)
i i
i≫−∞
X
extend by continuityeto BI(C) and foer any x ∈ BI(C) the sum ∞ pi[f (x)] converges
i=−∞ i
to x in BI(C); see [24, 2.5.1]. Let
P
e e
B† := B† (C) := B]0,r](C) .
rig e rig
r>0
[
e e e
The morphism ϕ extends to bicontinuous isomorphisms of rings B]0,pr](C) → B]0,r](C)
and B[ps,pr](C)→ B[s,r](C) defining an automorphism of B† (C). The rings B]0,r](C) and
rig
† e †,(p−1)/pre
B (C) were studied in detail by Berger [3, 2] who called the first B instead.
rig e e § e rig e
† alg
Note that the ring B (C) is denoted Γ by Kedlaya [24].
rig an,con
e e
There is a homomorphism θ : B(0,1](C) → C sending ∞ pi[x ] to ∞ pix(0)
i≫−∞ i i≫−∞ i
e
which extends by continuity to B]0,1](C). The series
P P
e
t := log[ε] = ∞ (−1)n−1πn ∈ B]0,1](C)
n=1 n e
generates kerθ. LPet
e
B+ := B+ (C) be the p-adic completion of B+(C) wn : w ∈ kerθ,n ∈ N and
cris cris n!
Bcris := Bcris(C) := B+cris(C)[1t]. e (cid:2) (cid:3)
The morphism ϕ extends to endomorphisms of B+ (C) and B (C). Let
cris cris
B+ := B+ (C) := ϕnB+ (C).
rig rig cris
n\∈N0
e e
One easily sees that B+ (C)⊂ B]0,r](C) for any r > 0. More precisely, B]0,r](C) equals the
rig
p-adic completion of B+ (C)[ p ] and is hence a flat B+ (C)-algebra.
rig [π¯] rig
e e e
e e
3 ϕ-MODULES 7
3 ϕ-Modules
Let K andC beas in theprevious section. To shorten notation wewill drop thedenotation
(C)fromtheringsintroducedthere. WerecallsomedefinitionsandfactsfromKedlaya [24].
Let a be a positive integer.
Definition 3.1. A ϕa-module over B† is a finite free B† -moduleM with a ϕa-semilinear
rig rig
†
map ϕM : M → M such that the image of ϕM generates M as B -module. The rank
rig
e e
of M as a B† -module is denoted rkM. A morphism of ϕa-modules is a morphism of the
rig
e
†
underlying B -modules which commutes with the ϕM’s. We denote the set of morphism
rig
between tweo ϕa-modules M and M′ by Homϕa(M,M′).
e
Thecategoryofϕa-modulesoverB† possessestensorproductsandduals. Forapositive
rig
integer b there is a restriction of Frobenius functor [b] from ϕa-modules over B† to ϕab-
∗ rig
e
modules over B†rig sending (M,ϕM) to (M,ϕbM).
e
Example 3.2. Let c,d ∈ Z with d > 0 and (c,d) = 1. Define the ϕa-module M(c,d) over
e
B† as M(c,d) = d B† e equipped with
rig i=1 rig i
e ϕM(eL1) =ee2, ... , ϕM(ed−1) = ed, ϕM(ed) = pce1. (3.1)
Lemma 3.3. ([24, Lemmas 4.1.2 and 3.2.4]) The ϕa-modules M(c,d) over B† satisfy
rig
(a) M(c,d)∨ ∼= M(−c,d) and
e
(b) [d] M(c,d) ∼= M(c,1)⊕d.
∗
Definition 3.4. For a ϕa-module M over B† we define the set of ϕa-invariants as
rig
H0ϕa(M) := {xe∈ M: ϕM(x) = x}.
It is a vector space over H0ϕa M(0,1) = W(Fpa)[p−1]; use [24, Proposition 3.3.4]. We
have Homϕa(M,M′) = H0ϕa(M∨(cid:0)⊗M′) fo(cid:1)r ϕa-modules M and M′ over B†rig.
Proposition 3.5. ([24, 4.1.3 and 4.1.4]) The ϕa-modules M(c,d) overeB† satisfy
rig
(a) Homϕa M(c,d),M(c′,d′) 6= (0) if and only if c/d ≥ c′/d′. e
(b) H0 M(cid:0)(c,d) 6= (0) if(cid:1)and only if c/d ≤ 0.
ϕa
Proposit(cid:0)ion 3.6.(cid:1)If c> 0 then the ϕa-module M(−c,1) over B† satisfies
rig
c−1 e
H0ϕa M(−c,1) = pcν pjϕ−aν([xj]) : x0,...,xc−1 ∈ E,vE(xj) > 0 .
(cid:0) (cid:1) nXν∈Z Xj=0 o
e
Proof. One easily verifies that the series on the right converges (even in B]0,r] for all r > 0)
and is a ϕa-invariant of M(−c,1).
Conversely let x ∈ B† satisfy x = p−cϕa(x). There is an r > 0 wieth x ∈ B]0,r]. Let
rig
f :B]0,r] → E be the functions from (2.1) which satisfy x= ∞ pi[f (x)]. The relation
i i=−∞ i
e e
P
e e
4 CONSTRUCTING ϕ-MODULES FROM FILTERED ISOCRYSTALS 8
x= p−cϕa(x)yieldsf (x)pa = f (x). Soxisdeterminedbyx := f (x)forj = 0,...,c−1.
i i−c j j
Since x∈ B]0,r] we must have lim v(0,r] pi[f (x)] = ∞. For i= j −kc we compute
i→−∞ i
e v(0,r] pi[fi(x)] = vE fi(x(cid:0)) + ri =(cid:1) pakvE(xj)+ j−rkc
(cid:0) (cid:1) (cid:0) (cid:1)
and so we must have vE(xj) > 0 for all j.
For ϕa-modules over B† Kedlaya proved the following structure theorem.
rig
Theorem 3.7. ([24, Theeorem 4.5.7]) Any ϕa-module M over B† is isomorphic to a
rig
direct sum of ϕa-modules M(c ,d ) for uniquely determined pairs (c ,d ) up to permutation.
i i i i
It satisfies ∧dM ∼= M(c,1) where c = c and d= rkM= d .e
i i i i
Definition 3.8. If detM := ∧rkMMP∼= M(c,1) we define thPe degree and the weight of M
as degM := c and wtM:= degM.
rkM
Proposition 3.9. ([24, Lemma 3.4.9]) Every ϕa-submodule M′ ⊂ M(c,d)⊕n over B†
rig
satisfies wtM′ ≥ wtM(c,d)⊕n = c.
d
e
Let us end this section with a remark on radii of convergence. Let M]0,r] be a free
B]0,r]-module and for 0 < s ≤ r let M]0,s] := M]0,r] ⊗ B]0,s] be obtained by base
B]0,r],ι
change via the natural inclusion ι: B]0,r] ֒→ B]0,s]. Let further
e e e
ϕ]0,rp−a] : M]0,r]e⊗ eB]0,rp−a] −∼−→ M]0,rp−a]
M B]0,r],ϕa
be an isomorphism. We say that a ϕa-meodule (eM,ϕM) over B† is represented by the pair
rig
(M]0,r],ϕ]M0,rp−a]) if (M,ϕM) = (M]0,r]⊗B]0,r] B†rig,ϕ]M0,rp−a]⊗eid).
Proposition 3.10. If M is representedeby (Me]0,r],ϕ]0,rp−a]) then
M
H0ϕa(M) = x ∈M]0,r] : ϕ]M0,rp−a](x⊗ϕa 1) = x⊗ι1 .
(cid:8) (cid:9)
Proof. The inclusion “⊃” follows from the inclusion M]0,r] ⊂ M.
To prove the opposite inclusion “⊂” let x ∈ H0 (M). Since B† = B]0,s] there
ϕa rig s>0
existsa0 < s ≤rwithx∈ M]0,s]. ChooseaB]0,r]-basisofM]0,r] andwritexwithrespectto
S
this basis as a vector v = (v ,...,v )T ∈ B]0,s] ⊕n. Let A ∈ GL eB]0,rp−a] beethe matrix
1 n n
]0,rp−a] e
by which the isomorphism ϕM acts o(cid:0)n this(cid:1)basis. The equati(cid:0)on x = ϕM(cid:1) (x) translates
into A−1v = ϕa(v ),...,ϕa(v ) T. Thues if s ≤ rp−a we find ϕea(v ) ∈ B]0,s], whence
1 n i
v ∈ B]0,spa]. Continuing in this way we see that in fact v ∈ B]0,r], that is x∈ M]0,r].
i (cid:0) (cid:1) i
e
e e
4 Constructing ϕ-Modules from Filtered Isocrystals
InthesituationwheretheHodge-Tateweightsare0and1wewillassociatewithanyanalytic
pointµ ∈ F˘anaϕ-moduleM overB† (C)whereC isthecompletionofanalgebraicclosure
µ rig
of H(µ). In case H(µ)/E˘ finitethis construction parallels Berger’s construction [4, II]that
works for arbitrary Hodge-Tate weeights and even produces an “H(µ)-rational” ve§rsion of
M . We begin with the following
µ
4 CONSTRUCTING ϕ-MODULES FROM FILTERED ISOCRYSTALS 9
Proposition 4.1. Let N be a ϕ-module over B† represented by a free B]0,1]-module N]0,1]
rig
and an isomorphism ϕN]0,p−1] : N]0,1]⊗B]0,1],ϕB]e0,p−1] −∼−→ N]0,p−1]. Let WeM be a C-subspace
of WN := N]0,1] ⊗B]0,1],θ C. Then theree existes a uniquely determined ϕ-submodule M ⊂ N
over B† with tN ⊂ M which is represented by a B]0,1]-submodule tN]0,1] ⊂ M]0,1] ⊂ N]0,1]
rig e
such that
e M]0,1]⊗B]0,1],θ C e= WM
inside WN.
e
Proof. If I is a closed subinterval of (0,1] set NI := N]0,1] ⊗ BI. Also if I = [s,r]
B]0,1]
we let pI := [ps,pr]. The isomorphism ϕ]0,p−1] induces an isomorphism ϕI : NpI ⊗
N e e N BpI,ϕ
BI −∼−→ NI. Let In := [p−n−1,p−n] for n ∈ N0. We define M′I0 as the preimage of WM
e
in NI0 under the canonical morphism NI0 → Ni0 ⊗BI0,θ C = WN and we let MI0 be the
ientersection of M′I0 with ϕ[Np−1,p−1] M′I0 ⊗BI0,ϕ B[p−e1,p−1] inside N[p−1,p−1], that is MI0
equals
(cid:0) (cid:1)
e e
ker M′I0 ⊂−→ N[p−1,p−1] −→→ N[p−1,p−1]/ϕ[p−1,p−1] M′I0 ⊗ B[p−1,p−1] .
N BI0,ϕ
(cid:16) (cid:17)
(cid:0) (cid:1)
Since BI0 is a principal ideal domain by [24, Proposition 2.6.8] weesee theat MI0 is a free
BI0-submoduleofNI0 offullrank. Forn ≥ 1wedefineMIn astheimageofMI0⊗ BIn
BI0,ϕn
uender ethe isomorphism ϕINn ◦...◦ϕIN1 : NI0 ⊗BI0,ϕn BIn −∼−→NIn. In the termienologyeof
Kedlaya [24, 2.8] thecollection MIn for n∈ N definesa vector bundle over B]0,1] whichby
0
[24, Theorem§2.8.4] corresponds to a free B]0,1]-esubmoeduleM]0,1] of N]0,1]. By construction
tN]0,1] ⊂ M]0,1] and ϕ]0,p−1] restricts to an isomorphism on M]0,1]. Thisemakes M :=
N
M]0,1] ⊗ B† into a ϕ-module with tehe desired properties. Clearly M]0,1] is uniquely
B]0,1] rig
determined by the subspace WM ⊂WN and the requirements that tN]0,1] ⊂ M]0,1] ⊂ N]0,1]
and ϕINn eMIn−e1 ⊗BIn−1,ϕBIn = MIn.
Now(cid:0)assume theat K isea ((cid:1)not necessarily finite) extension of K0. Let (D,ϕD) be an F-
isocrystal over Falg and let Fil0D be a K-subspace of D = D⊗ K. Let Fil−1D =
p K K K0 K
D and Fil1D = (0) and D = (D,ϕ ,Fil•D ). We set D]0,1] := D ⊗ B]0,1] and
K K D K K0
†
(D,ϕD):= (D,ϕD)⊗K0 Brig.
e
Definition 4.2. WeletM(D)betheϕ-submoduleofDover B† representedbytheB]0,1]-
e rig
moduleM]0,1] withM]0,1]⊗ C = (Fil0D )⊗ C insideD = D]0,1]⊗ C whose
B]0,1],θ K K C B]0,1],θ
existence was established in Proposition 4.1. e e
e e
Unfortunately we do not know how to construct M(D) for filtered isocrystals D with
Hodge-Tate weights other than 0 and 1. Therefore we cannot make D 7→ M(D) into a
tensor functor and so we cannot use Berger’s argument [4, Th´eor`eme IV.2.1] to prove the
next theorem.
Theorem 4.3. degM(D) = t (D)−t (D).
N H
Proof. By the Dieudonn´e-Manin classification [26] there exists an F-isocrystal (D′,ϕ )
D′
over Fpalg of rank one with ϕD′ = ptN(D)·ϕ which is isomorphic to det(D,ϕD). So by
construction degD= t (D). Moreover,
N
dim (D]0,1]/M]0,1])⊗ C = dim (D /Fil0D )⊗ C = −t (D).
C B]0,1],θ C K K K H
e
4 CONSTRUCTING ϕ-MODULES FROM FILTERED ISOCRYSTALS 10
The theorem is thus a consequence of the following lemma.
Lemma 4.4. Let M and M be ϕ-modules over B† represented by B]0,1]-modules M]0,1].
1 2 rig i
]0,1] ]0,1] ]0,1]
Assume that M ⊃ M ⊃ tM . Then
1 2 1
e e
]0,1] ]0,1]
degM −degM = dim (M /M )⊗ C.
2 1 C 1 2 B]0,1],θ
Proof. Clearly the equality holds for M = tM ∼= M ⊗M(1,1)esince degtM −degM =
2 1 1 1 1
rkM . We claim that it suffices to prove the inequality
1
degM −degM ≥ dim W (4.1)
2 1 C
]0,1] ]0,1]
where we abbreviate W := (M /M )⊗ C. Indeed we apply the inequality to
1 2 B]0,1],θ
the two inclusions M ⊃ M ⊃ tM and M ⊃ tM ⊃ tM and conclude using the exact
1 2 1 2 1 2
sequence of B]0,1]-modules e
]0,1] ]0,1] ]0,1] ]0,1] ]0,1] ]0,1]
0 −e→ M /tM −→ M /tM −→ M /M −→ 0.
2 1 1 1 1 2
To prove the inequality (4.1) we argue by induction on dim W. Let dim W = 1.
C C
Since detM ⊃ detM we know that degM ≥ degM from Proposition 3.5. If we had
1 2 2 1
degM = degM thenM = M by[24,Lemma3.4.2]. SodegM −degM ≥ 1 = dim W
2 1 2 1 2 1 C
as desired.
Let now dim W > 1 and choose a C-subspace W′ of dimension 1 of W. By Propo-
C
sition 4.1 there is a unique B† -submodule M ⊂ M ⊂ M corresponding to W′. By
rig 2 3 1
induction degM −degM ≥ dim (W/W′) and degM −degM ≥ dim W′ proving the
3 1 C 2 3 C
lemma. e
†
If K/K is finite one can check that M(D) equals the ϕ-module over B constructed
0 rig
by Berger [4, II]. One of Berger’s main theorems is the following criterion.
§
e
Theorem 4.5. ([4, IV.2]) Let K be a finite extension of K . Then D is admissible if
0
and only if M(D)∼= M§ (0,1)⊕dimD.
Let us make explicit what happens otherwise.
Proposition 4.6. Assume that t (D) = t (D). Then M(D) 6∼= M(0,1)⊕dimD if and only
N H
if for some (any) integer e ≥ (dimD)−1 there exists a non-zero x∈ H0 [e] D⊗M(1,1)
ϕe ∗
with θ ϕm(x) ∈ (Fil0D )⊗ C for all m = 0,...,e−1.
D K K
(cid:0) (cid:1)
No(cid:0)te that(cid:1)D can be represented by (D⊗ B]0,r],b·ϕ) for arbitrarily large r > 0. Hence
K0
by Proposition 3.10 the element x actually belongs to D ⊗ B]0,r] and the expression
K0
θ ϕm(x) makes sense. e
D
e
P(cid:0)roof. L(cid:1)etM(D)6∼= M(0,1)⊕dimD andlete ≥ (dimD)−1beanyinteger. ByTheorems4.3
†
and 3.7 there is a ϕ-module M(c,d) over B with c < 0 and d < dimD which is a
rig
summand of M(D). Thus by Proposition 3.5 there exists a non-zero morphism of ϕ-
modules f : M(−1,e) → M(c,d) ⊂ M(D)e⊂ D. Let e ,...,e be a basis of M(−1,e)
1 e
satisfying (3.1) and let x be the image of e in D under f. Then ϕe (x) = p−1x, that is
1 D
x∈ H0 [e] D⊗M(1,1) . Moreover, f(e ) = ϕm(x) in D for 0 ≤ m < e. Now the fact
ϕe ∗ m+1 D
that the morphism f factors through M(D) amounts by Proposition 4.1 to θ ϕm(x) ∈
D
(cid:0) (cid:1)
(Fil0D )⊗ C.
K K
(cid:0) (cid:1)