P -ADIC MONODROMY OF THE ORDINARY LOCUS OF

PICARD MODULI SCHEME

Dong Uk Lee

A Dissertation

in

Mathematics

Presented to the Faculties of the University of Pennsylvania in PartialFulfillment of the Requirements for the Degree of Doctor of Philosophy

2005

Ching-Li ChaiSupervisor of Dissertation

David HarbaterGraduate Group Chairperson

Acknowledgments

I could not express my gratitude to my advisor Ching-Li Chai enough. Many years

of his guidence, help and encouragement made this work possible.

Also, I would like to thank Steve Shatz, Ted Chinburg, David Harbater and

Florian Pop for their help and encouragement as well as mathematical teaching.

My life as graduate student at Penn would not have been much fun were it

not be for the fellowship in and outside math of my fellow graduate students, es-

pecially, Sukhendu Mehrota, Jimmy Dillies, Laurentiu Maxim, Cherng-tiao Perng,

and Shuichiro Takeda.

Finally, I would like to thank my parents for their support, and Ahram Seo for

new delight in my life.

ii

ABSTRACT

P -ADIC MONODROMY OF THE ORDINARY LOCUS OF PICARD MODULI

SCHEME

Dong Uk Lee

Ching-Li Chai

Let E be an imaginary quadratic number field, p be a rational prime splitting

in OE and m,n be distinct natural numbers. The naive p-adic monodromy of the

ordinary locus of the good reduction of a Shimura variety of U(m, n) type over Fp

is a subgroup of GLm(Zp)×GLn(Zp). In this paper, we prove that for any point in

the basic locus of the moduli space, the local monodromy is an open subgroup of

GLm(Zp)×GLn(Zp). From this local information, the global p-adic monodromy is

shown to be as big as possible, i.e. GLm(Zp)×GLn(Zp).

iii

Contents

1 Introduction 1

2 Shimura variety of U(m,n) type and Picard moduli scheme 5

2.1 Picard moduli scheme . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Basic locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . 15

3 Proof of the main theorem and the corollary 17

3.1 Some group theories . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Deformations of Dieudonne modules . . . . . . . . . . . . . . . . . . 23

3.3 Proof of Proposition 3.1.3. . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Determinant of the monodromy . . . . . . . . . . . . . . . . . . . . 40

3.5 Proof of Corollary 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 50

iv

Chapter 1

Introduction

Let X be a connected PEL-type modular variety over an algebraically closed field of

characteristic p, and A be its universal abelian scheme. Assume that the open sub-

scheme Xor of X whose points correspond to ordinary abelian varieties is nonempty.

Let Aor be the pullback of A over Xor, the maximal etale quotient of its Barsotti-

Tate group Aor[p∞] defines a lisse sheaf of free Zp modules of rank g (g being the

dimension of the abelian variety) and thus provides a representation of π1(Xor).

ρ : π1(Xor) → GLg(Zp)

We are interested in the image of this representation, which we will call the naive

(global) p-adic monodromy.

In the Siegel case, this image is well-known to be as big as possible, i.e. GLg(Zp).

Faltings-Chai proved this as an application of the minimal compactification [12]. In

fact, they proved that the local monodromy at 0-dimensional cusp is already quite

1

big, i.e. SLg(Zp). Ekedahl also proved the same result, analyzing deformations [11].

In this paper, we adopt this latter strategy.

In general, for Shimura varieties of PEL type with nonempty ordinary locus,

the naive global p-adic monodromy is expected to have a simple group theoretic

description in terms of the group G of the Shimura datum. This conjectured de-

scription implies, among other things, that the naive p-adic monodromy is reductive

([6], conjecture (7.4)(ii)). The results of this paper confirm this expectation for a

certain class of (PEL-type) Shimura varieties, including the Picard modular variety

of U(m,n) type provided that the prime p splits in OE.

On the other hand, often a substantial part of global monodormy already appears

in local monodromy. For example, Igusa considered a supersingular point s of a

certain modular curve X over an algebraically closed field k of characteristic p > 0

and showed that the local monodromy at that point is already as big as possible, i.e.

Z×p . In the higher dimensional Siegel case, other than the cusp used by Faltings and

Chai mentioned above, a superspecial point still works equally well ([7], [11]). Recall

that superspecial abelian variety is an abelian variety which after an extension to

an algebraically closed field, becomes isomorphic to the product Eg of copies of a

supersingular elliptic curve E.

In this paper, we study local monodromies of certain PEL-type Shimura varieties

which do not contain supersingular points. Therefore, we need to consider other

points which would do the same job as supersingular points did for Siegel moduli

2

scheme as far as the local monodromy is concerned.

On the other hand, there have been some evidences that the local monodromy

of formal Lie group at a point with different generic and closed slopes is big, for

exmaple [5]. Note that in the Siegel case, the biggest slope change occurs from a

supersingular abelian variety to an ordinary abelian variety; in the Newton polygon

stratification of Siegel modular variety, the supersingular strautm is minimal and

ordinary stratum is maxiaml. Every other slope stratum is between these. In

this paper, we also consider the minimal stratum (the basic locus) in the Newton

polygon stratification of some classes of Shimura varieties and show that the local

monodromies at these points are quite big as far as some obvious global restriction

can allow. Along this line, the result of this paper also can be regarded as another

evidence for this phenomenon.

Here is a brief sketch of the structure of the paper. In the first section of the

chapter 2, we first give a set-up for the Shimura varieties that we consider. In the

next two sections we describe a particular point in the basic locus of Picard moduli

scheme that will serve for the computation of the local monodromy. In the last

section, we state the main theorem about the local monodromy and its corollary

about the global monodromy. In chapter 3, we prove the main Theorem about the

local monodromies at the points defined in Chapter 2 using deformation theoretic

arguments. This consists of two parts. Firstly, we show that the local monomdromy

contains the derived group of the target and secondly we prove the determinant is

3

an open subgroup of Z×p × Z×p . In the last section, we derive the corollary using

another point in the moduli space.

4

Chapter 2

Shimura variety of U(m, n) type

and Picard moduli scheme

Let G be a quasi-split group over Q of type U(m, n), split by an imaginary quadratic

field E. The Shimura variety X defined by the Shimura datum associated to (G, h)

parametrizes (m+n)-dimensional polarized abelian varieties A having an endomor-

phism by OE with level structures such that the action of OE on Lie(A) has type

(m,n). For a scheme S over the localization OE⊗Z(p) of OE and an abelian scheme

over S equipped with an endomorphism by OE, the Lie algebra Lie(A/S) has two

OE ⊗ Z(p)-module structures, one via the base scheme OE ⊗ Z(p) and the other via

the action by OE. But, since p splits in OE, Lie(A/S) becomes a direct sum of two

factors M1, M2, where the two actions of OE coincide on M1 while they differ by

conjugation on M2. After fixing a prime of OE over p which also distinguishes the

5

two OE ⊗ Z(p) factors, we require M1 (resp, M2) to have rank m(resp. n).

Also, we assume there is given a conjugacy class h : C → Mm+n(E)R satisfying

the axioms (2.1.1.1-3) of [9].

When p is unramified in OE, the Shimura variety defined by (G, h) and of

sufficiently small level has good reduction at p. In general, when G is quasisplit

over Qp and split over the unramified extension E℘ (℘ is a prime over p), G has

a hyperspecial subgroup Kp ⊂ G(Qp). Then for a sufficiently small subgroup Kp

of G(Apf ), the PEL-type Shimura variety ShKpKp(G, h) is known to have a good

reduction at p. Moreover, there exists a smooth quasiprojective model over OF ⊗Z

Z(p) which is also a fine moduli scheme for a suitable PEL-type moduli functor [24],

[18].

Also, when p splits, it is well known that the ordinary locus is nonempty (For

this, see e.g. [33]).

In this paper, we assume that p splits in OE and the conditions of good reduction

be satisfied.

Let Xor be the ordinary locus of the reduction X mod p and x ∈ X(k)(k = k)

be a geometric point.

Let S = Spf(R) be the equicharacteristic deformation space of (Ax, λx, ιx); S is

the formal completion X/x of X at x. Let A → Spec(R) be the universal abelian

scheme over Spec(R) which is also the universal deformation of (Ax, λx, ιx) by the

Grothendieck algebraization theorem. Then we consider the associated Barsotti-

6

Tate group A[p∞] over Spec(R). This can also be constructed as follows. Let G

be the universal formal deformation over the formal scheme Spf(R) of the smooth

formal group Ax. If R is R = lim←Ri as an adic ring, we have a compatible system

of smooth formal groups Gi over Spec(Ri), each of which is the universal formal

deformation of Ax[p∞] over Spec(Ri). Then for fixed n, limi→∞Gi[p

n] becomes a

truncated Barsotti-Tate group over Spec(R) of level n and the inductive system

of finite locally free group schemes thus obtained defines a p-divisible group G[p∞]

over Spec(R). This defines an equivalence of the category of smooth formal groups

over Spf(R) and the category of connected BT groups over Spec(R). In the rest of

paper, this equivalence will be used without explicit mention.

If we let A[p∞]etη be the maximal etale quotient of the generic fiber of A[p∞],

then we get the assoiciated Galois representation

ρG : Gal(Ksep/K) → GL(Tp(G)) = GL(A[p∞]etη ).

The local monodromy is the image of this representation.

Note that the splitting pOE ' ℘1 × ℘2 of p induces the splitting of p-divisible

group Ax[p∞] ' Ax[℘

∞1 ]×Ax[℘

∞2 ] and similar splittings for any lifting of (Ax[p

∞], λx, ιx).

Also, the quasi-polarization λ : A[p∞] 7→ A[p∞]t induced by a given polarization of

A maps A[℘∞1 ] to A[℘∞2 ]t.

7

2.1 Picard moduli scheme

We review the construction of the Picard moduli scheme and the basic locus of its

closed fiber over a field of finite characteristic. For more details, one can see [20] or

[2]

Let E be an imaginary quadratic number field with discriminant D, H the

Hilbert class field of E and O = OE, OH respectively their rings of integers.

For positive integers m < n, let V0 = Om+nE be the free OE-module of rank m + n

and q0 be the skew-Hermitian form on the OE-module V0 defined by the diagonal

matrix whose first m diagonal entries are 1’s and the next n diagonal entries are

−1’s.

Let GU(m, n) be the algebraic group over Q whose R-rational points for Q-

algebra R are

GU(m,n)(R) = {g ∈ GLO⊗R(V0 ⊗R) : there exists a µ(g) ∈ R×such that

q0(gu, gv) = µ(g)q0(u, v)}.

Definition 2.1.1. [2] [20] Let S be a scheme over O[1/D]. One callsM-structure of

type (m, n) over S a triple (A, λ, ι) where A is an abelian scheme over S, of relative

dimension m+n, λ a principal polarization of A, and ι a ring homomorphism from

O to EndS(A), such that

(1)The group scheme A over S is of type (m, n).

8

(2)The Rosatti involution associated with φ acts as the complex conjugation on

ι(O).

(3)If D is even, there exists an isomorphism of O2 = O ⊗Z Z2-modules equipped

with hermitian form (T2(A), q) ' (V0, q0)⊗Z Z2.

Definition 2.1.2. [2] [20] A level N -structure over a M-structure of type (m,n)

(A, λ, ι) is defined to be a couple (σ, τ), where

(1)σ : V0 ⊗O/N → A[N ] is an isomorphism of O/N -module schemes over S.

(2)τ : Z/NZ → µN is an isomorphism of group schemes over S.

(3)τ(q(x, y)) = q0(σ(x), σ(y)).

Let MN be the category of M-structures of type (m,n) and level N over

O[1/DN ]-schemes. It is a fibered category in groupoids over O[1/DN ].

The following result is proved in [20] when (m, n) = (1, 2), but it is easily seen to

be true in general.

Theorem 2.1.3. The stack MN is an algebraic stack. It is connected, smooth and

of relative dimension mn over Spec(O[1/DN ]). For N ≥ 3, MN is an algebraic

space.

If N divides N ′, over Spec(O[1/DN ′]) one has a finite and etale forgetting mor-

phism,

MN ′ →MN

which identifies MN with the quotient of MN ′ by the group Γ(N ′, N), where

Γ(N ′, N) = Ker[GU(m, n)(Z/N ′Z) → GU(m,n)(Z/NZ)].

9

It is known that for a rational prime l 6= 2, an M-structure (A, φ, ι) over a

scheme S above O[1/Dl], (Tl(A), q) ' (V0, q0)⊗Z Zl.

2.2 Basic locus

The Newton stratification of the Siegel moduli space Ag which is defined in terms of

the formal isogeny type of the Barsotti-Tate group was generalized to the reductions

of general Shimura varieties by Kottwitz and many similar properties as in the Siegel

case were shown to hold by Rapoport, Richartz and Chai [19] [30] [4].

Here we give a brief review of the theory. For detailed discussion, we refer to

[19] [30] [4]. With the connection to Shimura varieties in mind, the base field is

assumed to be Qp. For more general situation, consult loc. cit.

Let K be the fraction field of the ring of p-adic Witt vectors W (Fp) with the

Frobenius automorphism σ of K/Qp, K an algebraic closure of K and let Γ =

Gal(Qp/Qp).

Let G be a connected reductive group over Qp. Let B(G) be the σ-conjugacy

classes of elements of G(K) : x ∼ y ⇐⇒ x = g · y · σ(g)−1 for some g ∈ G(K) and

let us define the Newton cone N (G) by N (G) = (Int G(K) \ HomK(D, G))<σ> ∼=

(W \X∗(T )Q)Γ, where D is the pro-algebraic group with character group Q and W

is the Weyl group of G with respect to a fixed maximal torus T of G.

Then Kottwitz defined the Newton map νG : B(G) → N (G) satisfying certain

properties ([19], Sect.4). We will be just contented with describing this map in the

10

case G = GLh ([30], Ex. 1.10) because in our case we have GU(m, n) ×Q Qp =

GLm+n × Gm. When G = GLh, B(G) is the set of all isomorphism classes of σ-

K-spaces of height h (i.e. h-dimensional K-vector spaces together with σ-linear

bijection) and the Newton map sends a σ-K-space of height h to its usual Newton

polygon determined by the decomposition into isotypical components according to

the Dieudonne-Manin classification. Also, in this case G = GLh the natural partial

ordering on the Weyl chamber C (i.e. for x, y ∈ C, x � y ⇐⇒ x − y ∈ C∨)

becomes the usual ordering on Newton polygons with same end points.

Let B(G)basic be the set of σ-conjugacy classes of the elements b ∈ G(K) whose

associated homomorphisms νb ∈ HomK(D, G) factor through the center of G. There

exists a functorial isomorphism γ : B(·)basic → π1(·)Γ, where for a connected

reductive group G over Qp, π1(G) is the common value of the Galois modules

π1(G, T ) = X∗(T )/∑

α∈Φ(G,T ) Zα∨ for various pairs (T, B) consisting of a maximal

torus T and a Borel subgroup B defined over Qp.

Now, we explain the connection of this theory with good reduction of Shimura

varieties. Let (G,X) be a PEL Shimura data, in particular G is a connected

reductive group and X is a G(R)-conjugacy class of a R-group homomorphism

h : C× → GR satisfying the axioms (2.1.1.1-3) of [9]. Assume that G is quasisplit

over Qp and splits over an unramified extension of Qp. This is the most well known

candidate of groups for which the associated Shimura varieties are conjectured to

have good reduction over places above p. Let Kp be a hyperspecial maximal compact

11

subgroup of G(Qp) and let ShKp(G,X) the tower of Shimura varieties attached to

these data. Assume that ShKp(G,X) has good reduction at a place v of the Shimura

reflex field above p.

Then we apply the above discussion to the connected reductive group G =

G ×Q Qp over Qp. Let µ be a minuscule dominant coweight of a maximal torus T

over Qp with respect to a Qp-rational Borel subgroup B of G with B ⊇ T , such

that the G-conjugacy class of µ corresponds to X.

Definition 2.2.1. Let b0 ∈ B(G)basic∼= π(G)Γ be the basic element in B(G) corre-

sponding to the image of µ in π(G)Γ. Then Sbasic is defined to be the locus in the

reduction of ShKp(G,X) at v consisting of points whose associated σ-K-space has

type b0.

Remark 2.2.2. (i) It was shown by Rapoport and Richartz that Sbasic is Zariski-

closed ([30], Thm. 3.6). In general, if for b ∈ B(G), we define Sb to be the subset of

the reduction of ShKp(G,X) at v consisting of points whose associated σ-K-space

has type b, they showed that Sb is locally closed ([30], Thm. 3.6) and thus we obtain

a generalized Newton stratification. For more about this stratification, we refer to

the Bourbaki article by Rapoport [29] and the references therein.

(ii) Chai gave a conjectural group theoretic description of the set of all Newton

points that are expected to appear in the good reduction of Shimura varieties ([4],

Remark 4.5) and showed that this set is a catenary poset, in particular it has a

unique maximal (minimal) element. Sbasic is then the minimal stratum which is

12

expected to appear in the good reduction of Shimura variety. There is also a purely

group theoretic description of the maximal stratum which is expected to appear in

the same situation. In the Siegel case the unique maximal (resp. minimal) stratum

corresponds to ordinary (resp. supersingular) abelian varieties.

(iii) Chai also gave a Lie-theoretic formula for the codimensions of generalized New-

ton strata (loc. cit., Question 7.6). These were verified in the Siegel case [22], [10].

For the following proposition, we use the setup and notations of Chapter 2. In

particular, MN is the moduli scheme over O[1/DN ] of M-structures of type (m,n)

and level N (for big enough N). For p such that p - DN , let MN be the reduction

of MN at a place v of O over p.

Proposition 2.2.3. When p splits in OE, the reduction MN of the Picard moduli

scheme of type U(m,n)(m < n) has nonempty basic locus.

Note that in our reduction case of Picard moduli scheme the reductive group

G = GU(m, n) ×Q Qp = GLm+n × Gm over Qp already splits over Qp because of

the assumption p being split in OE and we can take µ to be the unique minuscule

dominant cocharacter coming from the Shimura data because the reflex field is E

and E℘∼= Qp.

Let S be the maximal torus of GL2(m+n),Qp consisting of diagonal matrices and

let B be the Borel subgroup of GL2(m+n),Qp consisting of upper triangular matrices.

The splitting E ⊗Q Qp∼= Qp ⊕ Qp gives rise to an inclusion GU(m,n) ×Q Qp ↪→

13

GL2(m+n),Qp such that A B

C D

∈ GU(m, n)×Q Qp 7→ A ∈ GLm+n,Qp , A,B,C, D ∈ M(m+n)(Qp)

induces isomorphism from SU(m,n)×Q Qp to GLm+n,Qp . Also under this inclusion,

the subtorus

T = {diag(d1, · · · , d2(m+n)) | di d2(m+n)+1−i = const for all i = 1, · · · , m + n }

of S becomes a maximal torus and B ∩ GU(m, n) ×Q Qp is a Borel subgroup of

GU(m, n)×Q Qp containing T . Therefore we have

X∗(T ) = {2(m+n)∑

i=1

xiei | xi + x2(m+n)+1−i = const },

where {ei : i = 1, · · · , 2(m + n)} is the standard basis of X∗(S). The minuscule

dominant coweight µ of G = GU(m, n) ×Q Qp corresponding to the type U(m,n)

is µ = e1 + · · · + em + em+n+1 + · · · + em+2n and its corresponding basic element

b0 ∈ B(G)basic has the usual Newton slopes ( mm+n

, nm+n

) with same height ([30],

Prop.1.12). Therefore it suffices to prove the following claim.

There exists a principally polarized AV over Fp with an OE action of type

U(m, n) which has the slopes ( mm+n

, nm+n

) with the same height m + n.

Let pOE = ℘ ℘ be the splitting of pOE into two distinct prime ideals in OE.

Then for the class number h of E, there exist elements α, β of OE such that ℘h =

(α), ℘h = (β) and by multiplying a unit to β, we may assume that ph = α · β, from

which it easily follows that β = α, since (β) = (α). Let π = αmβn and q = ph(m+n).

14

Then π is a Weil q-number, i.e. ππ = q. Moreover, Q(π) = E , because otherwise

βn−m = p−hmπ ∈ Z = Q ∩ OE and so should be a power of p, contradicting

the assumption that p is unramified. Hence by Honda-Tate theory, there exists a

simple abelian variety A over Fq such that EndFq(A) is a central division algebra

over Q(π) = E. Moreover, by construction, we have v(π) = m = v(qm

m+n ) for the

valuation v of E such that v(℘) = 1, v(℘) = 0.

By changing in the isogeny class we may assume that EndFq(A) contains OE.

Also, there exists a OE-linear polarization λ : A → At, i.e. via ι : OE → EndFq(A),

the Rosati involution associated with the polarization λ induces the complex con-

jugation on OE ([18], Lemma 9.2). So by another isogeny, we obtain a principal

polarization λ : A× Fp → At × Fp with prescribed properties ([25], 23). �

Remark 2.2.4. The aforementioned conjecture ([4] Question 7.6) of Chai about the

codimensions of generalized Newton stratra was also verified in our Picard case (p

being split) by Oort ([29], Thm 5.3).

2.3 Statement of the main theorem

We continue to use the setup and notations of Chapter 2 for the Picard moduli

scheme. Moreover, we assume that p splits over OE.

Theorem 2.3.1. Let x = (Ax, λx, ιx) be a point in the basic locus of MN and let

15

us assume that p ≥ 5. Then the image of the local monodromy

ρG : Gal(Ksep/K) → GLm+n(Zp)

contains SLm(Zp)× SLn(Zp) and its determinant is an open subgroup of Z×p × Z×p .

In particular, the local monodromy is an open subgroup of GLm(Zp)×GLn(Zp).

Note that the Lie condition forces the universal deformation of x = (Ax, λx, ιx) to

decompose into the product of two formal groups of respective dimension m,n lifting

the product A[℘∞1 ]×A[℘∞1 ], hence the image of the local monodromy naturally lands

in GLm(Zp)×GLn(Zp).

Corollary 2.3.2. The naive global p-adic monodromy group of any connected com-

ponent of MN is GLm(Zp) × GLn(Zp). In particular, the Zariski closure of the

global p-adic monodromy group is a connected reductive algebraic group.

16

Chapter 3

Proof of the main theorem and

the corollary

We first note that it suffices to prove the main theorem 2.3.1 for any one point in the

basic locus. Indeed, since any two Barsotti-Tate groups with given extra structure

in the basic locus are isogenous, the local monodromy at a point in the basic locus

is an open subgroup of the target if and only if it is thus at any other point.

In the following, we describe the points in the basic locus that will serve for the

direct computation of local monodrmoy, in terms of its p-divisible group (equiva-

lently its (covariant) Dieudonne module) (Lemma 3.0.3).

First, we collect general facts about the Dieudonne module of a general polarized

abelian variety (A, λ, ι) equipped with endomorphism byOE whose p-divisible group

A[p∞] is connected, assuming only that p is split in OE.

17

Let σi : OE ↪→ Zp ⊂ W (k) (i = 1, 2) be the embeddings of OE. If M is

the covariant Dieudonne module of A[p∞] = A, then we have the decomposition

M = M1 ⊕M2 as the direct sum of OE-eigenspaces M1, M2, where OE acts on Mi

via σi i.e, ι(α)(mi) = σi(α)mi, mi ∈ Mi i = 1, 2: Mi is the Dieudonne module of

A[℘∞i ]. Also, Mi’s are isotropic spaces for the pairing < ·, · > defined by the given

polarization λ, i.e. < Mi, Mi >= 0 for i = 1, 2. Indeed, if α ∈ OE, x, y ∈ M1,

< σ1(α)x, y >=< ι(α)x, y >=< x, ι(α)y >=< x, σ2(α)y >, implying < x, y >= 0.

So the quasi-polarization λ gives the quasi-isogenies λ : M1 ' M t2 & M2 ' M t

1.

Lemma 3.0.3. There exists a principally polarized abelian variety x = (Ax, λx, ιx)

over Fp with an action of OE of type (m,n) with the following associated covariant

Dieudonne module (M, λx, ιx);

(1)M = M(Ax[p∞]) = M1 ⊕ M2, where M1 = W (k)[F, V ]/W (k)[F, V ](Fm −

V n), M2 = W (k)[F, V ]/W (k)[F, V ](F n − V m) with OE acting on Mi via σi i.e,

ι(α)(mi) = σi(α)mi, mi ∈ Mi (i = 1, 2). In other words, Ax[℘∞1 ] = Gm,n and

Ax[℘∞1 ] = Gn,m as introduced in [23].

(2)The principal quasi-polarization induces the canonical isomorphism between Gn,m

and its Serre dual Gtm,n.

In view of the proposition 2.2.3, there exists a principally polarized abelian

variety (A, λ, ι) over Fp in the basic locus whose covariant Dieudonne module N

satisfies

(i) N ⊗B(Fp) is generated over B(Fp)[F, V ] by two elements v1, v2 such that (Fm−

18

V n)v1 = (F n − V m)v2 = 0 (B(Fp) = FracW (Fp)),

(ii)OE acting on vi via σi i.e., ι(α)(vi) = σi(α)vi, (i = 1, 2)

(iii)The principal quasi-polarization induces a perfect paring between the Diedonne

submodules generated by v1 and v2.

Then there exist positive integers di such that pdivi ∈ N . So there exists an

injective homomorphism of (covariant) Dieudonne modules M → N : ei 7→ pdivi,

where ei (i = 1, 2) is a generator of Mi.

Recall [3] that for a p-divisible smooth formal group G over a perfect field k

of characteristic p, denoting the contravariant(resp. covariant) Dieudonne module

of G by M∗(G)(resp. M(G)), there exists a funtorial σ-linear isomorphism of left

W (k)[F, V ]-modules

M∗(Gt) = (M∗(G))t := HomW (M∗(G), W )∼→ M(G).

Accordingly, we have an exact sequence of left W (Fp)[F, V ]-modules

0 → M ′ → N ′ → N ′/M ′ → 0,

where M ′(resp. N ′) is the left W (Fp)[F, V ]-module such that M ′ ⊗(W,σ) W ∼= M

(resp. N ′⊗(W,σ)W ∼= N), and there exists a finite flat group scheme H over Fp whose

dual H t has the contravariant Dieudonne module M∗(H t) = N ′/M ′. Hence, if B

is the abelian variety whose dual Bt is the quotient At/H t, we have M∗(Bt) = M ′

19

and M(B) = M .

0 → H → B → A → 0,

0 → H t → At → Bt → 0,

0 → M∗(Bt) → M∗(At) → M∗(H t) → 0.

From the functoriality of the isomorphisms involved, it is clear that B has an action

by OE. Also, there exists a polarization λ : B → Bt which induces a principal

quasipolarization on B[p∞]. Then by a separable isogeny, one can find a principally

polarized abelian variety with an OE-action whose covariant Dieudonne module is

the given one. OE-linear �

We compute the local monodromy of the Picard moduli scheme at the point

described in Lemma 3.0.3 and show that the local monodromy at this point is an

open subgroup of GLm(Zp) × GLn(Zp). The proof of this fact (hence Thm 2.3.1)

consists of two parts. In the first part, we show that the image contains the derived

group SLm(Zp) × SLn(Zp) and then we prove that the determinant of the local

monodromy is an open subgroup of Z×p × Z×p .

Both of these parts will follow from some group theories once we have the in-

formation modulo suitable powers of p. For the first part, the computation modulo

p is enough, but for the second part, it is more subtle and it turns out that the

exact powers of p that we need depends more on the precise shape of the Dieudonne

module of the particular point that is used.

20

3.1 Some group theories

If one is only interested in global monodromy, somtimes the modulo p information

is enough to conclude the best one can hope for. For example, in [11], Ekedahl

used Igusa’ theorem alluded to in the introduction, to conclude that the global

monodromies of certain moduli spaces are as big as possible (In particular, his ar-

gument gives another proof of the same statement about the global naive p-adic

monodromy of Siegel moduli scheme). But we can still deduce quite strong conclu-

sion, once we know that the first layer (i.e. mod p) image of the local monodromy

is already quite big. In many cases, it is due to following fact.

Lemma 3.1.1. Let {ni} : 1 ≤ i ≤ r} be a set of natural numbers greater than 1. Let

p be a prime number bigger than 3 and let X be a closed subgroup of∏

i GLni(Zp).

If the image of X under reduction mod p contains∏

i SLni(Fp), then X contains∏

i SLni(Zp).

In the case of single SLn(Zp)(p ≥ 5), this is due to Serre ([31], IV-23 Lemma 3).

He proved that a closed subgroup X of SLn(Zp)(p ≥ 5) which maps onto SLn(Fp)

must be SLn(Zp). In a more geneal case as stated above as well, this is probably

well known. Though there is other argument which might work in more general

setting, we decided to present Serre’s original proof since it also proves our lemma

with no extra work.

Proof. We prove by induction on n that the image of X under reduction modulo

pn contains∏

i SLni(Z/pnZ). By our condition, this is true for n = 1. Assume it is

21

true for n, and let us prove it for n + 1. From the exact sequence

0 → Kn = Kerπn →∏

i

SLni(Z/pn+1Z) →

∏i

SLni(Z/pnZ) → 0,

it is enough to show that the image of X contains Kn = Kerπn, in other words,

for any s = (si)i ∈∏

i SLni(Fp) which is congruent to 1 mod pn, there exists

x ∈ X with x ≡ s mod pn+1. Write si = 1 + pnui; since det(si) = 1, one has

Tr(ui) ≡ 0 mod p. Then one can show that any such ui is congruent mod p to a

sum of matrices ui with u2i = 0. Hence, we may assume that u2

i = 0. By induction

hypothesis applied to (1 + pn−1ui)i ∈ Kn−1, there exists a single y ∈ X such that

y ≡ (1 + pn−1ui)i mod pn, i.e. y = (1 + pn−1ui)i + pnv, where v has coefficients in

Zp. If we put x = yp, then we have x ≡ (1+pnui)i mod pn+1 which can be checked

componentwise as done in the Serre’s original argument. �

For the second part, we need another lemma which is essentially due to R. Pink.

Lemma 3.1.2. Let Oλ be the ring of integers in a finite extension of Ql and let

{ni} : 1 ≤ i ≤ r} be a set of natural numbers. Let H be a closed subgroup of∏i GLni

(Oλ). Then there exists an integer ν, depending on K, with the following

property. for any closed subgroup H of K, H = K if (and only if) H and K have

the same image in∏

i GLni(Oλ/l

νOλ). In case ni = 1, one can take ν = 2.

The proof as presented in ([17], Key Lemma 8.18.3) is seen to carry over in this

case. Also in the special case ni = 1 one can verify the claim ν = 2 from the proof.

In view of Lemma 3.1.1, the following takes care of the first step to the proof of

the main theorem Thm. 2.3.1, namely the claim about the derived group.

22

Proposition 3.1.3. Let x = (Ax, λx, ιx) be as in Lemma 3.0.3 and let’s assume

that p ≥ 5. Then the image of the local monodromy under mod p reduction

ρG = π ◦ ρG : Gal(Ksep/K) → GLm(Zp)×GLn(Zp)π→ GLm(Fp)×GLn(Fp)

contains SLm(Fp)× SLn(Fp).

The main tool will be the Catier-Dieudonne theory. In the next section, we

review the theory briefly and give an explicit presentation of the Dieudonne module

of the universal deformation of (Ax, ιx) in terms of display as presented in [26], [34].

3.2 Deformations of Dieudonne modules

Our strategy for showing the assertion about local monodromy is computing the

local monodromies of the restrictions to various formal subschemes of the universal

deformation space of (Ax, ιx).

For the benefits of readers, we recall the main theorem of Cartier-Dieudonne

theory. For more details, we refer to [14].

Theorem 3.2.1. Let k be a commutative Z(p)-algebra with 1. Then there is a

canonical equivalence of categories, between the category of smooth commutative

formal groups over k and the category of V -flat V -reduced Cartp(k)-modules.

Let (A, ι) be the universal deformation over Spf(R) of the abelian variety (Ax, ιx)

that is described in Lemma 3.0.3 and let K be the field of fractions of R. Then

23

there is a natural pairing of Gal(Ksep/K)-modules

HomK(Gm, Aη)× Tp(Atη) → Zp,

where Gm is the formal multiplicative group and Aη (resp, Atη) is the formal group

(resp, the dual abelian variety) of the generic fiber Aη. Therefore, by the principal

polarization of A, the representation ρG : Gal(Ksep/K) → GLm+n(Zp) is dual to

the Galois representation attached to the cocharacter group of Aη,

X∗ = HomK(Gm, Aη)

By Cartier-Dieudonne theory (Theorem 3.2.1), this is isomorphic to

HomCartp(Ksep)(M0, M),

where M0 is the Dieudonne module of Gm, and M is the Dieudonne module of

A[p∞]◦η ' Aη. So it suffices to show Prop. 3.1.3 for this Galois representation.

Equicharacteristic deformations of a smooth formal group can be described in a

fairly simple way when its Dieudonne module is presented in Display.

Definition 3.2.2. ([26], [34]) A Cartp(R)-module M over arbitrary ring R of char-

acteristic p is displayed if M is given by generators ei, i = 1, · · · , g +h and relations

Fei =∑

aijej i = 1, · · · , g

ei = V (∑

aijej) i = g + 1, · · · , g + h

for an invertible matrix (aij) with entries in W (R).

24

We will call the matrix (aij) the display (matrix) of M when the basis is clear

in the context. Also note that if the display matrix of M is given in block forms

(aij) =

A B

C D

,

then the matrix giving the Frobenius F is

A pB

C pD

and the Hasse-Witt matrix

is A (mod p).

Theorem 3.2.3. ([26], [34]) Let G be a smooth formal group over k = k whose

Dieudonne module has a display matrix (aij). If R is an Artinian local ring of

characteristic p, residue field k, and maximal ideal m, then there is a one to one

correspondence between isomorphism classes of deformations of M over R and maps

d : V M/pM → m⊗k (M/V M),

ei 7→∑

dij ⊗ ej ,

where ei, ej are the images of ei, ej in V M/pM and M/V M respectively. Set dij =

(dij, 0, · · · ) ∈ W (R) for i > g, j ≤ g and dij = 0 otherwise. The Dieudonne module

corresdponding to the map d is defined by

Fei =∑

(aij(ej +∑

djkek)) i = 1, · · · , g

ei = V (∑

aij(ej +∑

djkek)) i = g + 1, · · · , g + h

Furthermore every Dieudonne module over R which restricts to M over k is iso-

morphic to exactly one Dieudonne module of these forms.

25

Now we return to our p-divisible group as defined in Lemma 3.0.3. If we set

ei =

F i−1 (1 ≤ i ≤ m)

V m+n−i+1 (m + 1 ≤ i ≤ m + n)

, fi =

F i−1 (1 ≤ i ≤ n)

V m+n−i+1 (n + 1 ≤ i ≤ n + m)

then {ei}1≤i≤m+n (resp, {fi}1≤i≤m+n) form a basis of M1(resp, M2). (We keep

using the same notations for the images of F and V in the quotients M1 =

W (k)[F, V ]/W (k)[F, V ](Fm − V n), M2 = W (k)[F, V ]/W (k)[F, V ](F n − V m) as

long as there is no fear of confusion).

The display matrices of the Dieudonne modules Mi (i = 1, 2) with respect to

these bases {ei}, {fi} are the same :

D1 = D2 =

1

1

. . .

1

,

where the lower left (m+n− 1)× (m+n− 1) matrix is the identity matrix and the

upper right entry is 1. Let us denote by M t2 the Dieudonne module of the Serre dual

A[℘∞2 ]t of A[℘∞2 ] with dual basis f ti such that < f t

i , fj >= δij ([26] p.502-504), then

one can easily check that the principal quasi-polarization λ : M1 → M t2 is given on

M by

λ(ei) =

f t

n+i (1 ≤ i ≤ m)

f ti−n (n + 1 ≤ i ≤ n + m)

.

Next, we construct the universal formal deformation space of (M, λ, ι). Let

(M, λ, ι) be the quai-polarized Dieudonne module of the point that was defined

26

in Lemma 3.0.3 equipped with OE-module structure ι and the quasipolarization

attached to λ. Let (M, λ, ι) be the universal local deformation of (M, λ, ι).

Then, one easily sees that M is the direct sum M = M1 ⊕ M2 such that

(1) Mi is a lifting of Mi;

(2) OE acts on Mi via σi;

(3) λ : M1 → M t2.

Proposition 3.2.4. Let M, M1, M2 be as above.

(1) M1 is given by generators {ei : 1 ≤ i ≤ m + n} satisfying

F (ei) =m+n∑j=1

ajiej (1 ≤ i ≤ m)

ei = V (m+n∑j=1

bjiej) (m + 1 ≤ i ≤ m + n), where

(aij) =

0 · · · 0 T11 T12 · · · T1n 1

T21 T22 · · · T2n 0

Im−1...

......

...

Tm1 Tm2 · · · Tmn 0

1 0 · · · 0 0

1...

. . ....

1 0

,

where Tij = (tij, 0, · · · )’s are Witt vectors in W (k[[tij]]).

27

(2) M2 is given by generators {fi : 1 ≤ i ≤ m + n} satisfying

F (fi) =m+n∑j=1

bjifj (1 ≤ i ≤ n)

fi = V (m+n∑j=1

bjifj) (n + 1 ≤ i ≤ m + n),

, where

(bij) =

U11 U12 · · · U1m 1

U21 U22 · · · U2m

In−1...

......

Un1 Un2 · · · Unm

1

1

. . .

1

,

Uij = (uij, 0, · · · ) are Witt vectors in W (k[[uij]]).

(3) The principal quasi-polarization λ lifts to a principal quasi-polarization of (M, λ, ι)

when Ti1 = −U1i (1 ≤ i ≤ m).

(1) One just needs to work out Theorem 3.2.3 (also see [26]). This theorem is

made more illuminating when one notices that the displayed matrix of the universal

deformation M1 of M1 with respect to the basis {ei : 1 ≤ i ≤ m + n} is given by

28

the product

Im T

In

1

1

. . .

1

. One can easily check that this

product gives the display matrix in (1) for the same T .

(2) This is completely analogous to (1).

(3) The theory of biextensions of formal groups by Mumford provides a recipe for

the display matrix of the dual of any Dieudonne module in terms of original display

([26] p.502-504, [27] p.420): if M is the Dieudonne module of a local-local p-divisible

group G with display matrix aij (with respect to a basis {ei}), then the Dieudonne

module M t of its dual Gt is generated by generators {fi : i = 1, · · · , n+h} satisfying

fi = V (∑

α′ijfj), i = 1, · · · , n

Ffi =∑

α′ijfj, i = n + 1, · · · , n + h,

with (α′ij)−1 = (aij)

t.

We apply this to find the dual display M t2 of M2. Observing that the display

matrix D2 of M2 is obtained from (bij) =

In U

Im

, by moving the first column

29

to the right end, one finds the inverse matrix D−12 of D2 is

1 −U21 · · · −U2m

. . ....

...

1 −Un1 · · · −Unm

1

...

1

1 −U11 · · · −U1m

,

which is similarly obtained from

In −U

Im

, by moving the first row to the

bottom. Therefore with respect to the dual basis f ti of fi (i.e. < f t

i , fj >= δij), one

has

f ti = V (

∑cijf

tj ) i = 1, · · · , n

Ff ti =

∑cijf

tj i = n + 1, · · · , n + m,

30

with (cij) =

1

0

In−1...

0

−U21 · · · −Un1 1 −U11

−U22 −Un2 1 −U12

......

. . ....

−U2m · · · −Unm 1 −U1m

,

If we change the basis

gi =

f t

n+i (1 ≤ i ≤ m)

f ti−n (n + 1 ≤ i ≤ n + m)

,

then we get a display matrix for

M t2 :

−U11 −U21 · · · −Un1 1

−U12 −U22 · · · −Un2

Im−1...

......

−U1m −U2m · · · −Unm

1

1

. . .

1

.

Note that the matrix appearing in the middle upper block is −U t. From this (3)

follows immediately. �

31

If we set the basis (X1, · · · , Xm+n, Y1, · · · , Ym+n) to be such that

Xi =

ei if 1 ≤ i ≤ m

fi−m if m + 1 ≤ i ≤ m + n

, Yi =

fi+n if 1 ≤ i ≤ m

ei if m + 1 ≤ i ≤ m + n

. ,

then the Dieudonne display of M with respect to this basis is given by

D =

A B

C D

,

where in particular

A =

T11

T21

Im−1...

Tm1

−T11

U21

In−1...

Un1

.

3.3 Proof of Proposition 3.1.3.

We prove Prop. 3.1.3. It suffices to prove that the image of

ρ : Gal(Ksep/K) → X∗/pX∗

contains both SLm(Fp)× 1 and 1× SLn(Fp).

32

Since the Dieudonne module M0 of Gm is generated as a left Cartp(Ksep)-module

by a single element e0 with relation Fe0 − e0 = 0, we have

X∗/pX∗ ∼={

φ ∈ HomF (M0/V M0, M/V M)}

= {e = φ(e0)|F (e) = F (φ(e0)) = φ(F (e0)) = e} .

X∗/pX∗ ∼= HomF (M0/V M0, M/V M)

=

{m+n∑i=1

ci Xi | (c1, · · · , cn+m)t = A (cp1, · · · , cp

n+m)t

}, where

A =

t11

t21

Im−1...

tm1

−t11

u21

In−1...

un1

.

Consider the closed formal subschemes

Y〉 = Spf(Ri) ↪→ Spf(R) = Spf(k[[tij, uij]])(i = 1, 2)

defined by

R1 = R/(ui1|2 ≤ i ≤ n), R2 = A/(ti1|2 ≤ i ≤ m).

33

The restrictions of our p-divisible group A[p∞] to Spec(Ri) are still ordinary as clear

from the Hasse-Witt matrix and so their maximal etale quotients A[p∞]et restrict

to the etale p-divisible groups of same ranks. Therefore to prove Prop. 3.1.3 for the

original representation ρ : Gal(Ksep/K) → X∗/pX∗, it suffices to consider the local

monodromies for these subschemes at the same point ρi : Gi → Spec(Ri) where Gi

is the restriction of A[p∞]et to Spec(Ri).

Proposition 3.3.1. Let Ri be as above and Li be their fraction fields. Let Yi =

HomL(Gm, Gi). Then the image of ρ1 : Gal(Lsep/L) → Y1/pY1 contains SLm(Fp)×

{1} and ρ2 : Gal(Lsep/L) → Y2/pY2 contains {1} × SLn(Fp).

We only consider the case i = 1. The proof of the other case is the same as that

of this case.

On the subscheme Spec(R1), the matrix equation in ci becomes the following two

sets of equations in c1, · · · , cm+n defined by

c1

...

cm

=

t11

t21

Im−1...

tm−11

cp1

...

cpm

,

cm+1 = −t11c

pn

m+1

cm+i = cpi−1

m (1 < i ≤ n)

The left matrix equation reduces to a nice single equation in c1 which is an example

34

of ”generic vectorial p-polynomial” [1]: all degrees of c1 occurring are powers of p.

c1 = cpmt11

c2 = cp1 + cp

mt21

...

cm = cpm−1 + cp

mtm−11.

Note that each of n variables ci(i ≥ 1) is hooked successively to cm and its previous

one ci−1. By taking the p-th power of the second equation and plugging that into cp2

in the first one, we get c2 = tp11cp2

m + t21cpm. Repeating this iteration, we get following

single equation in cm.

cm =m∑

i=1

tpi−1

m−i+1,1cpi

m

For simplicity, let us change the notation x = cm, ti = −ti+1,1(0 ≤ i ≤ m, tm = 1).

The given matrix equations are then

m∑i=1

tpi−1

m−ixpi

+ x = 0, y = t0ypn

.

The second equation is separable, so its roots generate a separable extension

L′ = L(y) = L(t−1/(pn−1)0 ) of L. Since our equation has no coefficient from the other

variables, we can assume that L is the fraction field of k[[t0, · · · , tm−1]]. Also note

that then

L′ = L(t−1/(pn−1)0 ) = Frac(k[[t0, · · · , tm−1]])(t

−1/(pn−1)0 )

= Frac(k[[t1/(pn−1)0 , t1, · · · , tm−1]])

35

Therefore we are reduced to proving the following statement.

Lemma 3.3.2. The equation

f(x) =m∑

i=1

tpi−1

m−ixpi

+ x = 0

has Galois group SLm(Fp) over L′ = Frac(k[[t1/(pn−1)0 , · · · , tm−1]]).

Let M be the splitting field of f(x) inside Lsep. We will first prove that for

general n, m ≥ 1, f(x) has Galois group GLm(Fp) over L = Frac(k[[t0, · · · , tm−1]])

(i.e. Gal(M/L) = GLm(Fp)) and then will show that [L′ ·M : M ] = pn−1p−1

, thereby

the fact [L′ ∩M : L] = p− 1. In view of the fact that Gal(L′ ·M/L′) is a subgroup

of SLm(Fp), this will prove the lemma since Gal(L′ · M/L′) = Gal(M/L′ ∩ M) is

then a subgroup of GLm(Fp) of index p− 1.

Since k((tm−i)) is purely inseparable over k((tpi−1

m−i)) and L′ is separable over L,

both of these questions can be answered equally legitimately for g(x) =∑m

i=0 tm−ixpi

=

0 (tm = 1)

We start with the first claim. This is also more or less well known, ([1], 3).

Indeed, if we let {s1, · · · , sm} is a Fp-basis of the solution space of this polynomial,

we have

t−10 g(x) =

∏(a1,··· ,am)∈Fn

p

(x− a1s1 − · · · − amsm)

and, since the coefficients are algebraically independent over k, we conclude that

the n elements s1, · · · , sm are also algebraically indenpendent over k. Therefore, ev-

ery Fp-linear automorphism of the Fp-vector space spanned by {s1, · · · , sm} defines

36

an automorphism of the splitting field. On the other hand, each such automor-

phism fixes the ground field Frac(k[[t0, · · · , tm−1]]) since each ti ∈ k((s0, · · · , sm))

is symmetric in s0, · · · , sm.

For the second assertion claiming [L′ ·M : M ] = pn−1p−1

, we need to prove that the

irreducible polynomial h(x) = Irred(t−1, k((s1, · · · , sm)), x) has degree pn−1p−1

. We

have the following identity

(t−1)pn−1 =∏

(a1,··· ,am)∈Fpm−(0)

(a1s1 + · · ·+ amsm)

= (∏

α∈F×pm

α) · (∏

[b1,··· ,bm]∈P(Fpm )

(b1s1 + · · ·+ bmsm))p−1

The first equality comes from the coefficients-roots relation of g(x) and in the sec-

ond product of the second line, the index runs over P(Fpm) and for each point

P ∈ P(Fpm), we choose a vector (b1, · · · , bm) ∈ F×pm − (0, · · · , 0) representing P

(i.e. P = [b1, · · · , bm] ∈ P(Fpm)). So we have (t−1)pn−1 − (p(s1, · · · , sm))p−1 = 0

for some polynomial p(s1, · · · , sm) in s1, · · · , sm(k = k. In particular, p(s) =

c ·∏

[b1,··· ,bm]∈P(Fpm )(b1s1 + · · · + bmsm) for some constant c ∈ k. Since we are in

an integral domain, we should have then (t−1)pn−1p−1 = βp(s1, · · · , sm) for a β ∈ F×p .

Let h(x) = xpn−1p−1 − βp(s1, · · · , sm). Since t−1 is a root of h(x), the lemma will be

proved if we show that h(x) is irreducible.

Since this polynomial is defined over k[[s1, · · · , sm]], it suffices to prove this over

a closed subscheme Spec(k[[s1, · · · , sm]]/(si − ri(s))) = Spec(k[[s]]) for a suitable

choice of ri(s) ∈ sk[[s]]. This can be done, for example, as follows. Let ri(s) = sei

37

such that {ei|1 ≤ i ≤ m} form a strictly increasing sequence of natural numbers

satisfying

gcd(pn − 1

p− 1, e1p

m−1 + e2pm−2 + · · ·+ em) = 1.

Then for this choice of ri(s), h(x) = xpn−1p−1 − βp(r1(s), · · · , rm(s)) is irreducible

over k[[s]]. Indeed, there are pm−1 points in P(Fpm) having nonzero first coordinate

a1 and for these points a1s1 + · · · + amsm = a1se1 + · · · + amsem has valuation

e1 for a valuation of which s is a uniformizer. Next, in the hyperplane a1 = 0 in

P(Fpm) there are pm−2 points whose second coordinates a2 are non zero and then the

corresponding linear factors a1s1+· · ·+amsm = a2se2+· · ·+amsem have valuation e2.

Continuing in an obvious way, we find that the constant term of p(x) has valuation

e1pm−1 + e2p

m−2 + · · · + em. Therefore, if ei’s satisfy gcd(pn−1p−1

, e1pm−1 + e2p

m−2 +

· · ·+em) = 1 (this can be always achieved because of em), then the Newton polygon

of p(x) over k[[s]] becomes a line segment so that xpn−1p−1 − βp(se1 , · · · , sem) becomes

irreducible over k[[s]]. This finishes the proof of Lemma. �

Clearly, the same argument will prove the corresponding assertion for the second

factor.

Finally, combining the facts about the original (for K, not just for Li) local mon-

odromy, we conclude that the image of local monodromy in GLm(Fp) × GLn(Fp)

contains SLm(Fp) × SLn(Fp). (Also, the projection of the image to each factor is

surjective as observed previouisly.)

Remark For our choice of the point Lemma 3.0.3, the local monodromy does

38

not become the entire GLm(Zp) × GLn(Zp). In fact, the projection of the local

monodromy to the first layer is not already the whole GLm(Fp) × GLn(Fp). More

precisely, we show that

ρ(Gal(Ksep/K)) ∩GLm(Fp)× {1} ⊆ SLm(Fp)× {1}.

Since our representation ρG : Gal(Ksep/K) → X∗/pX∗ is associated with the pull-

back of Lang torsor GLm+n → GLn+m : g 7→ g−1 · g(p) via

Spec(R) = Spec(k((ti1, u1j|1 ≤ i ≤ m, 2 ≤ j ≤ n)))φ→ GLn+m

{ti1, u1j} 7→ A ,

the given system of equations (c1, · · · , cn+m)t = A (cp1, · · · , cp

n+m)t has solutions

in Ksep(K = Frac(R)). These solutions form a Fp vector space of dimension

m + n and the representation is the tensor product of two representations of re-

spective dimensions m and n correponding to two block submatrices of A. Let

{ξi = (ξ1i, · · · , ξmi)|1 ≤ i ≤ m} be a basis for the solution space corresponding

to the upper left block A11 of A and let g =

ξ11 · · · ξ1m

......

ξm1 · · · ξmm

. Then we have

g = A11g(p). So det(g) = det(A11) det(g)p, i.e. det(g)p−1 = (−1)mt−1

11 . Similarly

we have det(h)p−1 = (−1)nt−111 for the lower right matrix. Here the columns of h

are solution vectors. Therefore if σ ∈ ρ(Gal(Ksep/K)) belongs to GLm(Fp) × {1},

namely acts trivially on h, then it will also act trivially on det(g), which implies

that σ is an element of SLm(Fp) × {1}. On the other hand, note that the same

39

argument also shows that the projections of ρ(Gal(Ksep/K)) to each factors are

surjective.

3.4 Determinant of the monodromy

In this section, we prove the rest of the proposition 2.3.1, in other words that the

image of local monodromy at the point in question is an open subgroup. More

precisely, we prove the following.

Proposition 3.4.1. Let x = (Ax, λx, ιx) be as in Lemma 3.0.3 and assume that

p ≥ 5. In particular, m < n. Then the image of the local monodromy under the

determinant map

det ◦ρG : Gal(Ksep/K) → GLm(Zp)×GLn(Zp)det→ Z×p × Z×p

contains (1 + pnZp)× (1 + pnZp).

The idea of proof of this lemma is to analyze the determinant of the local

monodromy, modulo high powers of p.

Recall that our representation

ρG : Gal(Ksep/K) → GLm(Zp)×GLn(Zp)

is given by the action of the Galois group Gal(K/K) on the free Zp-module MF−Id

of rank m + n which consists of the elements fixed by the Frobenius F . M is the

Dieudonne module of the universal deformation of the point and is a direct sum

40

MF−Id = MF−Id1

⊕MF−Id

2 of two Dieudonne submodules . Let

{vi =m+n∑j=1

ajiej| 1 ≤ i ≤ m, aji ∈ W (K)}

be a basis of the module MF−Id1 for the basis ej of M1 and let A = (aij) be the

(m + n)×m matrix whose (i, j) entry is aij.

Then, for an element τ of Gal(K/Kperf) and for the basis {ej, fj} of M , if

(g1, g2) = ρ(τ), we have

Aτ = A g1

where Aτ is the matrix obtained by applying τ to each entry of A. Hence, if we

define a (m×m) matrix A′ by truncating A up to m-th row, we have clearly

det A′τ = det A′ det ρ1(τ). (3.4.1)

Similar statements hold for MF−Id2 and in the following discussions we deal with

MF−Id1 and MF−Id

2 separately. First, we consider the determinant of the left factor

(of rank m). For this, we restrict the p-divisible group A[℘∞1 ] to the subscheme

Spec(R1) = Spec(k[[t11, · · · , tm1]]) of the universal deformation space and show that

the determinant of the local monodromy representation attached to the generic fiber

AK [℘∞1 ] contains 1 + pnZp. Here K is the field of fractions R1 and we continue to

use the same notations for the new Dieudonne module over K.

Then, each basis vector vj = (a1,j, · · · , am+n,j)tr(1 ≤ j ≤ m) satisfies vj = F (vj),

41

i.e.

a1,j

...

am,j

am+1,j

...

am+n,j

=

0 · · · 0 T11 0 · · · 0 p

T21 0 · · · 0 0

Im−1...

......

...

Tm1 0 · · · 0 0

1 0 · · · 0 0

p

. . ....

p 0

aσ1,j

...

aσm,j

aσm+1,j

...

aσm+n,j

.

Here, σ is the Frobenius on the Witt vecotrs in W (K). So for each j,

aij =

∑i

l=1 T σi−l

l1 aσi−l+1

m,j + pnaσn+i

m,j if 1 ≤ i ≤ m

pi−m−1aσi−m

m,j if m + 1 ≤ i ≤ m + n.

As before, this becomes a single equation f(am,j) = 0 in am,j, where

f(x) = x− Tm1xσ − T σ

m−11xσ2 − · · · − T σm−1

11 xσm − pnxσn+m

. (3.4.2)

Lemma 3.4.2. For each 1 ≤ i ≤ m, there exist elements bi and bi,n of W (K) such

that

(i) am,i = bi + pnbi,n;

(ii) bi satisfies g(bi) = 0, where

g(x) = f(x) + pnxσm+n

= x− Tm1xσ − T σ

m−11xσ2 − · · · − T σm−1

11 xσm

. (3.4.3)

Furthermore, bi is unique mod pn.

42

The uniqueness of bi mod pn is obvious.

For the existence, fixing i we inductively show that for each 1 ≤ l ≤ n, there exist

elements ˜bi,j (0 ≤ j ≤ l − 1) and bi,l of W (K) such that

(a)am,i = ˜bi,0 + p˜bi,1 + · · ·+ pl−1˜bi,l−1 + plbi,l;

(b)g(˜bi,j) = 0 1 ≤ j ≤ l − 1;

(c)g(bi,l) ≡ 0 mod p if l < n.

Then bi = ˜bi,0 + p˜bi,1 + · · · + pn−1˜bi,n−1 and the same bi,n will satisfy (i) and (ii) of

the lemma.

For the first term, we define bi,0 ∈ W (K)/pW (K) = K by bi,0 = am,i. Then

since g(bi,0) = g(am,i) = f(am,i) = 0 and both g(x) = 0 and g(x) = 0 have rank

m solution spaces, there exists a root ˜bi,0 of g(x) = 0 in W (K) such that the

image of ˜bi,0 in W (K)/pW (K) = K is bi,0. Also there exists an element bi,1 of

W (K) such that am,i = ˜bi,0 + pbi,1. On the other hands, since am,i is a root of

f(x), we have f(am,i) = f(˜bi,0 + pbi,1) = g(˜bi,0) + pg(bi,1) − pn(˜bσn+m

i,0 + pbσn+mi,1 ) =

pg(bi,1)−pn(˜bσn+m

i,0 +pbσn+mi,1 ) = 0. This implies that g(bi,1) ≡ 0 mod p and the first

step of the induction is done.

Next, assume that for 1 ≤ l ≤ n− 1 we found ˜bi,j( 1 ≤ j ≤ l − 1) satisfying (ii)

and bi,l ∈ W (K) such that am,i = ˜bi,0 + p˜bi,1 + · · ·+ pl−1˜bi,l−1 + plbi,l and g(bi,l) ≡ 0

mod p.

Since g(bi,l) = 0, as before, there exists a zero ˜bi,l of g(x) in W (K) such that

the image of ˜bi,l in W (K)/pW (K) = K is bi,l. Also there exists an element bi,1+1

43

of W (K) such that bi,l = ˜bi,l + pbi,1+1. Clearly, for this choice of bi,l+1, we have

am,i = ˜bi,0 + p˜bi,1 + · · · + pl˜bi,l + pl+1bi,l+1 and so to establish the (l + 1)-st step of

the inductive argument, it remains to show that g(bi,l) ≡ 0 mod p if l + 1 < n.

But f(am,i) = f(˜bi,0 + p˜bi,1 + · · ·+ ˜bi,l + pl+1bi,l+1) =∑l

r=0 prg(˜bi,r) + pl+1g(bi,l+1)−

pn(∑l+1

r=0 pr˜bσm+n

i,r + pl+1bσm+n

i,l+1 ) = pl+1g(bi,l+1) − pn(∑l+1

r=0 pr˜bσm+n

i,r + pl+1bσm+n

i,l+1 ) = 0.

Therefore, by reading modulo pl+2(note that l+1 < n), we find g(bi,l+1) ≡ 0 mod p.

Lemma 3.4.3. Let det ◦ρ1 : Gal(K/K) → Z×p be the determinant composed with

the first projection of the local monodromy representation. If Ki is an extension of

K satisfying ρ−11 (1 + piZp) = Gal(K/Ki) (i ≥ 1, K0 = K), then we have

Gal(Ki/Ki−1) =

F×p i = 1

1 2 ≤ i ≤ n

Fp i = n + 1

.

From Lemma 3.4.2 (i), det A′ is congruent, mod pn, to the determinant of the

matrix W whose i-th column Wi is

Wi = (T11bσi , T σ

11bσ2

i + T21bσi , · · · , T σm−1

11 bσm

i + · · ·+ Tm1bσi )tr.

Each column vector Wi(1 ≤ i ≤ m) satisfies the equation

Wi = F ′W σi , where F ′ =

0 · · · 0 T11

T21

Im−1...

Tm1

.

44

As for am,i, this is seen by solving X = F ′Xσ with X = (x1, · · · , xm)tr for xm = bi.

Hence, we have W = F ′W σ and det W (σ−1) = (det F ′)−1 = ((−1)m−1T11)−1. If

det W = (x0, x1, · · · ) ∈ W (K), then one has

((−1)m−1t11)−pj

xj = xpj (j ≥ 0).

Since bi(1 ≤ i ≤ m) are linearly independent over Fp and so det W 6= 0, x0 =

(−1)(m−1)/(p−1)t1/(p−1)11 for a (p − 1)-st root t

1/(p−1)11 of t11 and there exist elements

αj (j ≥ 0) of Fp such that xj = αj((−1)m−1p−1 t

1(p−1)

11 )−pj. Therefore Gal(K/K(t

1/(p−1)11 ))

acts trivially on det W and the Galois representation

ρ1 : Gal(K/K) → Z×p /(1 + pnZp)

τ 7−→ (det A′)τ/ det A′ (mod pn) = (det W )τ/ det W (mod pn)

has finite image of cardinality p − 1 with Kernel Gal(K/K(t1/(p−1)11 )). Namely,

Kn = K(t1/(p−1)11 ). But, det A′ ≡ det W ≡ x0 = α0((−1)

m−1p−1 t

1(p−1)

11 )−1 mod p and

so Gal(K/K) → Z×p /(1 + pZp) is surjective and K1 = K(t1/(p−1)11 ). This proves the

claim of Lemma up to i = n− 1.

To consider the case i = n, we analyze bi,n in Lemma 3.4.2 more closely.

The j-th column Aj of A′ is

Aj =

T11aσm,j + pnaσn+1

m,j

T σ11a

σ2

m,j + T21aσm,j + pnaσn+2

m,j

......

...

T σm−1

11 aσm

m,j + T σm−2

21 aσm−1

m,j + · · · + Tm1aσm,j + pnaσn+m

m,j

.

45

From am,j = bj + pnbj,n (Lemma 3.4.2 (i)), we can write det A′ = Cn + pnDn,

where Cn is the determinant of the matrix whose (i, j) entry cij is cij = gi(bj), with gi(x) =∑il=1 T σi−l

l1 xσi−l+1and

Dn ≡m∑

j=1

det

g1(b1) · · · g1(bj,n) + bσn+1

j · · · g1(bm)

......

...

gm(b1) · · · gm(bj,n) + bσn+m

j · · · gm(bm)

mod p

Then by elementary row operations, it is easily shown that

Cn = T 1+σ+···+σm−1

11 det

bσ1 · · · bσ

i · · · bσm

......

...

bσm

1 · · · bσm

i · · · bσm

m

.

Hence, for τ ∈ Gal(K/Kn),

(det A′)τ · (det A′)−1

= (Cn + pnDn)τ · (Cn + pnDn)−1

= Cn + pn(Dn)τ ) · (Cn(1 + pnC−1n Dn))−1 Cτ

n = Cn

≡ (Cn + pn(Dn)τ ) · C−1n (1− pnC−1

n Dn) mod pn+1

= (1 + pnC−1n (Dn)τ ) · (1− pnC−1

n Dn)

≡ 1 + pnC−1n ((Dn)τ −Dn) mod pn+1

Note that for τ ∈ Gal(K/Kn), we have Cτn = Cn, not just Cτ

n ≡ Cn mod pn

because Cn = T−(1+σ+···+σm−1)11 det W .

So under the canonical isomorphism (1 + pnZp)/(1 + pn+1Zp) ∼= Fp, the restriction

46

(still denoted by ρ1) of ρ1 to the subquotient Gal(Kn+1/Kn) becomes

ρ1 : Gal(Kn+1/Kn) → (1 + pnZp)/(1 + pn+1Zp) ∼= Fp

τ 7−→ C−1n ((Dn)τ − Dn)

On the other hand, from Lemma 3.4.2, we have

0 = f(am,j) = f(bj + pnbj,n)

= g(bj) + png(bj,n)− pn(bσm+n

j + pnbσm+n

j,n )

= png(bj,n)− pn(bσm+n

j + pnbσm+n

j,n )

Reading modulo pn, we obtain g(bj,n)− bjσm+n ≡ 0 mod p.

In other words, bj,n is a zero of polynomial

hj(x) = g(x)− apm+n

m,j = −apm+n

m,j + x− tm1xp − tpm−11x

p2 − · · · − tpm−1

11 xpm

.

with coefficients in K(am,j). This is of Artin-Schreier type; if αj is one root, then

every other root is αj +∑

clj am,l for some clj ∈ Fp. Also, for any choice of roots

{αj|1 ≤ j ≤ m} for each hj(x), {αj|1 ≤ j ≤ m} are algebraically independent

over k since {am,j|1 ≤ j ≤ m} are thus and hence for any (clj) ∈ Mm×m(Fp),

τ : αj 7→ αj +∑

clj am,l defines an element of Aut(K(α1, · · · , αm)) which fixes the

coefficients of hj(x)’s. Since Kn = · · · = K1 = K(t1/(p−1)11 ) is of degree p− 1 over K,

Gal(Kn(α1, · · · , αm)/Kn(apm+n

m,j ))

= Gal(K(α1, · · · , αm)/K(apm+n

m,j ))

∼= Mm×m(Fp) = glm,Fp.

47

If we let g = ρ(τ) ∈ Mn×n(Fp) such that τ(αj) = αj +∑

l gl,j am,l, we have

Dτn =

m∑j=1

det

g1(b1) · · · g1(bj,n) + bpn+1

j +∑

l glj g1(bl) · · · g1(bm)

......

...

gm(b1) · · · gm(bj,n) + bpn+m

j +∑

l glj gm(bl) · · · gm(bm)

= Dn + Tr(g)Cn

This shows that

Gal(Kn(α1, · · · , αm)/Kn+1(apm+n

m,i )) ∼= slm,Fp ,

Gal(Kn+1(apm+n

m,i )/Kn(apm+n

m,i )) = Fp, and

Gal(Kn+1/Kn) = Fp.

Now we consider the second factor and finish the proof of Prop 3.4.1. We observe

that by restriction ρ to the subscheme Y∈ = Spf(R2) ↪→ Spf(R) = Spf(k[[tij, uij]])

defined by R1 = R/(uij, tkl|2 ≤ j ≤ n, (k, l) 6= (1, 1)), we obtain completely analo-

gous results for the second factor of the original local monodromy representation.

If we let L be the fraction field of R2 and define Li similarly, K(T1/(p−1)11 ) ∩

L(T1/(p−1)11 ) = k(T

1/(p−1)11 ) and for the field F

F = Frac(k[[tij, urs]])(t−1/(p−1)11 )

= Frac(k[[t1/(p−1)11 , tij, urs]]),

it is easy to see that the two field compositums F · Kalg and F · Lalg are linearly

disjoint over F . Therefore, the determinant of ρ(Gal(F alg/F )) surject onto (1 +

48

pn−1Zp)× (1 + pn−1Zp)/(1 + pnZp)× (1 + pnZp) and by Lemma 3.1.2, we are done

with the proof of Lemma.

49

3.5 Proof of Corollary 2.3.2

We prove that the global p-adic monodromy is the entire GLm(Zp)×GLn(Zp).

Lemma 3.5.1. Let K be a field of characteristic p with an algebraic closure Kalg.

Let φ = (yo, y1, · · · ) be an invertible element of W (K) with components yi ∈ K.

Then, for a ∈ W (Kalg)× satisfying a = φ aσ, the Galois representation

ρ : Gal(Kalg/K) → Z×p

τ 7→ aτ · a−1

is surjective if and only if the following two conditions hold

(i) y0 /∈ Kp−1, where Kp−1 is the subset of K consisting of (p − 1)-th powers of

elements of K ;

(ii) Xp −X + y1y−p0 (equiv. Xp − yp−1

0 X + y1) is irreducible over K.

Note that this representation does not depend on the choice of a since any such

two differ by an element of Z×p . For proof, we resort again to the lemma 3.1.2 and

show that the Galois representation maps surjetively modulo p and p2 precisely

when the conditions (i) and (ii) hold.

We first claim that there exist elements a0, a1, b2 of W (Kalg) such that a =

a0 + pa1 + p2b2. Indeed, if a = (x0, x1, · · · ) with xi ∈ Kalg, let us define a0 =

(x0, 0, · · · ), a1 = (x1/p1 , 0, · · · ). Then since a ≡ a0 (mod p), we have a − a0 = pa

′1

for some a′1 ∈ W (Kalg). But a − a0 = (0, x1, ∗, · · · ) = p(x

1/p1 , ∗, · · · ), i.e. a

′1 ≡ a1

(mod p) and so a− a0 − pa1 ≡ 0 (mod p), which establishes the claim.

50

Solving a = φ aσ for the first two Witt components, we obtain

x0 = y0xp0, x1 = xp

1yp0 + y1x

p2

0 .

The condition (i) is clearly equivalent to that K1 = K(x0) = K(y−1/(p−1)0 ) is separa-

ble over K of degree p−1 and hence to the statement that the Galois representation

maps surjectively modulo p. Next, since aτ0 = a0 for τ ∈ Gal(Kalg/K1), one can

easily check that aτ ·a−1 ≡ 1+pa−10 (aτ

1−a1) (mod p) and the Galois representation

induced on the second level becomes

ρ2 : Gal(Kalg/K1) → (1 + pZp)/(1 + p2Zp) ∼= Fp

τ 7→ a−10 (aτ

1 − a1) = x−10 ((x

1/p1 )τ − (x

1/p1 ))

= x−p0 (xτ

1 − x1) = (yp/(p−1)0 x1)

τ − (yp/(p−1)0 x1)

But x1 = xp1y

p0 + y1x

p2

0 becomes the following Artin-Schreier equation in yp/(p−1)0 x1

(yp/(p−1)0 x1)

p − (yp/(p−1)0 x1) + y1y

−p0 = 0.

Then the condition (ii) is the necessary and sufficient condition for ρ2 to be surjective

and the conclusion of the lemma follows.

To prove the corollary, by Theorem 2.3.1, it suffices to show that the determinant

of the p-adic monodromy is Z×p ×Z×p and we can show this separately for each factor.

For this, we will find another point in our moduli space such that the determinant

of its local monodromy contains Z×p × 1. Similar argument will work for the other

factor.

51

Let us consider a point x = (Ax, λx, ιx) whose covariant Dieudonne module

M = M1 ⊕M2 has the following display.

M1 : F (ei) =

ei+1 (1 ≤ i < m)

ei+1 + ei+2 i = m

, ei =

V (ei+1) (m < i ≤ m + n)

V (e1) i = m + n

,

M2 = M t1.

Note that this Dieudonne module is isogenous to the Dieudonne module of the point

defined in Lemma 3.0.3, which guarantees the existence of similarity with the point

x = (Ax, λx, ιx).

The display matrix of the universal deformation of the corresponding formal

group is

0 · · · 0 T11 + T1n ∗

T21 + T2n ∗

Im−1... ∗

Tm1 + Tmn ∗

∗ ∗ ∗ ∗ ∗

.

The determinant φ of the m×m truncated matrix is

T11 + T1n = (t11 + t1n,1

p(tp11 + tp1n − (t11 + t1n)p), ∗, · · · ).

It suffices to show that y0 = s + t, y1 = 1p(sp + tp − (s + t)p) satisfies the condition

(i) and (ii) for the fraction field K of R = k[[s, t]](k = k). (i) is easy to verify. For

(ii), since the equation Xp− yp−10 X + y1 is defined over R, it is enough to show the

irreducibility over the quotient k[[s]] = k[[s, t]]/(t− cs) of R for c ∈ Z such that p2

52

does not divide 1 + cp − (1 + c)p. Then the Newton polygon of Xp − yp−10 X + y1

becomes the straight line joining (0, p) and (p, 0) and thus Xp − yp−10 X + y1 is

irreducible over the discrete valuation ring k[[s]].

The other factor can be dealt with similarly.

53

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