On Hyper-Symmetric Abelian Varieties Ying Zong A Dissertation in Mathematics Presented to the ...

59 Pages·2008·0.25 MB·English

On Hyper-Symmetric Abelian Varieties
Ying Zong
A Dissertation
in
Mathematics
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
2008
Advisor’s Name
Supervisor of Dissertation
Graduate Chair’s Name
Graduate Group Chairperson
Acknowledgments
The five years I spent in the graduate study has changed me a lot. Suddenly I feel
that I am no longer asleep in my dearest dream. Burdens and responsibilities drop
on my shoulders. Had no care and help from my wife Lei, I would not know where
to go. I dedicate this thesis to her.
This thesis is finished under the supervision of my advisor Ching-Li Chai. I
admire his pure spirit and I thank heartily for his patient and constant support.
I have been a dear student of all the mathematicians of the Univerisity of Penn-
sylvania, to whom I thank from the bottom of my heart. I thank in particular the
encouragement and support of Ted Chinburg.
I am grateful to Professors Paula Tretkoff and Steven Zucker; they gave me as
much support as they can when I met difficulties.
ii
ABSTRACT
On Hyper-Symmetric Abelian Varieties
Ying Zong
Ching-Li Chai, Advisor
Motivated by Oort’s Hecke-orbit conjecture, Chai introduced hyper-symmetric
points in the study of fine structures of modular varieties in positive characteristics.
We prove a necessary and sufficient condition to determine which Newton polygon
stratum of PEL-type contains at least one such point.
iii
Contents
1 Introduction 1
2 Notations and Generalities 6
2.1 The positive simple algebra Γ . . . . . . . . . . . . . . . . . . . . . 6
2.2 Brauer invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Isocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Dieudonn´e’s classfication of isocrystals . . . . . . . . . . . . . . . . 8
2.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Isocrystals with extra structure . . . . . . . . . . . . . . . . . . . . 9
2.8 Γ-linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9 Theory of Honda-Tate . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.10 Γ-linear abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . 12
2.11 A dimension relation . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.12 A variant of the Honda-Tate theory . . . . . . . . . . . . . . . . . . 13
iv
3 A Criterion of Hyper-Symmetry 14
3.1 A lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 A criterion of hyper-symmetry . . . . . . . . . . . . . . . . . . . . . 17
4 Partitions and Partitioned Isocrystals 19
4.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Partitioned isocrystals . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 A simply partitioned isocrystal s . . . . . . . . . . . . . . . . . . . 28
Γ
4.4 Partitioned isocrystals with (S)-Restriction . . . . . . . . . . . . . . 29
5 Main Theorem and Examples 30
5.1 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . 30
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Proof of the “only-if” part of (5.1.1) 39
6.1 Semi-simplicity of the Frobenius action . . . . . . . . . . . . . . . . 39
6.2 Proof of the only-if part . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Proof of the “if” part of (5.1.1) 43
7.1 Weil numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.2 Hilbert irreducibility theorem . . . . . . . . . . . . . . . . . . . . . 45
7.3 If F is a CM field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.4 If F is a totally real field . . . . . . . . . . . . . . . . . . . . . . . . 50
v
Chapter 1
Introduction
This work is to extend the study of hyper-symmetric abelian varieties initiated by
Chai-Oort [1]. The notion is motivated by the Hecke-orbit conjecture.
For the reduction of a PEL-type Shimura variety, the conjecture claims that
every orbit under the Hecke correspondences is Zariski dense in the leaf containing
it. In positive characterisitic p, the decomposition of a Shimura variety into leaves
is a refinement of the decomposition into disjoint union of Newton polygon strata.
A leaf is a smooth quasi-affine scheme over F . Its completion at a closed point is
p
a successive fibration whose fibres are torsors under certain Barsotti-Tate groups.
The resulting canonical coordinates, a terminology of Chai, provides the basic tool
for understanding its structure.
Fix an integer g ≥ 1 and a prime number p. Consider the Siegel modular variety
A in characteristic p. Denote by C(x) the leaf passing through a closed point x. By
g
1
applying the local stabilizer principle at a hyper-symmetric point x, Chai [3] first
gave a very simple proof that the p-adic monodromy of C(x) is big. Later, in their
solution of the Hecke-orbit conjecture for A , Chai and Oort used the technique of
g
hyper-symmetric points to deduce the irreducibility of a non-supersingular leaf from
theirreducibilityofanon-supersingularNewtonpolygonstratum, see[2]. Notethat
although hyper-symmetric points distribute scarcely, at least one such point exists
in every leaf [1].
Here we are mainly interested in the existence of hyper-symmetric points of
PEL-type. Let us fix a positive simple algebra (Γ,∗), finite dimensional over Q.
Following Chai-Oort [1], we have the definition:
Definition 1.0.1. A Γ-linear polarized abelian variety (Y,λ) over an algebraically
closed field k of characteristic p is Γ-hyper-symmetric, if the natural map
End0(Y)⊗ Q → End (H1(Y))
Γ Q p Γ
is a bijection.
For simplicity we denote by H1(Y) the isocrystal H1 (Y/W(k))⊗ Q. The goal
crys Z
of this paper is to answer the following question:
Question. Does every Newton polygon stratum contain a hyper-symmetric point?
The answer to the question in general is no; a Newton polygon stratum must
satisfy certain conditions to contain a Γ-hyper-symmetric point. See (5.2.2) for an
example when Γ is a real quadratic field split at p, and (5.2.6) when Γ is a division
algebra over a CM-field and the Γ-linear isocrystal M only has slopes 0,1.
2
In the main theorem (5.1.1), we characterize isocrystals of the form H1(Y) for
Γ-hyper-symmetric abelian varieties Y as the underlying isocrystals of partitioned
isocrystals with supersingular restriction (S).
Consider a typical situation. Let Y = Y0⊗ F be a Γ-simple hyper-symmetric
Fpa p
abelian variety over F , where Y0 is a Γ-simple abelian variety over a finite field
p
F . By the theory of Honda-Tate, up to isogeny, Y0 is completely characterized
pa
by its Frobenius endomorphism π . Let F be the center of Γ. Assume that F
Y0 pa
is sufficiently large. We show in (3.3.1) that Y is Γ-hyper-symmetric if and only if
the extension F(π )/F is totally split everywhere above p, that is,
Y0
F(π )⊗ F ’ F ×···×F ,
Y0 F v v v
for every prime v of F above p. Thus Y is Γ-hyper-symmetric if and only if it is
F-hyper-symmetric.
Denote by T the set of finite prime-to-p places ‘ of F where Γ is ramified. To
Γ
Y, one can associate its isocrystal H1(Y) as well as a family of partitions P = (P )
‘
of the integer N = [F(π ) : F] indexed by ‘ ∈ T . For each ‘ ∈ T , P is given by
Y0 Γ Γ ‘
P (‘0) = [F(π ) : F ]
‘ Y0 ‘0 ‘
with ‘0 ranging over the places of F(π ) above ‘. The pair (H1(Y),P) is the
Y0
partitioned isocrystal attachedtoY. Inparticular,wedenotebys thepairattached
Γ
to the unique Γ-simple supersingular abelian variety up to isogeny over F , see
p
(4.3.1).
3
To study the pair (H1(Y),P), it is more convenient to consider Y as an F-
linear abelian variety equipped with a Γ-action. Write ρ : Γ → End (H1(Y))
F
for the ring homomorphism defining the Γ-action induced by functoriality on its
isocrystal H1(Y). In essence, the definition (4.2.1) of partitioned isocrystals is a
purely combinatorial formulation of the conditions that Y is F-hyper-symmetric
and ρ factors through the endomorphism algebra End0(Y) of the F-linear abelian
F
variety Y.
The introduction of supersingular restriction (S) (4.4.1) has its origin in the
following example. Assume that F is a totally real number field. If a Γ-linear
isocrystal M contains a slope 1/2 component at some place v of F above p, but
not all, then there is no Γ-hyper-symmetric abelian variety Y such that H1(Y)
is isomorphic to M. In the proof of the main theorem (5.1.1), we treat specially
supersingular abelian varieties and isocrystals containing slope 1/2 components.
Given any pair y = (M,P) satisfying the supersingular restriction (S) and con-
taining no s component, the construction of a Γ-hyper-symmetric abelian variety
Γ
Y realizing y goes as follows. Let N be the integer such that P = (P ) is
‘ ‘∈TΓ
a family of partitions of N. The Hilbert irreducibility theorem [4] enables us to
find a suitable CM extension K/F of degree N, so that the family of partitions
(P ) given by
K/F, ‘ ‘∈TΓ
P (‘0) := [K : F ], ∀ ‘0 | ‘
K/F, ‘ ‘0 ‘
concide with (P ). Then a simple formula (7.1.1) gives directly a pa-Weil number
‘
4
π for a certain integer a ≥ 1, such that K = F(π) and the slopes of M at a
place v of F above p are equal to λ = ord (π)/ord (pa), for w|v. Let Y0 be the
w w w
unique abelian variety up to Γ-isogeny corresponding to π. For some integer e,
(Y0)e⊗ F equipped with a suitable polarization is a desired Γ-hyper-symmetric
Fpa p
abelian variety.
The organization of this thesis is as follows. In chapter 2 we set up the nota-
tions and review the fundamentals of isocrystals with extra structures, Dieudonn´e’s
theorem on the classification of isocrystals and the Honda-Tate theory. In chapter
3, we show that every Γ-hyper-symmetric abelian variety is isogenous to an abelian
variety defined over F (3.2.1). Then we prove a criterion of hyper-symmetry in
p
terms of endomorphism algebras (3.3.1). In the next chapter, we define partitions
and partitioned isocrystals. The main theorem (5.1.1) is stated in chapter 5. Sev-
eral examples are provided to illustrate how to determine which data of slopes are
realizable by hyper-symmetric abelian varieties. The proof of (5.1.1) is divided into
two parts. The “only-if” part, in chapter 6, shows that to every Γ-hyper-symmetric
abelian variety Y, one can associate a partitioned isocrystal y. We prove that y
satisfies the supersingular restriction (S). A key ingredient of the proof is that the
characteristic polynomial of the Frobenius endomorphism of H1(Y ) has rational
Fpa
coefficients. In chapter 7 we prove the inverse, the “if” part.
5