Moduli of abelian varieties

38 Pages·2017·1.03 MB·English

Moduli of abelian varieties
Ching-Li Chai and Frans Oort
Preliminary version May 2017.
Next: corrections, suggestions, etc. by Ching-LI,
many more references should be included in the text, t
then I will finish, and fill out all XX etc.
Add more questions?
f
Check bibliography, include references, or delete items.
We ask Rachel Pries for her comments.
a
Introduction
WediscussthreeQuestionsinthe1995manuscript[101]and(partial)answerstothese. More-
over we will describe other ideas connected with other problems on these topics.
r
0.1. [101], 8.3 Conjecture (irreducibility of Newton Polygon strata). Let β be a
symmetric NP, with β (cid:54)= σ, i.e. W is not the supersingular locus. Then for principally
β
polarized abelian varieties in the moduli space A ⊗ F of principally polarized abelian
D g,1 p
varieties in positive characteristic we conjectured in 1995:
the NP stratum W is irreducible. (?)
β,1
We knew at that moment that (for large g) the supersingular locus is reducible. In 1995 we
had good reasons to believe that the non-supersingular Newton Polygon strata in A ⊗F
g,1 p
should be geometrically irreducible. This is indeed the case, as was proved in [16], Theorem
3.1. For a sketch of a proof, for ingredients, and for other details, see Section 4.
0.2. [101], Section 12 (CM liftings). Tate proved that any abelian variety defined over
a finite field is a CM abelian variety, see [139] XX. Honda and Tate proved that any abelian
variety over F is isogenous with an abelian variety that can be CM lifted to characteristic
p
zero, see [42], basically contained in his Main Theorem, and [140], Th. 2 on page 102:
For any abelian variety A over a finite field κ = F there exists a finite extension κ ⊂ K
q
and an isogeny A⊗ K ∼ B and a CM lift of B to characteristic zero.
κ K 0 0
Is the field extension necessary? Is the isogeny necessary?
We knew, 1992, there exist abelian varieties defined over F that cannot be CM lifted to
p
characteristic zero (i.e. there are cases where an isogeny over F is necessary to make a CM
p
lift possible), see [98]: an isogeny is necessary in general.
Later, more precise questions were formulated:
1
0.2.1. Suppose A is an abelian variety over a finite field κ. Does there exists a κ-isogeny
A ∼ B suchthatB admitsaCMlifttocharacteristiczero? I.e. is a field extension necessary
0 0
in the Honda-Tate theory?
0.2.2. Suppose A is an abelian variety over a finite field κ. Does there exists a κ-isogeny
A ∼ B and a CM lifting of B to a normal mixed characteristic domain?
0 0
Complete answers to these questions are given in [12]. Below, in Section 5 we will describe
more precise questions, methods and results.
0.3. [101], Section 13 (generalized canonical coordinates). Around a moduli point
x = [X ,λ ] of an ordinary abelian variety over a perfect field the formal completion
0 0 0
t
(A )∧ ∼= (cid:0)(G )∧(cid:1)g(g+1)/2
g,1 x0 m
in mixed characteristic, is a formal torus, where the origin is chosen to be the canonical lift of
f
(X ,λ ); see [65], Chapter V; as proved by Drinfeld [25], see [54], also see [46]. XXX
0 0
0.4. In [101], 13A we ask for a
“canonical coordinates” around any moduli point in A ⊗F .
g p
Right-fully we said in 1995: “This question should be made much more precise before it can
be taken seriously.” We will show results and limitations:
r
• We do not know how to make canonical coordinates in mixed characteristic around any
point in the non-ordinary Newton Polygon locus, but
• we can construct theDright concept around a point y ∈ A ⊗ F on the central leaf
0 g p
passing through y .
0
Note that the central leaf through the moduli point of an ordinary abelian variety is dense-
open in the moduli space, and there Serre-Tate show us how to find coordinates in the mixed
characteristic case.
• For an almost ordinary abelian variety (i.e. the p-rank equals dim(A)−1) the central
leaf passing through is dense in the related Newton Polygon stratum in characteristic p;
• we do construct a generalization of canonical coordinates on this stratum, however in
characteristic p.
• Such a construction in mixed characteristic around the moduli point of an almost ordi-
nary abelian variety in positive characteristic does not exist if we impose the condition
the lifted stratum should be flat over the base and Hecke-invariant.
• For other moduli points in positive characteristic central leaves are smaller than the re-
lated Newton Polygon stratum; we do construct a generalization (analogue) of canonical
coordinates on (the formal completion of) the central leaf in characteristic p. But we
do not construct something like “canonical coordinates” on the whole Newton polygon
stratum if the p-rank is smaller than dim(A)−1. We will explain why. For references
see below. For much more information see [19].
2
1 p-divisible groups
In this section we briefly recall notations we are going to use.
1.1. For given integers d ∈ Z and h ∈ Z a Newton Polygon ζ (of height h and dimension
>0 ≥d
d) is a lower convex polygon starting at (0,0) and ending at (h,d) and having break points
in Z×Z; equivalently: we have slopes 0 ≤ s ≤ ··· ≤ s with s ∈ Q and s +···+s = d
1 h j 1 h
and if s < s we have s +···+s ∈ Z; the rational numbers s are called the slopes of ζ.
b b+1 1 b j
We only consider Newton Polygons with 0 ≤ s ≤ 1 for every j. A Newton Polygon is called
j
isoclinic if all slopes are equal; in this case s = d/h for every j.
j
AsymmetricNewtonPolygonisaNewtonPolygonξ withd = g, andh = 2g suchthatany
slope r ∈ Q with 0 ≤ r ≤ and the slope 1−r appear with the same multiplicity. An isoclinic
symmetric Newton Polygon consists of 2g slopes s = 1/2 this is called the supersingular
j
Newton Polygon, σ = σ . t
g
For Newton Polygons ζ , ζ of the same height and dimension we write
1 2 f
ζ ≺ ζ ⇐⇒ no point of ζ is below ζ
1 2 1 2
a
and we will say “ζ is above ζ ” (a bit strange? explanation: ζ defines a smaller stratum
1 2 1
than ζ in this case).
2
(cid:34)
(cid:34)(cid:0)
ζ1(cid:34)(cid:34)(cid:0)
r(cid:40)(cid:24)(cid:40)(cid:24)(cid:40)(cid:24)(cid:40)(cid:24)(cid:40)(cid:34)(cid:40)(cid:0)ζ2
Foranyfield K, a p-divisiblegroup X ofheight h over K isaninductivesystemoffinite group
schemes Gi over K, with mDaps
G −∼→ G [pi] ⊂ G , rk(G ) = pih,
i i+1 i+1 i
G = ∪ G , and ×p : G (cid:16) G is epimorphic.
i i
Crucial example: For any abelian variety A over a field K we see that ∪ A[pi] is p-divisible
i
group, denoted by X = A[p∞].
Thenotionofap-divisiblegroupcanbestudiedoveranybasefield. Ifpisnotthecharacteristic
of the base field, the dual notion of a Tate p-group is an equivalent datum; in this case to
notionofap-divisiblegroupoveraperfectK, andthenotionofaTatep-grouparedetermined
by a Galois representation.
If however p is the characteristic of the base field K we should be careful using the notion
ofa“Tatep-group”, butthenotionofap-divisiblegroupisveryuseful. Inthiscasethenotion
of a p-divisible group can be much more than just a Galois representation.
Over an arbitrary base scheme S one can define the notion of a p-divisible group over S; for
details see [46]. For an abelian scheme A → S indeed X = A[p∞] is a p-divisible group over S.
Thisenablesusto usethisnotioninmixedcharacteristic, evenwhentheresiduecharacteristic
is p.
An isogeny of p-divisible groups ψ : X → Y is a homomorphism with finite kernel and
Y = X/Ker(ψ). A p-divisible group X over a field is called simple if for every p-divisible
group subgroup Y ⊂ X either Y = 0 or Y = X (“iso-simple” would be a better terminology).
3
1.2. For the theory of Diedonn´e modules we refer to existing literature, e.g. XXmore
references [15], pp. 479–485. The result is that for any perfect field κ ⊃ F we have the
p
Dieudonn´e ring R generated over the ring Λ = Λ (κ) with automorphism σ ∈ Aut(Λ (κ)),
κ ∞ ∞
the unique lift to Frobenius automorphism x (cid:55)→ xp on κ and operators F and V subject to
0 0
relations
F·V = p = V·F, F·x = xσ·F, x·V = V·xσ, x ∈ Λ.
(For the Witt ring the usual notation is W; however this symbol we use for Newton Polygon
strata, see below, hence the notation Λ for the Witt ring.) The ring R is commutative if
K
and only if κ = F . For a finite commutative κ-group scheme N or for a p-divisible group
p
X over κ we define a module D(N), respectively D(G) (see the literature for definitions). In
particular D(G) is a left R -module, that is free of rank ht(G) = h over Λ (κ).
κ ∞
t
In [63] the contravariant theory is defined, used and developed. It has turned out that the
covariant theory is easier in use, especially with respect to theories like Cartier theory and
the theory of displays; up to duality these two different theories are the same, so for achieving
f
results it does not make much difference which is used.
Note that the Frobenius morphisms F : G → G(p)ainduces V on D(G), and the Verschiebung
V : G(p) → G induces F on D(G). For this reason we distinguish the morphisms F and V
(on group schemes) and the operation V and F (on covariant Dieudonn´e modules) by using
different symbols. XX see last section in Texel
Some examples. The p-divisiblergroup Q /Z over F corresponds with Λ·e with V·e = e
p p p
and F·e = pe.
The p-divisible group µ over F corresponds with Λ·e with V·e = pe and F·e = e.
p∞ p
For a p-divisible groupDX of dimension d the module D(X)/Rκ·V is a κ-vector space of
dimension d:
dim(X) = dim (D(X)/R ·V).
κ κ
For coprime integers m,n ∈ Z we define the p-divisible group G over F by the property
≥0 m,n p
D(Gm,n) = RFp/RFp·(Vn−Fm); we see: dim(Gm,n) = m, ht(Gm,n) = m+n.
This group G is simple over any extensions field. We write G instead of G ⊗K for
m,n m,n m,n
any K ⊃ F if no confusion is possible.
p
1.3. Theorem (Dieudonn´e, Manin). Suppose k = k ⊃ F is an algebraically closed field. For
p
any p-divisible group X over k there is an isogeny
X ∼ ⊕ G
k (mi,ni)∈S mi,ni
for some set S of pairs (repetitions are allowed).
For p-divisible groups of height h and dimension d over an algebraically closed field k ⊃ F
p
and Newton polygons with the same invariants the set of isogeny classes of p-divisible groups
and the set of Newton Polygons are “equal”
∼
{X}/ ∼ −→ {NP}, X (cid:55)→ N(X); height = h, dim = d.
XXexact reference
4
This correspondence is given by defining N(G ) to be the isoclinic Newton Polygon of
m,n
height h and slope m/(m+n). For X as in the theorem we order all slopes m /(m +n ) with
i i i
multiplicities m +n in non-decreasing order, resulting in N(X).
i i
We mention that the slopes of the Newton Polygon are given basically by the p-adic
valuations of the Frobenius of X (but we need Dieudonn´e module theory to make sense of
this). For example, on X = G over F we have Fn = Vm (or, if you want, (Vn−Fm)e = 0
m,n p
on the canonical generator of the Diedonn´e module); from Fn·Fm = Vm·Fm = pm, and we
see “the p-adic valuation of F = F is m/(m+n)”.
X
We illustrate this for a simple p-divisible X of height h over a finite field κ = F ; in this
q
case,q = pr,wehaveaFrobeniusπ : X → X,asther-timesiteratedabsoluteFrobenius; write
[Q(π) : Q] = d;considertheNewtonpolygonoftheeigenvaluesofπ,i.e. takethecharacteristic
polynomial of π, and define the slopes by taking p-adic values of the zeros; by stretching this
polynomial in horizontal and vertical direction by h/d and multtiplying vertically by 1/r we
obtain N(X). For the general case, we need the whole theory of Dieudonn´e modules to define
N(X) in general. XXcheck in example
f
For the theory of Diedonn´e modules, see [63], [84], [23], see [33], [5]; for a discussion of the
various approaches see [12], Appendix B3.
a
1.4. Minimal p-divisible groups. Onecanask,whenisap-divisiblegroupX determinedby
itsBT ? (HereBT standsfor“truncatedBarsotti-Tategroupatlevelone”.) Thisisanswered
1 1
in [113]. For coprime m,nZ define H over F by its Diedonn´e module M = D(H )
≥0 m,n p m,n
generated over RFp by a free Λ-basis {e0,e1,···,em+n−1} ⊂ M with relations
r
j = i+b(m+n) then e = pbe , F·e = e , V·e = e ;
j i i i+n i i+m
for simplicity here we write e = p·e ∈ D(H ). As (Fm−Vn)(e ) = 0 for every 0 ≤ i,
i+m+n i m,n i
we see that D
D(Gm,n) = RFp·e0 ⊂ D(Hm,n), hence Gm,n ∼ Hm,n.
This group H is simple over any extension field. We write H instead of H ⊗K for
m,n m,n m,n
any K ⊃ F if no confusion is possible, and we know G ∼ H over any extension field.
p m,n K m,n
Note there exists an endomorphism u ∈ End(D(H )) = End(H ) (already defined over
m,n m,n
F ) with the property u(e ) = e . We see that
p i i+1
End((H ) ) ⊂ End0((H ) ) = End0((G ) ) is the maximal order
m,n k m,n k m,n k
in the division algebra End0((G ) ); this is central simple of degree (m+n)2 over a field K
m,n k
that contains F checkXXX; over any algebraically closed field k ⊃ F this characterizes
pm+n p
(H ) .
m,n k
Suppose a Newton Polygon ζ is given by {(m ,n )}; in this case we define
i i
H = ⊕ H .
ζ i mi,ni
This is called the minimal p-divisible group in the isogeny class given by ζ.
Theorem ([113] , Theorem 1.2). Let k ⊃ F be algebraically closed, and let X be a p-divisible
p
group over k with N(X) = ζ. Then
∼ ∼
X[p] = H [p] =⇒ X = H .
ζ ζ
5
1.5. Duality. Suppose X is a p-divisible group over a field K, defined by X[pi] = G . The
i
exact sequence
0 → G −→ G −→ G → 0
i i+j j
using Cartier duality of finite group schemes gives the exact sequence
0 ← (G )D ← (G )D ← (G )D ← 0.
i i+j j
This defines a p-divisible group
Gt := ∪ (cid:0)(G )D (cid:44)→ (G )D(cid:1) = ∪ (G (cid:16) G )D,
j j+1 j+1 j
called the Serre dual of G (please do do not call this the Cartier dual of G; please do not write
GD).
Example/Remark. t
(G )t = G .
m,n n,m
f
Theorem (duality theorem), see [89], Theorem 19.1. For an isogeny ψ of abelian varieties
over any base scheme, with N = Ker(ψ), the exact sequence
a
ψ
0 → N −→ A −→ B → 0
and duality theory of abelian varieties and Cartier duality for finite group schemes give
0 → ND −→ Bt −ψ→t At → 0.
r
Corollary. For an abelian scheme A over any base scheme we have
D
At[p∞] = (A[p∞])t.
For any abelian variety A over K ⊃ F its Newton Polygon N(A) is symmetric.
p
2 Newton Polygon strata
In this section we briefly recall notations we are going to use. These strata can be defined in
deformation spaces, but we will not recall definitions in those case. All base schemes are in
characteristic p.
2.1. For any abelian scheme A → S, or for any p-divisible group X → S and a Newton
Polygon ξ (respectively ζ) we define
W0(A → S) = {x ∈ S | N(A) = ξ}.
ξ
Inside A we write W = W0(A ). For principally polarized abelian varieties we write
g,d ξ,d ξ g,d
W0 = W0 = W0(A ).
ξ ξ,1 ξ g,1
Discussion. This is a “point-wise” definition. We do not have a good functorial approach to
these strata.
Grothendieck and Katz showed these sets are locally closed in ∪ A , see [40], [52], 2.3.2.
d g,d
We consider these definitions either over F , or over a perfect field κ, and we consider these
p
6
as a scheme with the reduced scheme structure on this locally closed set.
Discussion. The proof by Grothendieck and by Katz does give a (non-canonical) scheme
structure (in the proof equations are given, but these depend on choices, and clearly we
obtain in general many more nilpotents than we want). It is possible that inside A , or more
g,1
generally for p not dividing d, we do obtain reduces schemes, and on can expect that some
other components (when a high power of p divides d) will give a non-reduced scheme, once a
good functorial description is available; part of this will follow from [19].
2.2. Here we consider closed Newton Polygon strata. There are two ways of doing this. We
can consider the Zariski closure of W0(A → S) or we can consider
ξ
W0(A → S)Zar ⊂ {x ∈ S | N(A) ≺ ξ}
ξ
(inclusion because, by Grothendieck, we know that under specialitzation, Newton Polygons go
up).
f
Fact. In the deformation space of unpolarized p-divisible groups, and in A where p does
g,d
not divide d we have equality:
a
W := W0(A )Zar = {x ∈ A | N(A) ≺ ξ}
ξ ξ g,1 g,1
ThisfollowsfromtheGrothendieckconjecture(forp-divisiblegroups, forprincipallypolarized
p-divisible groups, for principally polarized abelian varieties); see 7.4.
r
However, there are many examples where
(cid:0)W0(A )(cid:1)Zar (cid:36) {x ∈ A | N(A) ≺ ξ}
ξ g,d g,d
D
Here is an easy example, see [51]XXX; consider g = 3, and d = XX. Inside A the supersin-
3,d
gular locushas a component V of dimension three; this isone of the irreducible components of
the locus inside A of abelian varieties with p-rank equal to zero; consider ξ = (2,1)+(1,2);
3,d
the closure (W0(A ))Zar does not contain V, and we see inequality in this case. Below we
ξ g,1
discuss many other cases, see Section 3, and a systematic way to find them.
2.3. Supersingular abelian varieties. We say an elliptic curve E is supersingular if
E[p](k) = {0}. An elliptic curve is supersingular if and only if N(E) = σ = (1,1). As
1
there only two symmetric Newton Polygons for g = 1 this is clear. For every p there is at least
one elliptic curve in characteristic p.
We define
a(G) := dim Hom(α ,G),
κ p
where κ is a perfect field, and G a commutative group scheme over κ.
Theorem. Let k ⊃ be an algebraically closed field.
(1) For an abelian variety A of dimension g the following statements are equivalent
• N(A) = σ := g·(1,1);
g
• A[p∞] ∼ (G )g;
1,1
• for any supersingular elliptic curve E there exists an isogeny Eg ∼ A;
• Defintion. The abelian variety is supersingular.
7
(2)Supposea(A) = g > 1thenforeverysupersingularellipticcurveE thereisanisomorphism
Eg ∼= A; in this case the abelian variety is called superspecial.
XXX? give steps in proof
Note the fact that every supersingular abelian variety of dimension at least two is not geomet-
rically simple, but for every other Newton Polygon we can find a geometrically simple abelian
variety having that Newton Polygon, see HWLFOXX [60]
2.4. Description of components of ss locus
2.5. EO strata XX
3 Stratifications and foliations
t
In this section we briefly recall notations and constructions we are going to use. All base
schemes are in characteristic p. These constructions werefsuggested by the Hecke orbit prob-
lem: try to describe the Hecke orbit H(x) of a moduli point. In characteristic p the Newton
Polygondoesnotchangeundersuchactions. WeareinterestedintheZariskiclosureofthefull
a
Heckeorbit. Foliationsdescribedhere“separate”thetwoquidifferentcases, quasi-isogeniesof
degree prime to p on the one hand (“moving in a central leaf”) and α-isogenies (kernels have
filtrations where successive quotients are isomorphic with α , “moving in an isogeny leaf”).
p
3.1. Central leaves. Suppose [(B,µ)] = x ∈ A (κ), where κ is a perfect field. We write
g,d
r
C(x) = {[(A,ν)] = {y ∈ A | (B,µ)[p∞]⊗Ω ∼= (A,ν)⊗Ω};
d g,d
here Ω ⊃ κ is an algebraically closed field over which (A,ν) is defined; this is called the
D
central leaf passing through x; for d = 1 we simply write C(x) = C(x) . Theory about these
1
constructions and objects we find in [110]. The set C(x) ⊂ A ⊗κ is locally closed, in fact
d g,d
C(x) ⊂ (W ) is closed, where ξ = N(B). We consider this as a κ-subscheme with the
d ξ,d κ
reduced scheme structure. WE attache the index d in order to remind reader we are working
in A .
g,d
Discussion. This is a “point-wise” definition. At the time of writing [110] we did not have a
better way of approach. Now we develop a functorial approach, see [19]. It will turn out that
there is a canonical scheme structure on these C(x) . In case d = 1 (the principally polarized
d
case) the previous C(x) and the new leaves coincide. However there are cases (where p divides
d), where the new structure gives a scheme with nilpotents, over a perfect field having C(x)
d
as reduced scheme structure. Moreover, over non-perfect fields subtleties appear: the new
structure can give a reduced scheme, which over the perfection of the base field does have
nilpotents (the case of “hidden nilpotents”); hence the “old” definition and construction of
C(x) is full of difficulties over non-perfect fields. These phenomena, with many examples will
d
be discussed in [19].
3.2. Theorem. see [110], Theorem 2.3.
C (S) ⊂ W0 (S)
X N(X)
is a closed set.
8
3.3. Theorem. Isogeny correspondences, unpolarized case. Let ψ : X → Y be an
isogeny between p-divisible groups. Then the isogeny correspondence contains an integral
scheme T with two finite surjective morphisms
C (D(X)) (cid:17) T (cid:16) C (D(Y))
X Y
such that T contains a point corresponding with ψ.
3.4. For any two points x ∈ A , y ∈ A belonging to the same Newton Polygon ξ, we have
g,d g,e
dim(C(x) ) = dim(C(y) ),
d e
all central leaves in the same Newton Polygon stratum have the same dimension. Thisnumber,
depending only on ξ = N(A ) will be denoted by c(ξ).
x t
3.5. Isogeny correspondences, polarized case. Let ψ : A → B be an isogeny, and let λ
respectively µ be a polarization on A, respectively on B,fand suppose there exists an integer
n ∈ Z such that ψ∗(µ) = n·λ. Then there exist finite surjective morphisms
>0
a
C (A ⊗F ) (cid:17) T (cid:16) C (A ⊗F ).
(A,λ)[p∞] g p (B,µ)[p∞] g p
See [110], 3.16.
3.6. The dimension of C (A ⊗F ) only depends on the isogeny class of (X,λ).
(X,λ) g p
r
Remark/Notation. In fact, this dimension depends only on the isogeny class of X. We
write
c(ξ) := dim(cid:0)C (A ⊗F )(cid:1), X = A[p∞], ξ := N(X);
D(X,λ) g p
this is well defined: all irreducible components have the same dimension.
3.7. The central stream. Suppose [(A,λ)] = x ∈ A such that A[p∞] is the minimal
g,1
p-divisible group associated with ξ = N(A), i.e. A[p∞]⊗k ∼= H ⊗k. In this case we write
ξ
C(x) =: Z ⊂ W0 ⊂ A , called the central stream in W0.
ξ ξ g,1 ξ
We will see this central leaf plays an important role in many considerations.
3.8. Isogeny leaves. We say A··· → B is a quasi-isogeny if there is an integer q such that
×q
A −→ A··· → B
an isogeny. We say it is an α-quasi-isogeny if (perhaps over some extension field) kernels have
filtrations where successive quotients are isomorphic with α . Suppose [(B,µ)] = x ∈ A (κ),
p g,d
where κ is a perfect field. We write
I(x) = {[(A,ν)]y ∈ A | (A,ν)⊗Ω q∼ (B,µ)⊗Ω}.
d g,d
This is a closed set in W , we give it the reduced scheme structure, it is proper over Spec(κ).
ξ,d
Discussion. This is a “point-wise” definition.
Remark. As we see in Katz XXYYY complete this remark on functorial def of isogeny leaf
too big.
9
The almost product structure on W . For any W over k, with ξ not the ordinary
ξ,d ξ,d
Newton Polygon, there exist C(cid:48) and I(cid:48) and a surjective, finite, flat morphism C(cid:48) → C(x), a
surjective, finite morphism I(cid:48) → I(x), and a finite surjective morphism Φ : C(cid:48) ×I(cid:48) → W
ξ,d
such that for any x(cid:48) ∈ C(cid:48), with x(cid:48) (cid:55)→ x, the image
I(x) = Φ({x(cid:48)}×I(cid:48) ⊂ W
ξ,d
is the isogeny leaf I(x), and for any x(cid:48)(cid:48) ∈ C(cid:48), with x(cid:48)(cid:48) (cid:55)→ x the image
C(x) = Φ(C(cid:48)×{x(cid:48)(cid:48)}) ⊂ W
ξ,d
isthecentralleafC(x). Thisisthereasonweusetheword“foliation”inbothcasesinthesense
of a disjoint union of (sometimes) lower dimensional closed sets (although in some theories in
mathematicsthiswordisusedinamorestrictsense). Notethatontanyirreduciblecomponent
we have that dim(W(cid:48) ) and dim(I(x) ) = dim(W(cid:48) )−c(ξ) can depend on the choice of the
ξ,d d ξ,d
component W(cid:48) .
ξ,d f
Wedefinei(ξ) = dimW0−c(ξ); weseei(ξ) = dim(I(ξ))inA . Wewillseei(ξ) ≤ dim(I(ξ) ),
ξ g,1 d
and we will see (many) cases where the inequalityasign holds.
Some examples. For the ordinary symmetric Newton polygon ρ = (g,0) + (0,g) isogeny
leaves are empty and W is a central leaf.
ρ,d
For the almost ordinary symmetric Newton Polygon ξ = (g − 1,0) + (1,1) + (0,g − 1)
isogeny leaves are zero-dimensional,rand Wξ,d is a central leaf.
For the supersingular symmetric Newton Polygon σ = g·(1,1) central leaves in W are
σ,d
finite, and W is an isogeny leaf.
σ,d
Forξ = (g−1,1)+(1,g−1)isogenyleavesarerationalcurves, andc(ξ) = g(g+1)/2−g−1;
D
in this case all components of the whole Newton polygon stratum have the same dimension.
This example can be used to illustrate interesting phenomena.
3.9. ForeveryNewtonPolygonthedimensionoftherelatedstratum(inthedeformationspace
ofap-divisiblegroup)canbecomputedfromthedatadefiningthepolygon; alsothedimension
of W can be seen; also the possible dimensions of W and of I(x) can be computed; see
ξ ξ,d d
[118]. Here are the results.
Notation. Let ζ be a Newton polygon, and (x,y) ∈ Q×Q. We write
(x,y) ≺ ζ if (x,y) is on or above ζ,
(x,y) (cid:22) ζ if (x,y) is strictly above ζ,
(x,y) (cid:31) ζ if (x,y) is on or below ζ,
(x,y) (cid:23) ζ if (x,y) is strictly below ζ.
Notation (the unpolarized case). We fix integers h ≥ d ≥ 0, and we write c := h−d. We
consider Newton polygons ending at (h,d). For such a Newton polygon ζ we write
(cid:51)(ζ) = {(x,y) ∈ Z×Z | y < d, y < x, (x,y) ≺ ζ},
and we write
dim(ζ) := #((cid:51)(ζ)).
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