Example 5.2.1. (Siegel) Γ = Q. As TΓ is empty, the supersingular restriction (S)
is reduced to (S1) and (S2). A non-divisible simply partitioned isocrystal without slope 1/2 component is called balanced in the terminology of Chai-Oort [1]. In general, any simply partitioned isocrystal x can be expressed uniquely as
x=h.sΓ+m.y
with integers h, m≥0 and a balanced isocrystal y. One deduces that any Newton polygon of the form
ρ0.(1/2) +
X
i∈[1,t]
(ρi.(λi) +ρi.(1−λi))
can be realized by a hyper-symmetric abelian variety, where λi ∈[0,1/2) are pair-
wise distinct slopes, ρ0 = mult(1/2), ρi = mult(λi) are multiplicities. This example
recovers the Proposition (2.5) of Chai-Oort [1].
Example 5.2.2. Let F be a real quadratic field split at p, p = v1v2. The following
slope data 2.(1/2), atv1 1.(0) + 1.(1), atv2
admit no hyper-symmetric point.
Example 5.2.3. Let Γ =F be a totally real field of degree doverQ. The restriction (S) is reduced to (S1) and (S2).
The isocrystalsF is isotypic of slope 1/2 at every placev ofF. The multiplicity
is n(sF) = eF, the order of the class [Dp,∞⊗Q F] in the Brauer group of F, cf. (6.2.1).
Any simply partitioned isocrystal y without slope 1/2 component can be de- composed as a finite sum
y=y1 +· · ·+yn,
where each yi has two isotypic components at every place v|p.
Letzbe one of theyi’s , and let{λv,1−λv}be the two slopes of zatv. Then the
multiplicity n(z) is a common multiple of the denominators of [Fv :Qp]λv, where v
runs over the places of F above p.
As a consequence, anF-linear polarized isocrytalM of dimension 2doverK(Fp)
is realizable by anF-hyper-symmetric abelian variety overFpif and only if the slopes
of M has exactly one of the following two patterns:
(i) At every place v|p, there is only one slope 1/2 with multiplicity 2.
(ii) At every place v|p, there are two slopes {λv,1−λv}, each of multiplicity 1.
These λv are such that [Fv :Qp]λv ∈Z.
Example 5.2.4. Let Γ = F be a CM field, [F : Q] = 2d. The restriction (S) is reduced to (S1).
The isocrystal sF is isotypic of slope 1/2 at every place v of F above p. The
multiplicity is n(sF) = eF, the order of the class [Dp,∞⊗QF] in the Brauer group of F.
Any (S)-restricted simply partitioned isocrystaly is decomposed as a finite sum
y=y1 +· · ·+yn,
where eachyi has either one or two isotypic components. More explicitly, for a fixed
z =yi,
(i) if z has one isotypic component at every place v|p, the slopes are such that λv +λv = 1. In particular, λv = 1/2, if v = v. The multiplicity n(z) is a
multiple of the common denominator of [Fv :Qp]λv, for v|p.
(ii) if z has two isotypic components at every place v|p,
(a) if v =v, the slopes are {λv,1−λv}, with λv ∈[0,1/2).
(b) if v 6=v, the slopes are either
λv, 1−λv, at v λv, 1−λv, at v or µv, νv, atv 1−µv, 1−νv, atv with λv, λv ∈[0,1/2), µv 6=νv ∈[0,1].
Example 5.2.5. Let Γ be a definite quaternion division algebra over Q. We assume that Γ is ramified exactly at the infinity and a prime ` different from p. Hence TΓ={`} and inv`(Γ) = 1/2.
The partitioned isocrystalsΓis isotypic of slope 1/2 with multiplicityn(sΓ) = 2,
because the order eΓ of the class [Dp,∞]−[Γ] in the Brauer group of Q is 2. Lety be a simply partitioned isocrystal without slope 1/2 component. Let
P`: [1, d`]→Z>0
be the defining partition of y. The condition (SPI2) says that n(y).P`(i).1/2∈Z, for all i∈[1, d`].
Ify is (S)-restricted, then by (S3), its partition is of the following form
P`(2i−1) =P`(2i), i∈[1, c1(`)] P`(i) is even, i∈[2c1(`) + 1, d`]
for some integer c1(`)∈Z≥0.
Now letM be any effective Γ-linear polarized isocrystal satisfying the condition (SPI1) and without slope 1/2 component. We claim that M underlies an (S)- restricted simply partitioned isocrystal y. In fact, one can choose y = (M, Pl),
where d` = 1, P`(1) = N(y), and N(y) is the number of isotypic components of
M. Note thatN(y) is an even integer because M is polarized and has no slope 1/2 component.
With this choice of partitionPl, the simply partitioned isocrystaly decomposes
as a finite sum
y =y1+· · ·+ym,
where each yi has exactly two isotypic components with slopes {λi,1−λi}. The
For example, let us work out the slopes and multiplicities of all (S)-restricted partitioned isocrystals of dimension 12 over K(Fp). There are exactly five Newton
polygons which are realizable by 6-dimensional Γ-hyper-symmetric abelian varieties:
a. 3.(1/2).
b. 1.(0) + 1.(1) + 2.(1/2).
c. 2.(0) + 2.(1) + 1.(1/2).
d. 3.(0) + 3.(1).
e. 1.(1/3) + 1.(2/3).
The above notation, for example, 1.(0) + 1.(1) + 2.(1/2) means that the slopes are
{0,1,1/2}, with multiplicities {1,1,2}, respectively.
Example 5.2.6. Let F be a CM field, and Γ be a positive central division algebra over F. We make the following assumptions on Γ,
(i) [F :Q] = 4; [Fv1 :Qp] = 2, [Fv2 :Qp] = [Fv2 :Qp] = 1, v1, v2, v2 are above p.
(ii) Γ is ramified exactly atv1and a finite prime-to-pplace`,` =`; invv1(Γ) = 1/3,
The Brauer classc= [Dp,∞⊗QF]−[Γ]∈Br(F) has local invariants invν(c) = −1/3, if ν =v1 −1/2, if ν =v2, v2 −2/3, if ν =` 0, otherwise
Hence the order of c, as well as the multiplicity n(sΓ), is equal to 6.
Let y be a simply partitioned isocrystal. Let N(y) be the number of isotypic components, n(y) the multiplicity ofy at each place v ∈ {v1, v2, v2}. Denote by P`
the defining partition of y
P` : [1, d`]→Z>0.
In this case, the condition (SPI2) says that
n(y)P`(i).2/3∈Z, for all i∈[1, d`].
If y is (S)-restricted, then by (4.1.9), its partitionPl satisfies the condition
P`(2i−1) =P`(2i), ∀i∈[1, c1(`)],
for some integer c1(`), with 0≤2c1(`)≤d`.
We give another example of Newton polygon which admitsno hyper-symmetric point. ξ= 1.(0) + 1.(1), atv1 1.(0) + 1.(1), atv2 1.(0) + 1.(1), atv2
Note that if M has ξ as Newton polygon, then
dimK(Fp)(Mv1) = 12, dimK(Fp)(Mv2) = dimK(Fp)(Mv2) = 6.
At each place v ∈ {v1, v2, v2}, M has N = 2 isotypic components, the multiplicity
of every isotypic component is n = 1. But there is no partition P` of N = 2, such
that n.P`(i).2/3∈Z.
Now we compute the Newton polygons of all (S)-restricted partitioned isocrys- tals of dimension 72 over K(Fp). By (S1), we can write x = h.sΓ
W
y. Note that the dimension of sΓ is 72. One has eitherx=sΓ orx=y. Consider the case x=y
and write
y ={yb;b∈B},
where yb are the simple components of y. Comparing the dimension of yb and y,
one has
72 = [Γ :F]1/2[F :Q]X
b∈B
N(yb)n(yb),
where N(yb) denotes the number of isotypic components, n(yb) the multiplicity, of
yb at each place of F above p. Since [Γ : F]1/2 = 3, [F : Q] = 4, this equation is
reduced to
6 =X
b∈B
N(yb)n(yb).
One verifies that this condition forces that y is simply partitioned, N(y) = 2, and n(y) = 3. Here we list all the realizable Newton polygons as follows.
(i) The slopes at v1 are one of: 0, 1 1/3, 2/3 1/6, 5/6
(ii) The slopes at v2, v2, in this order, are one of:
0, 1; 1/3,2/3 0, 1/3; 1,2/3 0, 2/3; 1,1/3 1, 1/3; 0,2/3 1, 2/3; 0,1/3 1/3,2/3; 0,1
Chapter 6
Proof of the “only-if ” part of
(5.1.1)
6.1
Semi-simplicity of the Frobenius action
The following lemma is certainly well known and an analogous statement for `-adic cohomology can be found in Mumford’s book on abelian varieties.
Lemma 6.1.1. If X is an abelian variety over a finite field k, the Frobenius endo- morphism π acts in a semi-simple way on the isocrystal H1(X).
Proof. We may and do assume that X is a simple abelian variety. Let π = s+n be the Jordan decomposition of π considered as a linear endomorphism of H1(X).
By Katz-Messing [7], the characterisic polynomial det(T −π|H1(X)) has rational
such that the nilponent part n=f(π). The image of`n, for a sufficiently divisible integer `, is a proper sub-abelian variety ofX, thus equal to 0.