Newton Polygons as Lattice Points

Ching-Li Chai^{∗}^{†}

University of Pennsylvania

(version 2.1, 02/27/99)

Abstract

Let M be the reduction modulo p of a Shimura variety, and let 𝒩(M) be the partially ordered set predicted to be the set of all types occurring in the stratification of M by Newton polygons, in the context of F -isocrystals with additional structures ([7], [9], [14], [8]). We prove that 𝒩(M) is ranked (or catenary), in the sense that any two maximal chains have the same length. We also give a Lie-theoretic formula for the distance between two comparable elements in 𝒩(M). This formula can be used to give the expected dimension of the Newton strata of M.

1 Introduction

Let 𝒜_{g}^{ss} be the locus of supersingular abelian varieties inside 𝒜_{g}_{/𝔽_{p}}, the moduli space of g-dimensional principally polarized abelian varieties in characteristic p > 0. A main result in the monograph [10] by Li and Oort states that dim(𝒜_{g}^{ss}) = ⌊\frac{g^{2}}{4} ⌋, where ⌊·⌋ means “rounding down to the nearest integer”. Since 𝒜_{g}_{/𝔽_{p}} is the reduction modulo p of a Shimura variety, this beautiful result begs the following question: Extrapolate a group-theoretic formula from the numbers ⌊\frac{g^{2}}{4} ⌋, which makes sense for Shimura varieties.

∗This research is partially supported by grant DM95-02186 from the National Science Foundation

†email: chai@math.upenn.edu

We offer such a formula in this note; see §3, Theorem 3 and the Question at the end of §3. To give an idea of the formula, notice that dim(𝒜_{g}) = \frac{g(g+1)}{2}, and

\frac{g(g + 1)}{2} − ⌈\frac{g^{2}}{4} ⌉ = ⌈\frac{1}{2}⌉ + ⌈\frac{2}{2}⌉ + ⌈\frac{g}{2}⌉ = ⌈⟨ω_{1}, μ⟩⌉ + ⋯ + ⌈⟨ω_{g}, μ⟩⌉

where ω_{1} = x_{1}, …, ω_{g} = x_{1} + ⋯ + x_{g} are the fundamental weights for Sp_{2g}, μ = \frac{1}{2}(e_{1} + ⋯ + e_{g}) is the minuscule fundamental coweight, and ⌈·⌉ means “rounding up to the nearest integer”.

The general formula, in the same flavor as above, expresses the “expected dimension” of the Newton strata with a given type in the reduction of a Shimura variety. In the case of 𝒜_{g}, the expected dimension of the Newton strata is due to [13] in a different form, and [6] proves that the actual dimension is equal to the expected dimension.

Although we are motivated by questions in algebraic geometry, this note itself deals only with combinatorial properties of the set of generalized Newton polygons associated to a reductive group over a p-adic field. In the rest of the introduction we will trace through the evolution of the notion of Newton polygons, followed by a more detailed description of the results in this note.

Newton polygons appear in at least two different contexts. They are used to find the valuations of roots of polynomials, and they are convenient for organizing the slopes of a p-divisible group. Recall that a Newton polygon is a graphic representation of a sequence of rational numbers λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n}. These numbers are then encoded in the graph of either a convex or a concave piecewise-linear function on [0, n], which passes through the origin and has the λ_{i}’s as its slopes. We use the concave one below, so that the region below the graph is convex. One can also bring in Lie theory: a sequence λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n} of non-increasing rational numbers corresponds to a rational point in the Weyl chamber of the group GL_{n}. The partial ordering “lying above” for Newton polygons with the same end points translates into the inclusion relation between convex hull of Weyl-orbits: Say we have a Newton polygon NP_{1} with slopes λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n} and a Newton polygon NP_{2} with slopes μ_{1} ≥ μ_{1} ≥ … ≥ μ_{n}, corresponding to points x_{1}, x_{2} in the Weyl chamber of GL_{n}. Then the convex hull of the Weyl orbit of x_{1} contains the convex hull of the Weyl orbit of x_{2} if and only if

∑_{j=1}^{k} λ_{j} ≥ ∑_{j=1}^{k} μ_{j}, ∀ k = 1, …, n − 1, and ∑_{j=1}^{n} λ_{j} = ∑_{j=1}^{n} μ_{j} .

Graphically, this means that NP_{1} lies above NP_{2}. We say that x_{1} ⪰ x_{2} if this is the case; this defines a partial ordering on the set 𝒩 of all Newton polygons. The group-theoretic point of view has the advantage that it generalizes to every reductive groups; a good reference is [1, §12]. (Although our results may be relevant for questions about families of principal bundles, we do not explore the possible connection here.)

The Newton polygons one encounters in application usually satisfy an extra integrality condition, namely the denominator of each ‘slope’ λ_{i} divides its multiplicity. If this condition is satisfied, we say that the Newton polygon is integral. Denote by 𝒩_{ℤ} the set of all integral Newton polygons (for a fixed n).

A lattice often refers to a discrete subgroup of finite covolume in a Lie group. A lattice in a finite dimensional real vector space V_{ℝ} defines an integral structure V_{ℤ} on V_{ℝ}. In a different context though, a lattice means a partially ordered set (or poset) in which any two elements have a supremum (or join) and an infimum (or meet). Happily this dual meaning of lattice applies to Newton polygons: on the one hand, the integrality condition means that the vertices of a Newton polygon have integer coordinates. Thus an integral Newton polygon, viewed as an element of the Weyl chamber, is in some sense an integer point for some integral structure of the Weyl chamber. On the other hand, the poset 𝒩 is a disjoint union of connected components, indexed by the end point of the Newton polygons. Each connected component is a lattice as a poset. For any two integral Newton polygons in the same connected component of 𝒩, their join is again integral, although their meet may not. This is easy to see if we view Newton polygons as graphs. As we shall discuss next, the above properties for the group GL_{n} generalize to all reductive groups over p-adic fields. Incidentally, the poset of all integral Newton polygons in a given connected component of 𝒩 actually forms a lattice, but the “meet” operation is different from the restriction of the meet operation on 𝒩.

For a reductive group G over a p-adic field F, the set B(G) of σ-conjugacy classes was defined and investigated in [7]. The Newton map

ν̄_{G} : B(G) → 𝒩(G)

defined in [7] and [14] generalizes the classical notion of the Newton polygon associated to an F -isocrystal. The target 𝒩(G) can be identified with the set of all rational points of a Weyl chamber C of G which are fixed under the Galois group Γ = Gal(\overline{F} , F). In particular it is the set of all rational points of a rational polyhedral cone C_{F} in a finite dimensional vector space over ℝ. The inclusion relation between the convex hull of Weyl orbits gives a partial ordering on 𝒩(G).

Just as in the case for GL_{n}, only a discrete subset of 𝒩(G) can be actual Newton points, that is points in the image of ν̄_{G}. It is natural to think of these points as giving some suitable integral structure to the Newton cone 𝒩(G). We use both 𝒩(G)_{ℤ} and C_{F,ℤ} to denote the set of all Newton points. Contrary to what one may expect, these points are not the intersection of the Newton cone with a lattice in the underlying real vector space. (However this “naive” expectation is correct when one passes to the dual and consider the integral structure on the dual cone.) The effect is that there tend to be more “integral points” on a face than what the integral structure of the open Newton cone induces on that given face.

In [14] Rapoport and Richartz generalized Grothendieck’s theorem that the Newton polygon of an F -crystal goes down under specialization (according to our convention on Newton polygons). They further proved a congruence condition on the effect on Newton points under specialization, see [14, 3.11]. They also formulated a generalization of a conjecture of Grothendieck, to the effect that “these are the only restrictions”. Applying this to the reduction of a Shimura variety Sh(G, X), one obtains a subset 𝒩(G, X)_{ℤ} of 𝒩(G)_{ℤ} consisting of all Newton points which are expected to appear in the reduction of a Shimura variety Sh(G, X).

On a formal level this note is devoted to studying the combinatorial properties of the above posets, especially 𝒩(G, X)_{ℤ}. For simplicity we assume that G is quasisplit over F. In §2 we put together materials in [7] and [14], and make explicit the set 𝒩(G)_{ℤ} (resp. 𝒩(G, X)_{ℤ}) of all possible Newton points (resp. the subset of all Newton points that are expected to appear in the reduction of a Shimura variety Sh(G, X).) In fact for each ν ∈ C_{F,ℤ} = 𝒩(G)_{ℤ} we construct a subset C_{F,R^{∨}}^{ν} of C_{F,ℤ}, defined as the subset of all elements in 𝒩(G)_{ℤ} bounded above by ν and satisfies a congruence relation with ν. For a suitable ν_{0} ∈ C_{F,ℤ} ≅ 𝒩(G)_{ℤ}, C_{F,R^{∨}}^{ν_{0}} is the poset 𝒩(G, X)_{ℤ} in the previous paragraph.

Many properties of the poset 𝒩_{ℤ} hold for the posets C_{F,R^{∨}}^{ν}, especially 𝒩(G, X)_{ℤ}. We list some notables:

• Every finite subset of C_{F,R^{∨}}^{ν}, viewed as a subset of C_{F,ℤ} has a join (supremum) which lies in C_{F,R^{∨}}^{ν}.

• The poset 𝒩(G, X)_{ℤ} has a unique maximal element ν̄_{max} and a unique minimal element ν̄_{basic}. The maximal element ν̄_{max} is the expected Newton point for the generic stratum of the reduction of Sh(G, X); for 𝒜_{g} it is the stratum of ordinary abelian varieties. The minimal element ν̄_{basic} is the Newton point for the basic locus, also expected to appear; for 𝒜_{g} it is the locus of supersingular abelian varieties.

• The poset 𝒩(G, X)_{ℤ} consists of all projections of the Weyl orbit of the coweight μ associated to the Shimura variety to the faces of the relative Weyl chambers C_{F} for G. (This gives some supporting evidence for the conjecture of Grothendieck-Rapoport-Richartz from the philosophy of motives; see the Remark after Theorem 1 in §3 for a Hodge-theoretic interpretation of this fact.)

• For every ν ∈ C_{F,ℤ} the poset C_{F,R^{∨}}^{ν} is ranked (or catenary) in the sense that any two maximal chains in between two comparable elements have the same length.

• There is a simple formula for the length of any segment in C_{F,R^{∨}}^{ν}.

The above are the main results in this note; see §3 Theorems 1, 2, 3 for the precise statements. Some examples illustrating the poset 𝒩(G, X)_{ℤ} are given in §4. From the examples we also see that the properties listed above may not hold for the poset N(G)_{ℤ}

The fact that 𝒩(G, X)_{ℤ} is ranked supports the expectation that the codimension of the basic locus in the reduction of Sh(G, X) is equal to the length of 𝒩(G, X)_{ℤ}, that is the length of any maximal chain in 𝒩(G, X)_{ℤ}. (A chain x_{0} ≺ x_{1} ⋯ ≺ x_{k} has length k.) Going one step further, one may even expect that the codimension of a Newton strata of type ν̄ ∈ 𝒩(G, X)_{ℤ} in the reduction of a Shimura variety is equal to the length of the segment [ν̄, ν̄_{max}] in 𝒩(G, X)_{ℤ}; see the Question at the end of §3. This expectation is actually a consequence of the conjecture of Grothendieck-Rapoport-Richartz in view of the purity theorem in [6]; it has been verified for 𝒜_{g} in [6].

This note is inspired by [10] and by a question of F. Oort on the purity of the Newton strata, which has since been proved by him and J. de Jong in[6].

It is a great pleasure to thank him for stimulating discussions and continuing encouragement. My intellectual debt to Kottwitz, Rapoport and Richartz are all too obvious and gratefully acknowledged.

2 Lattice points on the Newton cone

Here is a (long) list of notations, most of them are standard in Lie theory. We generally follow those in [7], [8] and [14].

k  — an algebraically closed field of characteristic p > 0.
K  — the fraction field of the ring of p-adic Witt vectors W (k).
\overline{K}  — an algebraic closure of K.
F  — a finite extension of ℚ_{p} in \overline{K}.
L  — the compositum of K and F in \overline{K}.
σ  — the Frobenius automorphism of L/F.
Γ  — the Galois group of \overline{F} /F.
𝔻  — the proalgebraic torus with character group ℚ.
G  — a connected reductive group quasisplit over F.
B(G) — the set of all σ-conjugacy classes of G(L).
S  — a maximally F -split torus in G.
T  — a maximal torus of G over F which contains S, i.e. Z_{G}(S).
B  — a Borel subgroup of G over F which contains T.
X_{∗}(T) — cocharacters of T.
X^{∗}(T) — characters of T.
Φ  — Φ(G, T), the root system of T -roots of G.

Φ^{∨} — Φ^{∨}(G, T), the dual root system of Φ(G, T) consisting of

coroots.

Φ^{+} — Φ^{+}(G, T), the B-positive roots in Φ.
Φ_{F} — Φ(G, S), the relative root system of S-roots of G.
Φ_{F}^{∨} — Φ^{∨}(G, S), the relative dual root system of S-coroots of G.
Φ_{F}^{+} — the B-positive roots in Φ_{F}.
Δ  — the simple roots in Φ.
Δ^{∨} — the simple coroots in Φ^{∨}.
Δ_{F} — the simple roots in Φ_{F}.
Δ_{F}^{∨} — simple coroots in Φ_{F}^{∨}.
W  — the Weyl group of Φ(G, T).
W_{F} — the Weyl group of Φ_{F}.

C — the Weyl chamber in X_{∗}(T)_{ℝ}, with edges given by the fundamental coweights if G is semisimple. In general it is the inverse image of the Weyl chamber for G^{ad}.

C^{∨} — the obtuse Weyl chamber in X_{∗}(T)_{ℝ}, or the dual cone to C. It has the simple coroots as edges if G is semisimple. In general it is the image of the obtuse Weyl chamber for G^{der}.

C_{F} — the Weyl chamber in X_{∗}(S)_{ℝ}.
C_{F}^{∨} — the obtuse Weyl chamber in X_{∗}(S)_{ℝ}.

We assumed for simplicity that G is quasisplit over F. For applications we are mostly interested in the case when F = ℚ_{p}, because we will be dealing with the usual Frobenius-isocrystals when this holds, and also because the reductive groups attached to Shimura varieties are defined over ℚ.

Since the Borel subgroup B is F -rational, the action of the Galois group Γ action on X^{∗}(T) restricts to an action on Δ. Since G is quasisplit over F, the natural surjection X^{∗}(T) ↠ X^{∗}(S) induces a surjection Δ ↠ Δ_{F}, which identifies Δ_{F} as the set of Γ-orbits of Δ.

Recall that the Newton cone 𝒩(G) is defined to be

𝒩(G) = (Int G(L)\Hom_{L}(𝔻, G))^{⟨σ⟩} ≅ (X_{∗}(T)_{ℚ}/W)^{Γ}

Since G is quasisplit over F, one can also identify 𝒩(G) with X_{∗}(S)_{ℚ}/W_{F}, or with the set of all rational points in C_{F}. Thus ℕ(G) has a canonical structure as the set of all rational points in a simplicial cone. Its faces are indexed by subsets of Δ_{F}, or equivalently Γ-stable subsets of Δ. To be even more intrinsic, the indexing set is the G(F)-conjugacy classes of F -rational Levi subgroups of G, or equivalently the Γ-invariant G(\overline{F})-conjugacy classes of Levi subgroups of G. In the following we shall use the standard Levi subgroups.

Let θ_{F} be a subset of Δ_{F} and let θ be the corresponding subset of Δ. Let T_{θ} = ⋂_{α∈θ}(Ker χ_{α})^{0} be the largest subtorus of T killed by all characters χ_{α} with α ∈ θ, and let S_{θ_{F}} = (T_{θ} ∩ S)^{0}. Let M_{θ} = Z_{G}(T_{θ}) = Z_{G}(S_{θ_{F}}), the standard Levi subgroup indexed by θ; the torus T_{θ} is the neutral component of the center of M_{θ}. Reflections about the root hyperplanes in X_{∗}(T)_{ℝ} indexed by elements in θ generate a subgroup W_{θ} of W, which is canonically isomorphic to the Weyl group of M_{θ}.

The interior of C_{F} ∩ X_{∗}(S_{θ_{F}})_{ℝ} in X_{∗}(S_{θ_{F}})_{ℝ} is an open face of C_{F}; denote it by C_{F,θ}^{0}. The closed face C_{F} ∩ X_{∗}(S_{θ_{F}})_{ℝ} will be denoted by C_{F,θ}.

Definition 2.1. For each subset θ_{F} of Δ, we have a canonical projection

π_{θ_{F}} : X_{∗}(T)_{ℚ} ↠ X_{∗}(S_{θ_{F}})

This projection can be defined in several equivalent ways.

• The Galois group Γ operates on X_{∗}(T)_{ℚ} via a finite quotient; the subspace of fixed vectors is X_{∗}(S)_{ℚ}. This gives us a projection pr^{Γ} : X_{∗}(T)_{ℚ} ↠ X_{∗}(S)_{ℚ}. The finite reflection group W_{θ} also operated on X_{∗}(T)_{ℚ}, with X_{∗}(T_{θ})_{ℚ} as the subspace of fixed vectors. So we have a projection pr^{W_{θ}} : X_{∗}(T)_{ℚ} ↠ X_{∗}(T_{θ})_{ℚ}. Clearly X_{∗}(S_{θ_{F}})_{ℚ} = X_{∗}(S)_{ℚ}∩X_{∗}(T_{θ})_{ℚ}, and X_{∗}(T_{θ})_{ℚ} is stable under Γ since θ is. In fact the action of W_{θ} on X_{∗}(T)_{ℚ} is normalized by the action of Γ. We define π_{θ_{F}} to be

π_{θ_{F}} = pr^{Γ} ∘ pr^{W_{θ}} = ∫_{W_{θ}·Γ} .

• The ℚ-vector subspace of X_{∗}(T)_{ℚ} generated by coroots {α^{∨} | α ∈ θ} and {γ · β − β | β ∈ Δ, β ∉ θ} is a complement to X_{∗}(S_{θ_{F}})_{ℚ}. We define π_{θ_{F}} to be the projection to X_{∗}(S_{θ_{F}}) with respect to this direct sum decomposition.

• Choose and fix an admissible inner product (·|·) on X_{∗}(T)_{ℝ}, i.e. it is invariant under both Γ and W. Then we define π_{θ_{F}} to be the orthogonal projection to X_{∗}(S_{θ_{F}}) with respect to (·|·).

We collect some easy properties in the following lemmas. The proofs are omitted.

Lemma 1. (i) Let i : X_{∗}(S)_{ℚ} ↪ X_{∗}(T)_{ℚ} be the natural inclusion. Then i(C_{F}^{∨}) ⊆ C^{∨}, and i(C_{F}) ⊆ C. If G is semisimple, then the fundamental F -coweight (resp. simple F -coroot) in X_{∗}(S)_{ℚ} corresponding to a Γ-orbit is identified via i with the sum (resp. a positive multiple of the sum) of the fundamental coweights (resp. simple coroot) in X_{∗}(T)_{ℚ} attached to elements of the Γ-orbit.

(ii) The relative Weyl group W_{F} is canonically isomorphic to the normalizer subgroup of X_{∗}(S)_{ℚ} inside W, and it is also canonically isomorphic to N_{G}(T) ∩ N_{G}(S).

Lemma 2. Let θ_{F} ⊆ Δ_{F} and let π_{θ_{F}} : X_{∗}(T)_{ℝ} → X_{∗}(S)_{ℝ} be the projection as in Definition 2.1.

(i) The projection π_{θ_{F}} maps the Weyl chamber C onto the closed face C_{F,θ} of C_{F}.

(ii) Every coroot α^{∨} with α ∈ θ is projected to zero under π_{θ_{F}}. On the other hand, if β ∈ Δ − θ, then π_{θ_{F}} (β^{∨}) generates the edge corresponding to j(β) ∈ Δ_{F} − θ_{F} of the dual cone in X_{∗}(S_{θ_{F}})_{ℝ} of the face C_{F,θ} ⊂ X_{∗}(S_{θ_{F}})_{ℝ} with respect to the admissible inner product.

Recall that there is a natural partial ordering on the Weyl chamber C: For x, y ∈ C,

x ⪰ y ⇔ x − y ∈ C^{∨} .

One can also characterize this partial ordering more intrinsically by convexity: For x̄, ȳ ∈ X_{∗}(T)_{ℝ}/W, denote the corresponding Weyl orbits by W · x, W · y. Then x̄ ⪰ ȳ ⇔ the convex hull of W · x in X_{∗}(T)_{ℝ} contains the convex hull W · y in X_{∗}(T)_{ℝ}. Similarly for the relative Weyl chamber C_{F}. We refer the readers to [1, §12] for a thorough discussion of this partial ordering, where one can also find the closely related notion about convexity for compact Lie groups.

Lemma 3. Let θ_{F} be a subset of Δ_{F}, and let θ be the corresponding subset of Δ. Then for any x ∈ C, pr^{W_{θ}}(x) ⪯ x. Similarly for any y ∈ C_{F}, π_{θ_{F}} (x) ⪯ x.

Definition 2.2. (i) For each subset θ_{F} of Δ_{F}, let Λ_{θ_{F}} = π_{θ_{F}} (X_{∗}(T)) be the projection of the coweight lattice of X_{∗}(T) under π_{θ_{F}}, and let R_{θ_{F}}^{∨} be the projection of the coroot module R^{∨}(G, T) in X_{∗}(T)_{ℚ} under π_{θ_{F}}. Here R^{∨}(G, T) is the ℤ-submodule of X_{∗}(T) generated by Φ^{∨}. Clearly R_{θ_{F}} ⊆ Λ_{θ_{F}}, and Λ_{θ_{F}} is a lattice in X_{∗}(S_{θ})_{ℚ}, while R_{θ_{F}}^{∨} is a lattice in X_{∗}(S_{θ})_{ℚ} if Z(G)^{0} is an anisotropic over torusF.

(ii) Define C_{F,θ,ℤ}^{0} (resp. C_{F,θ,R^{∨}}^{0}) to be the intersection of Λ_{θ_{F}} (resp. R_{θ_{F}}^{∨}) with the open face C_{F,θ}^{0} of C_{F}. Similarly let C_{F,θ,ℤ} (resp. C_{F,θ,R^{∨}}) be the intersection of Λ_{θ_{F}} (resp. R_{θ_{F}}^{∨}) with the closed face C_{F,θ} of C_{F}.

(iii) Define C_{F,ℤ} (resp. C_{F,R^{∨}}) to be the disjoint union of all C_{F,θ,ℤ}^{0} (resp. C_{F,θ,R^{∨}}^{0}), with θ running over all subsets of Δ stable under Γ. Since 𝒩(G) is canonically isomorphic to the set of all rational points of C_{F}, C_{F,ℤ} (resp. C_{F,R^{∨}}) can be identified with a discrete subset of 𝒩(G); denote it by 𝒩(G)_{ℤ} (resp. 𝒩(G)_{R^{∨}}).

Remark. If we want to be more intrinsic, then we can define 𝒩(G)_{ℤ} (resp. 𝒩(G)_{R^{∨}}) to be the union of all points of 𝒩(G) represented by points of Λ_{θ_{F}} (resp. R_{θ_{F}}^{∨}), where θ_{F} runs through all subsets of Δ_{F}.

Proposition 1. Let ν̄_{G} : B(G) → 𝒩(G) be the Newton map as defined in [7, §4] and [14, 1.9], which gives every σ-conjugacy class b̄ ∈ B(G) its associated Newton point ν̄(b̄) ∈ 𝒩(G). Then Im(ν̄_{G}) = 𝒩(G)_{ℤ}.

Proof. Other than the definition of ν̄_{G}, this is an immediate consequence of Kottwitz’s description of B(G) in [7, §4]; see also [14, 1.18]. To elaborate a bit: Kottwitz’s theorem implies that

Im(ν̄_{G}) = ⋃_{θ} Im (π_{1}(M_{θ})_{Γ} → 𝒩(G))

where θ runs through all subsets of Δ stable under Γ. The map π_{1}(M_{θ})_{Γ} → 𝒩(G) factors through π_{1}(M_{θ}^{ab})_{ℚ}^{Γ}. Now π_{1}(M_{θ}) is the quotient of X_{∗}(T) by the subgroup generated by the coroots of M_{θ}. So its image in π_{1}(M_{θ}^{ab})_{ℚ}^{Γ} = X_{∗}(S_{θ_{F}})_{ℚ} is exactly equal to pr_{θ_{F}} (X_{∗}(T)).

Definition 2.3. For a give element ν ∈ 𝒞_{F,ℤ}, define

C_{F,ℤ}^{ν} = {x ∈ C_{F,ℤ} | x ⪯ ν}

C_{F,R^{∨}}^{ν} = ⋃_{θ_{F} ⊆Δ_{F}} {x ∈ C_{F,θ,ℤ}^{0} | x ⪯ ν, x − π_{θ_{F}} (ν) ∈ π_{θ_{F}} (R^{∨}(G, T))}

The corresponding subsets in 𝒩(G)_{ℤ} will be denoted by 𝒩(G)_{ℤ}^{ν} and 𝒩(G)_{ℤ,R^{∨}}^{ν} respectively.

Proposition 2. Let b_{1} ∈ B(G) be a σ-conjugacy class, ν̄(b_{1}) ∈ C_{F,ℤ} be the representative of the Newton point of b in C_{F}. Then C_{F,R^{∨}}^{ν̄(b_{1})} is equal to the image of

{b ∈ B(G)|ν̄(b) ⪯ ν̄(b_{1}), γ(b) = γ(b_{1})}

under the Newton map ν̄_{G} : B(G) → 𝒩(G)_{ℤ} ≅ C_{F,ℤ}. Here γ : 𝒩(G) → π_{1}(G)_{Γ} is the map defined in [9, 6.1] and [14, 1.15].

Proof. This is essentially a tautology. First notice that the projection

pr_{θ_{F}} : X_{∗}(T) → X_{∗}(S_{θ_{F}})_{ℚ}

and

X_{∗}(T) → π_{1}(G)_{Γ} ≅ X_{∗}(T)/(R^{∨} + I_{Γ} · X_{∗}(T))

both factor through X_{∗}(T)/ (R^{∨}(M_{θ_{F}}) + I_{Γ} · X_{∗}(T)). Here R^{∨}(M_{θ_{F}}) denotes the ℤ-module generated by the coroots of M_{θ}, and I_{Γ} denotes the augmentation ideal in the integer group ring of Γ. Then one uses the fact that a point x ∈ X_{∗}(S_{θ_{F}})_{ℚ} is the image of a point of R^{∨}(G, T) if and only if ⟨χ, x⟩ ∈ ℤ for all characters χ ∈ X^{∗}(T) orthogonal to R^{∨}(M_{θ_{F}}) + I_{Γ} · X_{∗}(T).

The generalization [14, 3.6] of Grothendieck’s specialization theorem for F -crystals by Rapoport and Richartz says that for a family ℳ of F -isocrystal with G-structure whose generic Newton point is ν for some element ν ∈ C_{F,ℤ}, all Newton points of ℳ are elements of C_{F,R^{∨}}. In the situation of a Shimura variety Sh(G, X), X corresponds to a W -orbit W μ, where μ is a dominant minuscule coweight. The expected maximal Newton point is the Γ-average μ^{♮} = pr^{Γ}(μ) of μ, and the predicted set of all Newton points that appears in the reduction of Sh(G, X) is C_{F,R^{∨}}^{μ^{♮}}.

3 Properties of the Newton lattice

Theorem 1. Let μ be a minuscule dominant coweight, that is ⟨α, μ⟩ ∈ {0, 1} for each root α ∈ Φ^{+}. Let μ^{♮} ∈ C_{F,ℤ} be the projection of μ to C_{F}, that is the average of Γ · μ. Then every y ∈ C_{F,R^{∨}}^{μ^{♮}} is the projection of some element of the Weyl orbit W · μ under π_{θ_{F}} for a suitable subset θ_{F} ⊆ Δ_{F}.

Proof. We imitate the technique in [4, VIII §7 n^{∘} 2]. Suppose that y = π_{θ_{F}} (ν) ∈ C_{F,θ,ℤ}^{0}, where ν is a coweight. That y ∈ C_{F,θ}^{0} means that (ν | α^{∨}) ≥ 0 for all α ∈ Δ such that α|_{S_{θ}} ≠ trivial; that is α ∉ θ. Since y ∈ C_{F,R^{∨}}^{μ^{♮}}, we have

π_{θ_{F}} (μ) = π_{θ_{F}} (ν) + ∑_{α_{i}∈θ} c_{α_{i}} π_{θ_{F}} (α_{i}^{∨}) , c_{α_{i}} ∈ ℤ_{≥0}

for suitable nonnegative integers c_{α_{i}}. Moreover we may and do assume that not all c_{α_{i}} are 0. Then

(μ | ∑_{i} c_{α_{i}} α_{i}^{∨}) = (ν | ∑_{i} c_{α_{i}} α_{i}^{∨}) + (∑_{i} c_{α_{i}} α_{i}^{∨}| ∑_{i} c_{α_{i}} α_{i}^{∨}) > 0 .

Hence there exists an i_{0} such that (μ | α_{i_{0}}^{∨}) > 0 and c_{α_{i_{0}}} > 0. Because μ is minuscule, W · μ is saturated with respect to Φ^{∨}, see [4, VIII §7 n^{∘} 3]. So there exists an element w_{0} ∈ W such that w_{0}μ = μ − α_{i_{0}}. Thus we have

π_{θ_{F}} (w_{0}μ) = π_{θ_{F}} (ν) + (c_{α_{i_{0}}} − 1) π_{θ_{F}} (α_{i_{0}}^{∨}) + ∑_{α_{i}∈θ,i≠i_{0}} c_{α_{i}} π_{θ_{F}} (α_{i}^{∨}) .

Now we are in a position to repeat the same argument above, with μ replaced by w_{0} μ, and the sum of the c_{i}’s is decreased by one. Induction finishes the proof.

Remark. In the situation of a Shimura variety Sh(G, X), the group G is defined over ℚ and X corresponds to a minuscule dominant T -coweight μ of G. This is exactly the situation considered in the above with F = ℚ_{p}. Suppose now that we have a special point h_{1} : 𝕊 → G in X, where 𝕊 = Res_{ℂ/ℝ}(𝔾_{m}). Let μ_{1} : 𝔾_{m} → G be the Hodge coweight corresponding to h_{1}. Let H be the smallest ℚ_{p}-subtorus of G containing μ_{1}, and let M be the centralizer in G of the maximal ℚ_{p}-split subtorus of H. So the image of μ_{1} in the fundamental group of M gives a basic element b_{M,1} ∈ B(M)_{b} ≅ π_{1}(M), which maps to an element of b_{1} ∈ B(G). The Newton point ν̄_{M}(b_{M,1}) is the image of μ_{1} in π_{1}(M)^{Γ} ⊗ ℚ, also equal to the Γ-average of μ_{1}. The Levi subgroup M is conjugate to a standard Levi subgroup M_{θ_{F}} of G; then μ_{1} corresponds to an element in the Weyl-orbit W μ. So the Newton point ν̄_{G}(b_{1}) of b_{1} is represented by an element of C_{F,R^{∨}}^{μ^{♮}}, where μ^{♮} is the Γ-average of μ. Of course b_{1} is expected to be the F -crystal with G-structure of the reduction of a CM-motive attached to the reduction of a point in Sh(G, X) represented by a point in X × G(𝔸_{f}) with X-component μ_{1}. Since one expects that every point of the reduction of Sh(G, X) defined over a finite field can be lifted to a CM-point of Sh(G, X), the subset of C_{F,ℤ} consisting of all projections of elements in W · μ is the set of all Newton points which are expected to occur from the “philosophy” of motives, especially Hodge theory. Theorem 1 says that this prediction coincides with that of the generalized Grothendieck conjecture in [14, §3].

The focus of the rest of this section is to show that C_{F,R^{∨}}^{ν} is ranked as a poset for any ν ∈ C_{F,ℤ}. This will follow from the existence of the join (or supremum) of finite subsets of C_{F} over a point of X_{∗}(G^{ab})_{ℝ} and the general properties of the join operation. Although we are primarily interested in cones coming from a root system, our proof requires a more general setting for the induction to go through. The following paragraph establishes the necessary notations.

Let V be a finite dimensional vector space over ℝ, and let (· | ·) be an inner product on V. Let e_{1}, …, e_{l} be a basis of V such that (e_{i} | e_{j}) ≤ 0 ∀i ≠ j. Define e_{1}^{∨}, …, e_{l}^{∨} to be the basis of V such that (e_{i}^{∨}|e_{j}) = δ_{i,j} ∀i, j. We have (e_{i}^{∨}|e_{j}^{∨}) ≥ 0 ∀i, j by [4, V §3 n^{o}5 Lemma 6]. Let C be the closed cone spanned by e_{1}^{∨}, …, e_{l}^{∨}, i.e.

C = {x ∈ V | (x|e_{i}) ≥ 0 ∀i = 1, …, l} .

For a while the notations in §2 are temporarily suspended, so C is not to be confused with the absolute Weyl chamber here. We define a partial ordering on C by

x ⪰ y iff (e_{i}^{∨} | x) ≥ (e_{i}^{∨} | y) ∀i = 1, …, l

For each i = 1, …, l, let pr_{(i)} : V → ℝ e_{i}^{∨} be the orthogonal projection to ℝ e_{i}^{∨}. For each x ∈ C, let

J(x) = {j | (x|e_{i}) > 0}

For a subset J ⊆ {1, …,; }, let

C_{(J)}^{0} = ∑_{j∈J} ℝ_{>0} e_{j}^{∨} = {x ∈ C | J(x) = J}

The closure of C_{(J)}^{0} will be denoted by C_{(J)}.

Lemma 4. Let J be a subset of {1, …, l}, and let pr_{(J)} : V → ∑_{j∈J} ℝ e_{j}^{∨} be the orthogonal projection from V to ∑_{j∈J} ℝ e_{j}^{∨}.

(i) The projections pr_{J}(e_{j}) of e_{j}, j ∈ J, form a basis of ∑_{j∈J} ℝ e_{j}^{∨} such that

( pr_{(J)}(e_{j_{1}}) | pr_{(J)}(e_{j_{1}}) ) ≤ 0 ∀j_{1} ≠ j_{2}, j_{1}, j_{2} ∈ J

(ii) For each i, pr_{(J)}(e_{i}^{∨}) ∈ C_{(J)}. Consequently pr_{(J)}(C) = C_{(J)}.

(iii) The partial orderting ⪰ on C_{(J)} defined by the its obtuse dual basis {pr_{(J)}(e_{j}) | j ∈ J} is equal to the restriction to C_{(J)} of the partial ordering on C.

(iv) We have x ⪰ pr_{(J)}(x) for each x ∈ C.

Proof. It suffices to prove this for the case Card(J) = l − 1; the general case follows by induction. We may and do assume that J = {1, …, l − 1}, and we abbreviate pr_{(J)} by pr. Clearly (pr(e_{j_{1}})|e_{j_{2}}^{∨}) = δ_{j_{1},j_{2}} for all j_{1}, j_{2} ∈ J with j_{1} ≠ j_{2}. Hence pr(e_{1}), ⋯, pr(e_{l−1}) form a basis of ∑_{j=1}^{l−1} ℝ e_{j}^{∨}, dual to e_{1}^{∨}, …, e_{j−1}^{∨}. For each x ∈ C, we have

pr(x) = x − \frac{(x|e_{l})}{(e_{l}|e_{l})}e_{l}

Statement (iv) is immediate from this formula. We use this formula to compute

(pr(e_{1})|pr(e_{2})) = (e_{1}|e_{2}) − \frac{(e_{1}|e_{l}) (e_{2}|e_{l})}{(e_{l}|e_{l})} ≤ (e_{1}|e_{2}) ≤ 0

Similarly for any other pair of elements of J. This proves (i). We have

(pr(x)|e_{i}) = (x|e_{i}) − \frac{(x|e_{l}) (e_{l}|e_{i})}{(e_{l}|e_{l})} ≥ (x|e_{i}) ≥ 0 ∀i = 1, …, l − 1, ∀x ∈ C

which proves (ii). Finally statement (iii) is a consequence of (ii), in fact equivalent to it.

For each x ∈ C, let

C_{⪰x} = { y ∈ C | y ⪰ x } .

More generally for a subset E ⊆ C, let

C_{⪰E} = { y ∈ C | y ⪰ x ∀x ∈ E } .

For x ∈ C, let

E(x) = { pr_{(j)}(x) | j = 1, …, l }
E_{red}(x) = {pr_{(j)}(x) | (x|e_{j}) > 0 }

Lemma 5. Let x be a point of C.

(i) We have C_{⪰x} = C_{⪰E(x)} = C_{⪰E_{red}(x)}.

(ii) For each e ∈ E_{red}(x), we have C_{⪰E(x)} ⫋ C_{⪰E_{red}(x)−{e}}.

Proof. (i) Clearly C_{⪰x} ⊆ C_{⪰E(x)} ⊆ C_{⪰E_{red}(x)}. Assume now that z ∈ C, z ⪰ pr_{(j)}(x) for all j ∈ J(x). Then z_{1} = pr_{(J(x))}(x) is an element of C_{(J(x))}, and z_{1} ⪰ x by Lemma 4 (iii) Since z ⪰ z_{1} by Lemma 4 (iv), we conclude that z ⪰ x for all z ∈ C_{⪰E_{red}(x)}.

(ii) Suppose that e = pr_{(j)}(x), j ∈ J(x). Let x_{1} = x − \frac{(x|e_{j})}{(e_{j}|e_{j})}e_{j} be the orthogonal projection of x to the subspace of V spanned by E_{red}(x) − {e}. Then x_{1} ∈ C_{⪰E_{red}(x)−{e}} and x_{1} ∉ C_{⪰E(x)}.

Definition 3.1. (i) A subset E of C is an edge subset if it is contained in the union of all edges ℝ_{≥0} e_{i}^{∨} of C, i = 1, …, l.

(ii) An edge subset E ⊆ C is reduced if C_{⪰E} ⫋ C_{E−{e}} for each e ∈ E. (iii) It is easy to see that if E is a reduced edge subset of C, then there

exists a unique subset J(E) ⊂ {1, …, l} and elements v_{j} ∈ ℝ_{>0} e_{j}^{∨} such that E = {v_{j} | j ∈ J(E)}.

Remark. (i) Lemma 5 (i) shows that every element x ∈ C is the join of the finite edge subset E(x), and is also the join of the reduced edge subset E_{red}(x).

(ii) More generally, for every finite subset A of C, there exists a finite edge subset E of C such that C_{⪰A} = C_{⪰E}.

(iii) Trivially, for every finite edge subset E of C, there exists a reduced edge subset E′ ⊂ E such that C_{⪰E} = C_{⪰E′}.

Theorem 2. Let E = {v_{j} | j ∈ J(E)} be a reduced edge subset of C. Then C_{⪰E} has a unique minimal element sup E ∈ C with C_{⪰E} = C_{⪰sup E}. Moreover sup E is the unique element x ∈ C_{J}^{0}(E) such that

(x|e_{j}^{∨}) = (v_{j}|e_{j}^{∨}) ∀j ∈ J(E) .

Proof. We shall prove this theorem by induction on l = dim(V); the case l = 1 is trivial. Assume by induction that the statement is true for dim(V) < l.

Suppose that Card(E) = m < l, say J = J(E) = {1, …, m}. Consider the cone C_{(J)} in ℝ e_{1}^{∨} + ⋯ + ℝ e_{m}^{∨}. The dual cone of C_{(J)} in ℝ e_{1}^{∨} + ⋯ + ℝ e_{m}^{∨} is generated by pr_{(J)}(e_{1}), …, pr_{(J)}(e_{m}), and (pr_{(J)}(e_{i}) | pr_{(J)}(e_{j})) ≤ 0 for all 1 ≤ i, j ≤ m with i ≠ j, by Lemma 4 (i). Hence E is a reduced edge subset of C_{(J)}. By induction there exists an element x ∈ C_{J}^{0} with

(x|e_{j}^{∨}) = (v_{j}|e_{j}^{∨}) ∀j ∈ J

which is the minimal element of

{y ∈ C_{J} | y ⪰ e ∀e ∈ E} .

If z is an element of C such that z ⪰ e for all e ∈ E. Consider the element z_{1} = pr_{J}(z). Then z_{1} ⪰ e for each e ∈ E since (z_{1} | e_{j}^{∨}) = (z | e_{j}^{∨}) for all j ∈ J. Hence z ⪰ z_{1} ⪰ x. So x is also the unique minimal element of C_{⪰E}.

Assume now that #E = l. Let x_{E} be the element of V such that (x_{E} | e_{i}^{∨}) = (v_{i} | e_{i}^{∨}) for all i = 1, …, l. We have to show that x_{E} lies in the open cone C^{0}, i.e. (x_{E} | e_{i}) > 0 for any i = 1, …, l. By symmetry we may assume that i = 1. It is not too hard to check that the edge subset {v_{2}, …, v_{n}} is reduced: If not, there exists a proper subset, say {v_{3}, …, v_{n}}, such that sup{v_{3}, …, v_{n}} = sup{v_{2}, …, v_{n}}. Here the supremum is taken in the face C_{{2,... ,n}}, and exists by the induction hypothesis; by Lemma 4 (iv) it is equal to the supremum taken in C. So C_{⪰E−{v_{2}}} = C_{⪰E} and E is not reduced, a contradiction.

Consider the element y = sup{v_{2}, …, v_{l}} ∈ C_{{2,... ,l}}^{0}, which exists by induction and satisfies (y | e_{j}^{∨}) = (v_{j} | e_{j}^{∨}) for j = 2, …, l. Therefore there exists an element a ∈ ℝ such that x_{E} = y + a e_{1}. Since E is reduced, the inequality (pr_{1}(y) | e_{1}^{∨}) < (v_{1} | e_{1}^{∨}) holds; otherwise we would have C_{⪰E} = C_{⪰E−{v_{1}}}. So

a = (x_{E} | e_{1}^{∨}) − (y | e_{1}^{∨}) = (x_{E} | e_{1}^{∨}) − (pr_{1}(y) | e_{1}^{∨}) > 0

and therefore (x_{E} | e_{1}) = a (e_{1} | e_{1}) > 0.

Corollary 1. If E is a reduced edge subset of C, then J(sup E) = J(E), and E = E_{red}(sup E). Conversely, for any x ∈ C, E_{red}(x) is a reduced edge suset of C, J(x) = J(E_{red}(x)), and x = sup E_{red}(x).

We now resume the notation in §2. Since the canonical homomorphism G ↠ G^{ad} = G/Z(G) induces an isomorphism C_{F,R^{∨}}^{ν}(G) ⟶^{∼} C_{F,R^{∨}}^{ν′}(G^{ad}) of posets, where ν′ is the image of ν in C_{F} (G^{ad}). We may and do assume without loss of generality that G is semisimple, or even that G is adjoint.

Now we apply the above results to the relative Weyl chamber C_{F}. The above corollary says that elements of C_{F} are in one-to-one correspondence with reduced edge subsets of C_{F}. Let ω_{F,1}^{∨}, …, ω_{F,l}^{∨} be the fundamental F - coweights; similarly let ω_{F,1}, …, ω_{F,l} be the fundamental F -weights. Let ν be an element of C_{F,ℤ}. Suppose that x ∈ C_{F} and x ⪯ ν, then x belongs to C_{F,R^{∨}}^{ν} if and only if

⟨ω_{F,j}, v_{j}⟩ ≡ ⟨ω_{F,j}, ν⟩ (mod ℤ) ∀v_{j} ∈ E_{red}(x) , v_{j} ∈ ℝ_{>0} ω_{F,j}^{∨}

Equivalently, an element x ∈ C_{F} is in C_{F,R^{∨}}^{ν} if and only if its reduced edge subset E_{red}(x) is a subset of C_{F,R^{∨}}^{ν}.

Notation 3.2. Let ν be an element of C_{F,ℤ}. For any element x ∈ C_{F}, let

\begin{aligned}E_{F,R^{∨},i}^{ν}(x) & = & sup \{ y ∈ C_{F,R^{∨}}^{ν} ∩ ℝ_{>0} ω_{i}^{∨} | y ⪯ x \} i = 1, …, l \\ E_{F,R^{∨}}^{ν}(x) & = & \{ E_{F,R^{∨},i}^{ν} | i = 1, …, l \} \end{aligned}

Lemma 6. (i) For any element x ∈ C_{F,R^{∨}}^{ν}, we have x = sup E_{F,R^{∨}}^{ν}(x). (ii) If E ⊆ C_{F,R^{∨}}^{ν}, then sup E ∈ C_{F,R^{∨}}^{ν} as well.

Proof. (i) Since x ∈ C_{F,R^{∨}}^{ν}, we have E_{red}(x) ⊆ C_{F,R^{∨}}^{ν}. We have

x = sup E_{red}(x) ⪯ sup E_{F,R^{∨}}^{ν}(x) ⪯ x

and the equality follows.

(ii) By (i), we can find a finite edge subset E_{1} consisting of element in C_{F,R^{∨}}^{ν} such that sup E = sup E_{1}. Let E_{2} be a subset of E_{1} which is reduced. Then sup E = sup E_{2} ∈ C_{F,R^{∨}}^{ν} by the discussion after Corollary 1.

Theorem 3. (i) Let y, z be elements of C_{F,R^{∨}}^{ν} with y ≺ z, and suppose that # (E_{F,R^{∨}}^{ν}(z) − E_{F,R^{∨}}^{ν}(y)) ≥ 2. Then length_{C_{F,R^{∨}}^{ν}}([y, z]) ≥ 2; i.e. there exists an element x ∈ C_{F,R^{∨}}^{ν} strictly between y and z. Here

[y, z] = {x ∈ C_{F,R^{∨}}^{ν} | y ⪯ x ⪯ z}

and length_{C_{F,R^{∨}}^{ν}}([y, z]) is the length of the seqment [y, z] ⊆ C_{F,R^{∨}}^{ν}.

(ii) The poset C_{F,R^{∨}}^{ν} is ranked. In other words every maximal chain between two comparable elements have the same length.

(iii) Let y, z be as in (i). Then

\begin{matrix}length_{C_{F,R^{∨}}^{ν}}([y, z]) = # (E_{F,R^{∨}}^{ν}(z) − E_{F,R^{∨}}^{ν}(y)) \\ = ∑_{i=1}^{l} (⌈⟨ω_{F,i}, ν⟩ − ⟨ω_{F,i}, y⟩⌉ − ⌈⟨ω_{F,i}, ν⟩ − ⟨ω_{F,i}, z⟩⌉) \end{matrix}

Especially

length_{C_{F,R^{∨}}^{ν}}([x, ν]) = ∑_{i=1}^{l} ⌈⟨ω_{F,i}, ν⟩ − ⟨ω_{F,i}, x⟩⌉

for any x ∈ C_{F,R^{∨}}^{ν}.

Proof. Statements (ii) and (iii) both follow from (i). It remains to prove (i). Since y ≺ z and y ≠ z, E_{red}(y) ⫋ E_{red}(z). Let e be an element of E_{red}(z) − E_{red}(y), and suppose that e ∈ ℝ_{>0} ω_{j_{1}}^{∨}. We know that E_{F,R^{∨}}^{ν}(y) ⫋ E_{F,R^{∨}}^{ν}(z)−{e}. Let E_{1} be a reduced edge subset contained in E_{F,R^{∨}}^{ν}(z)−{e}. Consider the element x = sup(E_{F,R^{∨}}^{ν}(z) − {e}) in C_{F,R^{∨}}^{ν}. Clearly y ⪯ x ⪯ z. It is clear that y ≠ x since E_{F,R^{∨}}^{ν}(x) ⊇ E_{F,R^{∨}}^{ν}(z) − {e} ⫌ y. Moreover x ≠ z since e ∉ E_{1} = E_{red}(x). This finishes the proof.

Let (G, X) be a Shimura data, especially G is a conneced reductive algebraic group over ℚ and X is a G(ℝ)-conjugacy class of ℝ-homomorphism from Res_{ℂ/ℝ} 𝔾_{m} to G. Assume that G is quasisplit over ℚ_{p} and splits over an unramified extension of ℚ_{p}. Let S be a maximally ℚ_{p}-split subtorus in G, let T ba a maximal torus of G containing S. Let μ be a dominant coweight of T with respect to a ℚ_{p}-rational Borel subgroup B of G with B ⊇ T, such that the G-conjugacy class of μ corresponds to X. Let μ^{♮} be the Gal(\overline{ℚ_{p}}/ℚ_{p})-average of μ, and let M_{1} = Z_{G}(μ^{♮}). Let b_{M_{1}} be the basic element in B(M_{1})_{basic} ≅ π_{1}(M_{1}) which corresponds to the image of μ in π_{1}(M_{1}). Denote by b_{1} ∈ B(G) the image of b_{M_{1}} under the canonical map B(M_{1}) → B(G). Notice that the Newton point ν̄(b_{1}) of b_{1} is represented by μ^{♮}. Let b_{0} ∈ B(G)_{basic} ≅ π_{1}(G) be the basic element in B(G) corresponding to the image of μ in π_{1}(G).

Let K_{p} be a hyperspecial maximal compact subgroup of G(ℚ_{p}), and let Sh_{K_{p}}(G, X) be the tower of Shimura varieties attached to these data. Assume furthermore that Sh_{K_{p}}(G, X) has good reduction at a place v of the Shimura reflex field E(G, X) over p. Let 𝒮_{basic} be the locus in the reduction of Sh_{K_{p}}(G, X) at v consisting of points with type b_{0} ∈ B(G), cf. [14, §3]. More generally, for each b ∈ B(G) such that b ⪯ b_{1} and γ(b) = γ(b_{1}), let 𝒮_{b} be the stratum of the reduction of Sh_{K_{p}}(G, X) at v consisting of points with type b. Here we have used the notation in [14]. Let ν̄(b) denote the representative of the Newton point of b in C_{ℚ_{p},ℤ}.

Question. (i) Is the codimension of 𝒮_{basic} in the reduction of the Shimura variety Sh_{K_{p}}(G, X) equal to length (C_{ℚ_{p},R^{∨}}^{μ^{♮}})?

(ii) More generally, suppose that b is an element of B(G) such that b ≤ b_{1} and γ(b) = γ(b_{1}). Is codim (𝒮_{b}) equal to length_{C_{ℚp,R^{∨}}^{μ^{♮}}} ([ν̄(b), μ^{♮}])?

Remark. (i) The answer to question (i) (resp. (ii)) is “yes” in the Siegel case according to [10] (resp. [6]).

(ii) It is expected that the generic stratum of the reduction of the Shimura variety Sh_{K_{p}}(G, X) at v has type b_{1}.

(iii) The stratum with type b_{1} generalizes the ordinary locus in the moduli space of principally polarized varieties. It has long been observed that abelian varieties of dimension g with f-rank g − 1 share many desirable properties with ordinary abelian varieties of dimension g. They are called “almost ordinary” abelian varieties by some authors. An analog of “almost ordinary” type in the present setting in terms of the Newton points exists when μ^{♮} is an edge element: If this is the case, then there exists a unique maximal element in C_{ℚ_{p},R^{∨}}^{μ^{♮}} − {μ^{♮}}, namely sup (E_{F,R^{∨}}^{μ^{♮}}(μ^{♮}) − {μ^{♮}}). This occurs often; for instance when G is absolutely simple, or when G is ℚ-simple and the reflex field is equal to ℚ.

(iv) Along the same train of thought, one may ask whether there exists a unique minimal element in C_{ℚ_{p},R^{∨}}^{μ^{♮}}−{0}. The answer is yes in some situations, for instance when G is absolutely simple of type B or C in the Dynkin classification. But the answer is no in many cases, for instance when G is absolutely simple of type A.

4 Examples

4.1 split A_{1}

Let ω^{∨} be the fundamental coweight, and let α^{∨} = 2ω^{∨} be the simple coroot. Then C_{ℤ} = C_{F,ℤ} = ℤ_{≥0} · ω^{∨}. For any positive integer d > 0,

C_{F,R^{∨}}^{d·ω^{∨}} = {a · ω^{∨}|0 < a ≤ d, a ≡ d (mod 2)} ∪ {0}

4.2 splict C_{2}, C_{3}, C_{4}

In the next several examples, we consider the split C_{n} case for n = 2, 3, 4. We use the standard notation for the coroots and coweights for C_{n}. The fundamental coweights are

ω_{1}^{∨} = e_{1}, ω_{2}^{∨} = e_{2}, …, ω_{n−1}^{∨} = e_{1} + ⋯ + e_{n−1}, ω_{n}^{∨} = \frac{1}{2}(e_{1} + ⋯ + e_{n})

and the simple coroots are

α_{1}^{∨} = e_{1} − e_{2}, …, α_{n−1}^{∨} = e_{n−1} − e_{n}, α_{n} = e_{n}

where e_{1}, …, e_{n} is an orthonormal basis for an admissible inner product on X_{∗}(T)_{ℝ}. Let x_{1}, …, x_{n} denote the coordinate functions with respect to this basis. Let μ = ω_{n}^{∨} be the fundamental coweight corresponding to the unique special simple root α_{n}. Given an element ν ∈ C_{ℤ}, the sequence

⟨x_{1}, ν⟩ + \frac{1}{2}, …, ⟨x_{n}, ν⟩ + \frac{1}{2}, −⟨x_{n}, ν⟩ + \frac{1}{2}, … − ⟨x_{1}, ν⟩ + \frac{1}{2}

will be called the slopes of ν. For an element of ν̃ of C(GSp_{2n})_{ℤ} which lifts ν, this is the slopes sequence of ν̃ under the standard representation of GSp_{2n} up to a shift by a constant. The choice of \frac{1}{2} makes the above slopes sequence compatible with the case of abelian varieties.

In each case among C_{2}, C_{3}, C_{4}, we shall give a table which lists the elements of C_{F,ℤ}^{μ} in its first column. The second column gives the corresponding slopes. The third column gives an element of the Weyl orbit W · μ which projects to the element of C_{F,ℤ}^{μ} in question, if one exists.

4.2.1 split C_{2}

x ∈ C_{ℤ}^{μ} slopes of x y ∈ W · μ with π_{θ_{F}} (y) = x
μ = ω_{2}^{∨}  1,1,0,0  μ
\frac{1}{2}ω_{2}^{∨}  1, \frac{1}{2} , \frac{1}{2} ,0  μ
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. in W · μ

We have

μ ≻ \frac{1}{2}ω_{2}^{∨} ≻ 0

In the present case C_{F,R^{∨}}^{μ} coincides with C_{F,ℤ}^{μ}.

4.2.2 split C_{3}

x ∈ C_{ℤ}^{μ} slopes of x y ∈ W · μ with π_{θ_{F}} (y) = x
μ = ω_{3}^{∨}  1,1,1,0,0,0  μ
\frac{1}{2}ω_{2}^{∨}  1,1, \frac{1}{2} , \frac{1}{2} ,0,0  μ
\frac{1}{2}ω_{1}^{∨}  1, \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} ,0  μ
\frac{2}{3}ω_{3}^{∨}  \frac{5}{6} , \frac{5}{6} , \frac{5}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6}  does not exist
\frac{1}{3}ω_{3}^{∨}  \frac{2}{3} , \frac{2}{3} , \frac{2}{3} , \frac{1}{3} , \frac{1}{3} , \frac{1}{3}  \frac{1}{2}(e_{1} + e_{2} − e_{3})
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. in W · μ

We have

μ ≻ \frac{1}{2}ω_{2}^{∨} {≻ \frac{1}{2}ω_{1}^{∨} \atop ≻ \frac{2}{3}ω_{3}^{∨}} ≻ \frac{1}{3}ω_{3}^{∨} ≻ 0

Except \frac{2}{3}ω_{3}^{∨}, all other elements of C_{F,ℤ}^{μ} are in C_{F,R^{∨}}^{μ}. Clearly the Newton point \frac{2}{3}ω_{3}^{∨} does not appear in the moduli space 𝒜_{3} of principally polarized abelian threefolds (when F = ℚ_{p}), since the denominator (6) of the two slopes does not divide the multiplicity (3).

4.2.3 split C_{4}

x ∈ C_{ℤ}^{μ}  slopes of x y ∈ W · μ with π_{θ_{F}} (y) = x
μ = ω_{4}^{∨}  1,1,1,1,0,0,0,0  μ
\frac{1}{2}ω_{3}^{∨}  1,1,1, \frac{1}{2} , \frac{1}{2} ,0,0,0  μ
\frac{1}{2}ω_{2}^{∨}  1,1, \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} ,0,0  μ
\frac{1}{3}ω_{1}^{∨} + \frac{1}{3}ω_{4}^{∨} 1, \frac{2}{3} , \frac{2}{3} , \frac{2}{3} , \frac{1}{3} , \frac{1}{3} , \frac{1}{3} ,0  \frac{1}{2}(e_{1} − e_{2} + e_{3} + e_{4})
\frac{1}{2}ω_{1}^{∨}  1, \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} ,0  μ
\frac{1}{2}ω_{4}^{∨}  \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4}  \frac{1}{2}(e_{1} − e_{2} + e_{3} + e_{4})
\frac{1}{6}ω_{3}^{∨}  \frac{2}{3} , \frac{2}{3} , \frac{2}{3} , \frac{1}{2} , \frac{1}{2} , \frac{1}{3} , \frac{1}{3} , \frac{1}{3}  \frac{1}{2}(e_{1} − e_{2} + e_{3} − e_{4})
\frac{1}{3}ω_{3}^{∨}  \frac{5}{6} , \frac{5}{6} , \frac{5}{6} , \frac{1}{2} , \frac{1}{2} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6}  does not exist
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. of W · μ

The partial ordering of C_{F,ℤ}^{μ} is as follows:

μ ≻ \frac{1}{2}ω_{3}^{∨} ≻ \frac{1}{2}ω_{2}^{∨} ≻ \frac{1}{3}ω_{1}^{∨} + \frac{1}{3}ω_{4}^{∨} {≻ \frac{1}{2}ω_{1}^{∨} \atop ≻ \frac{1}{3}ω_{3}^{∨} ≻ \frac{1}{2}ω_{4}^{∨}} ≻ \frac{1}{6}ω_{3}^{∨} ≻ 0

The element \frac{1}{3}ω_{3}^{∨} is not in C_{F,R^{∨}}^{μ}; all others are. Notice that C_{F,ℤ}^{μ} is not ranked as a partially ordered set: Both \frac{1}{3}ω_{1}^{∨} + \frac{1}{3}ω_{4}^{∨} ≻ \frac{1}{2}ω_{1}^{∨} ≻ \frac{1}{6}ω_{3}^{∨} and \frac{1}{3}ω_{1}^{∨} + \frac{1}{3}ω_{4}^{∨} ≻ \frac{1}{3}ω_{3}^{∨} ≻ \frac{1}{2}ω_{4}^{∨} ≻ \frac{1}{6}ω_{3}^{∨} are maximal chains between \frac{1}{3}ω_{1}^{∨} + \frac{1}{3}ω_{4}^{∨} and \frac{1}{6}ω_{3}^{∨}. But C_{F,R^{∨}}^{μ} is ranked, once the offending element \frac{1}{3}ω_{3}^{∨} is removed from C_{F,ℤ}^{μ}.

4.3 Res_{K/F} P GL_{2}, K/F a finite field extension

Let g = [K : F]. Let ω^{∨} be the fundamental F -coweight, and α^{∨} = 2ω^{∨} be the simple F -coroot. The simple coroots are of the form α_{i}^{∨}, indexed by i ∈ Hom_{F -alg}(K, \overline{F}). Similarly the fundamental coroots are of the form ω_{i}^{∨}, i ∈ Hom_{F -alg}(K, \overline{F}). We have α^{∨} = ∑_{i} α_{i}^{∨}, ω^{∨} = ∑_{i} ω_{i}^{∨}. We take μ to be μ = ω^{∨}. Then

C_{F,ℤ} = ℤ_{≥0} · \frac{1}{g}ω^{∨}
C_{F,R^{∨}}^{μ} = \{a · \frac{1}{g} ω^{∨} | 0 < a ≤ g, a ≡ g (mod 2)\} ∪ \{0\}

Notice that the length of the partially ordered set C_{F,R^{∨}}^{μ} is ⌈\frac{g}{2}⌉.

4.4 quasisplit non-split A_{2}, A_{3}, A_{4}

The next batch of examples are about the adjoint form of quasisplit unitary groups of type A_{n}, n = 2, 3, 4. We first fix the notation about roots and weights. The simple coroots are

α_{1}^{∨} = e_{1} − e_{2}, α_{2}^{∨} = e_{2} − e_{3}, …, α_{n}^{∨} = e_{n} − e_{n−1}

where the coroots are all contained in an n + 1 dimensional Euclidean space with orthonormal basis e_{1}, …, e_{n+1}. The coordinate functions with respect to this basis will be denoted by x_{1}, …, x_{n+1}. So the coroots are all contained in the hyperphane x_{1} + ⋯ + x_{n+1} = 0. The fundamental coweights are

ω_{1}^{∨} = e_{1} − \frac{1}{n+1}(e_{1} + ⋯ + e_{n+1})
ω_{2}^{∨} = e_{1} + e_{2} − \frac{2}{n+1}(e_{1} + ⋯ + e_{n+1})
⋮  ⋮
ω_{n}^{∨} = e_{1} + ⋯ + e_{n} − \frac{n}{n+1}(e_{1} + ⋯ + e_{n+1})

We shall be interested in the case when the group is not split over F. So we assume that the group splits over a quadratic extension field K of F, and the non-trivial element ι ∈ Gal(K/F) operates on the root system via the opposition involution. Thus ι·ω_{i}^{∨} = ω_{n+1−i}^{∨}, ι·α^{∨} = α_{n+1−i}^{∨}. For simplicity, we will only deal with the fundamental coweight μ = ω_{1}^{∨}, or which is equivalent to the case μ = ω_{n}^{∨}. This corresponds to the hermitian symmetric space which is isomorphic to the n-dimensional unit ball. Let ν_{1} = \frac{1}{2}(ω_{1}^{∨} + ω_{n}^{∨}) = e_{1} − e_{n}. For an element ν ∈ C_{F,ℤ}, the set with multiplicity

\{±⟨x_{i}, ν⟩ + \frac{1}{2} | 0 ≤ i ≤ n + 1\}

will be called the slopes of ν. Shifting by \frac{1}{2} made it compatible with the case of abelian varieties of dimension n + 1.

4.4.1 quasisplit A_{2}

x ∈ C_{ℤ}^{ν_{1}} slopes of x y ∈ W · μ with π_{θ_{F}} (y) = x
ν_{1}  1,1, \frac{1}{2} , \frac{1}{2} ,0,0  μ
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. in W · μ

4.4.2 quasisplit A_{3}

x ∈ C_{ℤ}^{ν_{1}} slopes of x y ∈ W · μ with π_{θ_{F}} (y) = x
ν_{1}  1,1, \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} ,0,0  μ
\frac{1}{2}ω_{2}^{∨}  \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4}  μ
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. in W · μ
ν_{1} ≻ \frac{1}{2}ω_{2}^{∨} ≻ 0

4.4.3 quasisplit A_{4}

x ∈ C_{ℤ}^{ν_{1}}  slopes of x  y ∈ W · μ with π_{θ_{F}} (y) = x
ν_{1}  1,1, \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} ,0,0  μ
\frac{1}{4}(ω_{2}^{∨} + ω_{3}^{∨}) \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{3}{4} , \frac{1}{2} , \frac{1}{2} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4} , \frac{1}{4}  μ
0  \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , \frac{1}{2}  any elt. in W · μ
ν_{1} ≻ \frac{1}{4}(ω_{2}^{∨} + ω_{3}^{∨}) ≻ 0

4.5 Res_{K/F} (PSp_{4}), K a finite extension field of F.

This is the restriction of scalars from a K to F of the split adjoint group of type C_{2}. Let f = [K : F]. As before the fundamental F -coweights are denoted by ω_{1}^{∨} = e_{1}, ω_{2}^{∨} = \frac{1}{2}(e_{1} + e_{2}), and μ = ω_{2}^{∨}. We have ω_{i}^{∨} = ∑_{τ} ω_{i,τ}^{∨}, where τ runs through Hom_{F -alg}(K, \overline{F}), and the ω_{i,σ}^{∨}’s are the fundamental coweights of Res_{K/F} (PSp_{4}). We have

\begin{aligned}C_{F,ℤ} & = & \{0\} ∪ \{a\frac{1}{2f} ω_{1}^{∨} | a ∈ ℤ_{>0}\} ∪ \{b\frac{1}{f}ω_{2}^{∨} | b ∈ ℤ_{>0}\} \\ & & ∪ \{a\frac{1}{f}ω_{1}^{∨} + b\frac{1}{f}ω_{2}^{∨} | a, b ∈ ℤ_{>0}\} \end{aligned}

C_{F,R^{∨}}^{μ} = \{0\} ∪ \{a\frac{1}{2f} ω_{1}^{∨} | a ∈ ℤ, 0 < a ≤ f, a ≡ f (mod 2)\}
∪ \{b\frac{1}{f}ω_{2}^{∨} | b ∈ ℤ, 0 < b ≤ f\}
∪ \{a\frac{1}{f}ω_{1}^{∨} + b\frac{1}{f}ω_{2}^{∨} | a, b ∈ ℤ_{>0}, a + 2b ≤ f, a ≡ f (mod 2)\}

Here aω_{1}^{∨} + bω_{2}^{∨} ⪰ a′ω_{1}^{∨} + b′ω_{2}^{∨} iff a + \frac{1}{2}b ≥ a′ + \frac{1}{2}b′ and a + b ≥ a′ + b′. One can check directly that C_{F,R^{∨}}^{μ} is a ranked partially ordered set, and length(C_{F,R^{∨}}^{μ}) = ⌈\frac{3f}{2} ⌉.

References

[1] M. F. Atiyah and R. Bott. The Yang Mills equations over Riemann surfaces. Phil. Trans. R. Soc. London. A, 308:523–615, 1982.

[2] P. Berthelot, L. Breen, and W. Messing. Théorie de Dieudonné Cristalline II, volume 930. Springer-Verlag, 1982.

[3] P. Berthelot and A. Ogus. F -isocrystals and de rham cohomology I. Invent. Math., 72:155–199, 1983.

[4] N. Bourbaki. Groupes et Algèbres de Lie. Ch. 1, 2nd ed. Herman, 1971; Ch. 2–3, Hermann, 1972; Ch. 4–6, Hermann, 1968; Ch. 7–8, Hermann, 1974; Ch. 9, Masson, 1982.

[5] C.-L. Chai. Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli. Invent. Math., 121:439–479, 1995.

[6] A. J. de Jong and F. Oort. Purity of the stratification by Newton Polygons. preprint, 1998.

[7] R. E.Kottwitz. Isocrystals with additional structure. Compos. Math., 56:201–220, 1985.

[8] R. E.Kottwitz. Isocrystals with additional structure. II. Compos. Math., 109:255–339, 1997.

[9] R. E. Kottwitz. Shimura varieties and λ-adic representations. In L. Clozel and J. Milne, editors, Automorphic Forms, Shimura Varieties and L-functions, part 1, volume 10 of Perspectives in Mathematics, pages 161–209. Academic Press, 1990.

[10] K.-Z. Li and F. Oort. Moduli of Supersingular Abelian Varieties, volume 1680 of Lecture Notes in Math. Springer-Verlag, 1998.

[11] J. S. Milne. The points on a Shimura variety modulo a prime of good reduction. In R. Langlands and D. Ramakrishnan, editors, The Zeta Function of Picard Modular Surfaces, pages 153–255, Montréal, 1992. Les Publications CRM.

[12] F. Oort. Moduli of abelian varieties and Newton polygons. C. R. Acad. Sci. Paris, 312:385–389, 1991.

[13] F. Oort. Newton polygons and formal groups: conjectures by manin and grothendieck. Utrecht preprint 995, January 1997.

[14] M. Rapoport and M. Richartz. On the classification and specialization of F -isocrystals with additional structure. Compos. Math., 103:153–181, 1996.