Density of Hecke orbits for abelian varieties of p-corank one

Ching-Li Chai^{∗} University of Pennsylvania ^{†}

(version 2.0, 5/15/98)

Abstract

Let k be a field of characteristic p > 0 and let 𝒜_{g} be the moduli space of principally polarized abelian varieties over k of dimension g. Suppose that (A, λ) is a principally polarized abelian variety over k such that Card(A[p](k^{alg})) = p^{g−1}. Then the symplectic separable isogeny class of (A, λ) is Zariski dense in the zero locus of the Hasse invariant in 𝒜_{g}.

1 Introduction

Let k be an algebraically closed field of characteristic p > 0. We denote by 𝒜_{g} the moduli space of principally polarized abelian varieties over k of dimension g. It was proved in [3] that for every prime number ℓ ≠ p, the ℓ-power Hecke orbit of every ordinary point (A, λ) of 𝒜_{g} is Zariski dense in 𝒜_{g}. In this paper we study the Hecke orbits of certain non-ordinary points in 𝒜_{g} which are quite close to being ordinary.

Clearly the separable Heck correspondences preserve all stratifications of 𝒜_{g} defined using the p-power torsion points of the universal abelian variety. On the other hand, the equivalence relation given by Zariski closure of prime-to-p Hecke orbits is strictly finer than the above stratifications if g > 1, because the prime-to-p Hecke orbit of every super-singular principally polarized abelian variety is finite; see [3, Prop. 1]. This suggests that a complete understanding of the Zariski closure of all prime-to-p Hecke orbits in 𝒜_{g} may be difficult.

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

†email: chai@math.upenn.edu

For an abelian variety A of dimension g over an overfield of k, the prank f(A) of A is defined to be the height of the maximal étale quotient of the Barsotti-Tate group attached to A. The p-rank f(A) of A is an integer between 0 and g, and we shall call g − f(A) the p-corank of A. The main result of this paper says that for every prime number ℓ ≠ p, every ℓ-power Hecke orbit in the p-corank-one locus of 𝒜_{g} is Zariski dense. This of course is the strongest possible statement for the p-corank-one locus, just as the case of the ordinary locus. For higher p-corank, our method can be used to show that the codimension of the Zariski closure of the ℓ-power Hecke orbit of a point (A, λ) with p-corank r is at most \frac{r(r+1)}{2}. This is clearly not the definitive statement though. So we have chosen not to present the higher p-corank case, in favor of keeping the exposition shorter than it would be otherwise. This generalization to the case of higher p-corank is left to the interested readers “as an exercise”—a nontrivial one which the author has not properly done, and there may be potential pitfalls.

Here is an outline of the argument in four steps. Much of the ideas already appeared in [3]; the precise definition of the Hecke orbits can also be found there. Let Z be the Zariski closure of the ℓ-power Hecke orbit of a point x_{0} = [(A_{0}, λ_{0})] in 𝒜_{g}(k). As in the beginning of §3 of [3], it suffices to deal with the case when (A_{0}, λ) is defined over a finite field 𝔽_{q}. Therefore we may assume from the beginning that k = 𝔽, the algebraic closure of 𝔽_{p}.

1) Use embedded Hilbert-Blumenthal varieties, c.f. [3, §3].

It is well-known that an abelian variety defined over a finite field has sufficiently many complex multiplications; see [18]. Especially End(A)^{∘} := End_{k}(A) ⊗_{ℤ}ℚ contains a product F of totally real number fields, such that [F : ℚ] = g and F is fixed under the Rosati involution ∗ on End(A)^{∘}. This allows us to construct an algebraic correspondence f from a Hilbert-Blumenthal moduli space ℳ_{F} associated to F to 𝒜_{g}, given by a proper morphism π : ℳ̃_{F} → ℳ_{F} which is finite over the locus with p-corank at most one and a morphism f̃: ℳ̃_{F} → 𝒜_{g}. Moreover there is a point x_{1} in ℳ_{F}, represented by an abelian variety A_{1} with multiplication by 𝒪_{F} which is F -isogenous to A_{0}, such that x_{1} corresponds to x_{0} under f. The ℓ-power Hecke orbit of x_{1} in ℳ_{F} is “mapped into” the ℓ-power Hecke orbit of x_{0} in 𝒜_{g} in the obvious sense.

2) Solve the problem for Hilbert-Blumenthal varieties.

Let F be a product of totally real number fields with [F : ℚ] = g. Let A_{1} be an abelian variety of dimension g over k = 𝔽 with multiplication by 𝒪_{F} such that the p-rank of A_{1} is equal to g − 1. So the Barsotti-Tate group A_{1}[p^{∞}] splits into a direct sum of two Barsotti-Tate groups, A_{1}[p^{∞}]_{ord} and A_{1}[p^{∞}]_{ss}; A_{1}[p^{∞}]_{ord} has height 2g − 2 and slopes 0 and 1, while A_{1}[p^{∞}]_{ss} has height 2 and slope \frac{1}{2}. Moreover there exists a degree-one prime ℘ of 𝒪_{F} over p, such that A_{1}[p^{∞}]_{ss} = A_{1}[℘^{∞}].

Using the method of [3, §4, §5], one shows that the Zariski closure Z_{1} of the ℓ-power Hecke orbit of A_{1} in ℳ_{F} is equal to the locus in ℳ_{F} where the Barsotti-Tate group A[℘^{∞}] of the ℘-power torsion points of the universal abelian variety over ℳ_{F} has slope \frac{1}{2}. The dimension of Z_{1} is therefore equal to g − 1. We note that the theory of linear subvarieties as explained in [2] can be applied to simplify the proof of [3, §4], as well as the corresponding part in this step.

3) Explore the action of the stabilizer subgroup at a super-singular point.

The main result of [8] implies that Z_{1} contains a supersingular point z_{1}. This is used in the previous step, just as in [3, §5]. Let z_{0} be an image of z_{1} under f, and let (A_{z}, λ_{z}) be the principally polarized abelian variety corresponding to z_{0}. As explained in [3, §1], since Z is stable under the prime-to-ℓ Hecke operators, the formal completion 𝒜_{g}^{/z_{0}} of 𝒜_{g} at z_{0} is stable under the natural action of an open subgroup K of the group of automorphisms of (A_{z}[p^{∞}], λ_{z}). The latter group is a compact open subgroup of the group of ℚ_{p}-points of an inner form of Sp_{2g}(ℚ_{p}).

Our commutative semisimple algebra F is embedded in End(A_{z})^{∘}, fixed by the Rosati involution ∗; the embedding is denoted by j. Also recall the F has a degree-one place ℘. Let K be an open subgroup of the ℚ_{p}-points of the unitary group attached to (End(A_{z})^{∘}, ∗). One shows that there exists an element u ∈ K such that u · j(F ⊗_{ℚ} ℚ_{p}) · u^{−1} is the ℚ_{p}-span of a ℚ-subalgebra F′ of End(A_{z})^{∘} and u · j(F_{℘}) · u^{−1} is the ℚ_{p}-span of a ℚ-subalgebra F′′ of F′. One deduces that Z contains a point x_{2} such that the corresponding principally polarized abelian variety (A_{2}, λ_{2}) is such that A_{2} is isogenous to a product of an ordinary abelian variety with a supersingular elliptic curve.

4) At this point one can finish the proof without much difficulty. First, using the part of the ℓ-power Hecke correspondences which preserves the splitting of A_{2} up to isogeny, one sees that by “varying the ordinary factor of A_{2} by symplectic isogenies” and then take the Zariski closure, one gets a subvariety W of 𝒜_{g} of dimension \frac{g(g−1)}{2}. It is known that the local completion D of the p-corank-one locus at x_{2} has a canonical structure as a T -torsor over a formal group G, where T is a formal torus of dimension \frac{g(g−1)}{2} and G is non-canonically isomorphic to g − 1 copies of the formal completion of a supersingular elliptic curve; see [4, Thm. 4.4, Prop. 4.7]. The subvariety W corresponds to the fiber T over the origin of G. It is easy to see that W is a proper subset of Z. Changing the point x_{2} to another point of Z if necessary, we may assume that the ordinary factor of A_{2} is the (g −1)st self-product of an ordinary elliptic curve over 𝔽. Since the formal completion Z^{/x_{2}} of Z at x_{2} is stable under the natural action of the group of automorphisms of (A_{2}[p^{∞}], λ_{2}), this forces Z^{/x_{2}} to be equal to D. In other words dim(Z) = dim(D) = \frac{g(g+1}{2} − 1, and Z is equal to the zero-locus of the Hasse invariant on 𝒜_{g}.

In the exposition in the rest of this paper, the quaternionic linear algebra involved in step 3) is explained in §2; see the statement of Theorem 1. Step 2) is carried in §3. The rest of the proof are in §4. Roughly, the first two paragraphs of §4 set up the scene as sketched in step 1). In the next two paragraphs the results of §2 is applied to complete step 3). The rest of §4 completes the proof as sketched in step 4). Actually our presentation deviates a little from the above outline. We found it convenient to use an isogeny correspondence between 𝒜_{g,d} and 𝒜_{g}, and a morphism i from ℳ_{F} to 𝒜_{g,d}. Here 𝒜_{g,d} denotes the moduli stack of abelian varieties of dimension g with a polarization of degree d^{2}, where d is a suitable positive integer. So in §4 the morphism i plays the role of f in the steps above.

Although the situation in this paper is different from that in [3], the ideas for steps 1), 2), 4) already appeared there. Only the method used in step 3) is really new. We also formulated a slight generalization, which can be proved by the same method; see Prop. 4.

This paper owes its existence to questions of F. Oort on relations between Hecke orbits and stratifications of the moduli space 𝒜_{g}, see [15, 16A_{?}] He also went through a first draft, pointed out many errors and inaccuracies, and suggested improvements. It is a great pleasure to thank him for stimulating discussions and continuing encouragement.

2 Linear algebra for the stabilizer subgroup at supersingular abelian varieties

Let (A_{z}, λ_{z}) be a polarized supersingular abelian variety over 𝔽 of dimension g. It is well-known that A_{z} is isogenous to a self-product of a supersingular elliptic curve over 𝔽. Therefore E = End_{𝔽}(A_{z})^{∘} is isomorphic to a matrix algebra M_{g}(B), where B is the quaternion division algebra over ℚ ramified exactly at p and ∞. The Rosati involution attached to λ_{z} is a positive definite involution on E. Clearly E is a central simple algebra over ℚ. We fix a simple left E-module N and let D = End_{E}(N) be the centralizer of E operating on N. If one chooses an isomorphism of E with M_{g}(B), then N is isomorphic to the set B^{⊕g} of all g × 1 column vectors with entries in B, and D is isomorphic to B^{opp} ≅ B; the action of D on the left of N becomes the right multiplication of B on B^{⊕g}. It is a general fact that every positive definite involution on E comes from a positive definite D-valued hermitian form on N; we denote by h the hermitian form on N corresponding to the Rosati involution ∗. Thus h : N × N → D satisfies

h(d_{1} · n_{1}, d_{2} · n_{2}) = d_{1} · h(n_{1}, n_{2}) · d_{2}^{∗}

for all n_{1}, n_{2} ∈ N and all d_{1}, d_{2} ∈ D, where d_{2}^{∗} denotes the image of d_{2} under the standard involution of D. Moreover

h(n_{1}, n_{2}) = h(n_{2}, n_{1})^{∗} ∀n_{1}, n_{2} ∈ N; h(n, n) > 0 ∀0 ≠ n ∈ N .

We recall that the standard involution is the unique positive definite involution on D ≅ B; abusing the notation, ∗ will also denote the standard involution on D ≅ B. Let D_{p} = D ⊗_{ℚ} ℚ_{p}, N_{p} = N ⊗_{ℚ} ℚ_{p}, E_{p} = E ⊗_{ℚ} ℚ_{p}. Let GU(E, ∗) = GU(h) be the group of unitary similitudes defined by (E, ∗) or equivalently (h, N). For any commutative ℚ-algebra R, GU(E, ∗)(R) consists of all elements u ∈ (E ⊗_{ℚ} R)^{×} such that u · u^{∗} = u^{∗} · u ∈ R^{×} · 1; it is also the group of D-linear automorphisms of N which preserve the hermitian form h up to multiples.

Suppose that F is a commutative semisimple algebra over ℚ, and let j : F ↪ E be an embedding of F into E such that j(F) is fixed by the involution ∗ on E. Since the restriction to F of the trace form for E is a positive linear functional on F with respect to the involution ∗, F is isomorphic to a product of totally real number fields. Clearly every conjugate of j(F ⊗_{ℚ} ℚ_{p}) by an element of GU(E, ∗)(ℚ_{p}) is a commutative semisimple subalgebra of E_{p} of dimension g over ℚ fixed by ∗. Our goal is to show that one can conjugate j(F) by a suitable element u of GU(E, ∗)(ℚ_{p}) such that u · j(F ⊗_{ℚ} ℚ_{p}) · u^{−1} is the ℚ_{p}-span of a commutative semisimple subalgebra F′ of E of dimension g over ℚ, which is “as split as possible”.

Theorem 1. As above let j : F ↪ E be an embedding of a commutative semisimple algebra F into E which is fixed under the involution ∗ on E. Suppose that F ⊗_{ℚ}ℚ_{p} = F_{℘_{1}} × ⋯ × F_{℘_{a}}, where each F_{℘_{i}} is a finite extension field of ℚ_{p}. Then there exist commutative semisimple subalgebras F′_{1}, …, F′_{a} of E and an element u ∈ GU(E, ∗)(ℚ_{p}) such that u·j(F_{℘_{i}})·u^{−1} is the ℚ_{p}-span of F′_{i} for each i = 1, …, a.

Remark. (a) Clearly each F′_{i} is a totally real number field.

(b) Since the reduced norm Nm_{D_{p}/ℚ_{p}}^{red} : D^{×} → ℚ_{p}^{×} is surjective, one can replace the group of unitary similitudes GU(E, ∗)(ℚ_{p}) by the unitary group U(E, ∗)(ℚ_{p}) = U(h)(ℚ_{p}). We formulated the theorem with GU(E, ∗)(ℚ_{p}) since this is more natural, but the element u we construct in the proof actually belongs to U(E, ∗)(ℚ_{p}).

(c) Of course the totally real fields F′_{1}, …, F′_{a} are not uniquely determined by j(F); one only knows that F′_{i} ⊗_{ℚ} ℚ_{p} ≅ F_{℘_{i}} for each i.

Corollary 2.1. Under the hypotheses of Theorem 1, let K be an open subgroup of U(E, ∗)(ℚ_{p}). Then there exist commutative semisimple subalgebras F′_{1}, …, F′_{a} of E and an element u_{1} ∈ K such that the conclusion of Theorem 1 holds for u_{1}.

Proof of Corollary 2.1. The group of ℚ-rational points U(E, ∗)(ℚ) is dense in the p-adic group U(E, ∗)(ℚ_{p}) by the weak approximation theorem. By the remark above, there exists an element u ∈ U(E, ∗)(ℚ_{p}) such that each u · j(F_{℘_{i}}) · u^{−1} is the ℚ_{p}-span of a ℚ-subalgebra F′_{i} of E. Pick an element v ∈ U(E, ∗)(ℚ) such that v^{−1} · u ∈ K. Then u_{1} = v^{−1} · u does the job.

Proof of Theorem 1. The image under j of the identity element in each F_{℘_{i}} is an idempotent e_{i} in E_{p}. For each i, the D_{p}-submodule e_{i} · N_{p} of N_{p} is a free D_{p} module of rank

r_{i} = dim_{ℚ_{p}} (F_{℘_{i}}) = Tr_{E_{p}/ℚ_{p}}^{red}(e_{i}) .

This assertion is easily checked after base change to \overline{ℚ_{p}}. Clearly the e_{i}N_{p}’s are orthogonal to each other with respect to the hermitian form h. One sees quickly that for each i, the subalgebra e_{i}E_{p}e_{i} of E_{p} is stable under the involution ∗, e_{i}N_{p} is a simple left e_{i}E_{p}e_{i}-module, and e_{i}E_{p}e_{i} = End_{D_{p}}(e_{i}N_{p}). The subalgebra j(F_{℘_{i}}) of e_{i}E_{p}e_{i} is fixed by the involution.

It is an easy fact that every nondegenerate D_{p}-valued hermitian form on the D_{p}-module N_{p} is diagonal. Since the reduced norm Nm_{D_{p}/ℚ_{p}}^{red} : D_{p}^{×} → ℚ_{p}^{×} is surjective, any two non-degenerate D_{p}-valued hermitian form on N_{p} are equivalent under U(h, ∗)(ℚ_{p}). This argument already appeared in the proof of [7, §1, Thm. 3.1]. It follows that there exists an element u_{2} ∈ U(h, ∗)(ℚ_{p}) such that each u_{2} · e_{i}N_{p} is the ℚ_{p} span of a D-submodule N_{i} of N. In other words e′_{i} := u_{2} · j(e_{i}) · u_{2}^{−1} ∈ E for i = 1, …, a.

For each i = 1, …, a, let E′_{i} = e′_{i}Ee′_{i} = End_{D}(N_{i}); it is stable under the involution ∗ on E. The subfield F_{i} := u · F_{℘_{i}} · u^{−1} of E′_{i} ⊗_{ℚ} ℚ_{p} is fixed under the involution. Its has dimension r_{i}, while E_{i} is non-canonically isomorphic to M_{r_{i}}(B). Let S_{i} denote the subspace of all symmetric elements of E′_{i}, that is elements of E_{i} fixed by the involution. Consider the algebraic map

C_{i} : U(h|N_{i} ⊗ ℚ_{p}) × F_{i} → S_{i} ⊗ ℚ_{p}

over ℚ_{p} defined by conjugating elements of F_{i} by elements of the unitary group. Pick a generator y_{i} of the field F_{i}. One easily checks that the differential of C_{i} at the point (e′_{i}, y_{i}) is an isomorphism. Hence the image of C_{i} contains an open neighborhood of y_{i} with respect to the p-adic topology of S_{i} ⊗ ℚ_{p}. Clearly S_{i} is dense in S_{i} ⊗ ℚ_{p}. So there exists an element s_{i} ∈ U(h|N_{i})(ℚ_{p}) such that s_{i} ·y_{i} ·s_{i}^{−1} ∈ E′_{i}. The elements s_{1}, …, s_{a} together define an element s ∈ U(h, N)(ℚ_{p}) = U(E, ∗)(ℚ_{p}). Take u = s·u_{2} and let F′_{i} be the subfield of E′_{i} generated by s_{i} · y_{i} · s_{i}^{−1} for i = 1, …, a. The theorem is proved.

3 The case for Hilbert-Blumenthal modular varieties

In this section we solve the Hecke orbit problem for the p-corank-one locus of a Hilbert-Blumenthal modular variety. We first restrict ourself to the essential case of one totally real number field. The more general case of a product of a finite number of totally real number fields can be easily deduced from the essential case.

Let F be a totally real number field, and let 𝒪_{F} be the ring of integers of F. We refer the readers to [16] and [6] for background material on the Hilbert-Blumenthal modular variety; generally we use the notation in [6]. For any invertible 𝒪_{F} -module Λ with a notion of positivity, let ℳ_{F}^{Λ} be the moduli stack of abelian varieties A of dimension g = [F : ℚ] with endomorphisms ι : 𝒪_{F} → End(A) and a Λ-polarization ϕ : A ⊗_{𝒪_{F}} Λ ⟶^{∼} A^{t}, where A^{t} is the dual abelian variety of A; see [6, §2]. Let GL_{2}^{red} be the reductive algebraic group over ℚ whose ℚ-rational points consist of all elements of GL_{2}(F) with determinant in ℚ^{×}. Denote by 𝔸_{f}^{(p)} the prime-to-p part of the finite adèles for ℚ. Elements of GL_{2}^{red}(𝔸_{f}^{(p)}) give rise to prime-to-p Hecke correspondences on ℳ_{F}^{Λ}; similarly elements of GL_{2}^{red}(ℚ_{ℓ}) give ℓ-power Hecke correspondences for any prime number ℓ ≠ p. Geometrically, the prime-to-p Hecke orbit of a point (A_{1}, ι_{1}, ϕ_{1}) of ℳ_{F}^{Λ} is the set consisting of all points (A, ι, ϕ) such that there exists a prime-to-p 𝒪_{F} -linear isogeny α : A → A_{1} such that the pullback of ϕ_{1} is a multiple of ϕ; similarly for the ℓ-power Hecke orbit. Sometimes we will use the term ‘F -Hecke orbit’ to emphasize that we are considering the Hecke orbits for ℳ_{F}^{Λ}.

Suppose that (A_{1}, ι_{1}, ϕ_{1}) is a point of ℳ_{F}^{Λ}(𝔽) such that its p-rank f(A_{1}) is equal to g − 1 = [F : ℚ] − 1. We assume that g > 1 to avoid the trivial case F = ℚ. The Barsotti-Tate group A_{1}[p^{∞}] of A_{1} decomposes canonically into a direct sum of its toric part A_{1}[p^{∞}]^{toric}, its étale part A_{1}[p^{∞}]^{ét}, and a third factor isomorphic to the Barsotti-Tate group of a supersingular elliptic curve over 𝔽; denote the third factor by A_{1}[p^{∞}]^{ss}. The 𝒪_{F} -action on A_{1} extends canonically to an action of 𝒪_{F} ⊗_{ℤ} ℤ_{p}, and the latter preserves the decomposition above. Since the action of 𝒪_{F} ⊗_{ℤ} ℤ_{p} on the character group of A_{1}[p^{∞}]^{toric} is dual to its action on the p-adic Tate module of A_{1}[p^{∞}]^{ét}, it follows that the ideal of 𝒪_{F} ⊗_{ℤ}ℤ_{p} which operates trivially on both A_{1}[p^{∞}]^{toric} and A_{1}[p^{∞}]^{ét} is a direct summand of 𝒪_{F} ⊗_{ℤ}ℤ_{p} corresponding to a degree-one place ℘_{1} of F above p. Let ℘_{2}, …, ℘_{r} be the remaining places of F above p. We have 𝒪_{F} ⊗_{ℤ}ℤ_{p} = 𝒪_{℘_{1}} × ⋯ × 𝒪_{℘_{r}}. The character group of A_{1}[p^{∞}]^{toric} and the the p-adic Tate module of A_{1}[p^{∞}]^{ét} are both free 𝒪_{℘_{2}} ×⋯×𝒪_{℘_{r} x}-modules of rank one.

The Serre-Tate theorem says that deforming (A_{1}, ι_{1}, ϕ_{1}) is equivalent to deforming the corresponding Barsotti-Tate group (A_{1}[p^{∞}], ι_{1}[p^{∞}], ϕ_{1}[p^{∞}]). The decomposition 𝒪_{F} ⊗_{ℤ} ℤ_{p} = 𝒪_{℘_{1}} × ⋯ × 𝒪_{℘_{r}} of 𝒪_{F} therefore induces a decomposition of the deformation space of (A_{1}, ι_{1}, ϕ_{1}) in ℳ_{F}^{Λ} into a product D_{1} × ⋯ × D_{r} of r factors, where each D_{i} is the deformation space of the triple (A_{1}[℘_{i}^{∞}], ι_{1}[℘_{i}^{∞}], ϕ_{1}[℘_{i}^{∞}]). Clearly D_{1} is isomorphic to the deformation space of a supersingular elliptic curve, while for each i = 2, …, r D_{i} is a formal torus of dimension g_{i} = [F_{℘_{i}} : ℚ_{p}] because A_{1}[℘_{i}^{∞}] is ordinary.

Let ℓ ≠ p be a prime number, and let Z_{1} be the Zariski closure of the ℓ-power Hecke orbit of x_{1} = [(A_{1}, ι_{1}, ϕ_{1})] ∈ ℳ_{F}^{Λ}(𝔽) in M_{F}^{Λ} over 𝔽. It is clear that the one-dimensional Barsotti-Tate group of height two attached to the ℘_{1}-power torsion point of any point [(A, ι, ϕ)] in Z_{1} is supersingular in the sense that \frac{1}{2} is its only slope. Therefore the formal completion Z_{1}^{/x_{1}} of Z_{1} at x_{1} is contained in (the base point of D_{1}) × D_{2} × ⋯ D_{r}. Since x_{1} is defined over a finite field, it has sufficiently many complex multiplications. As in [3, §4], this implies that Z_{1}^{/x_{1}} is stable under the action under an open subgroup of 𝒪_{℘_{2}} ×⋯×𝒪_{℘_{r}}. By [3, Prop. 4], there exists a non-empty subset I ⊆ {2, …, r} such that Z_{1}^{/x_{1}} is equal to the product

∏_{i∈I} D_{i} × ∏_{i∉I}(the base point of D_{i}) .

As in [3, Prop. 5], this subset J of the set of all places of F above p is independent of the choice of points of p-corank one on of Z_{1}.

As in [3, Prop. 6], since Z_{1} is stable under all ℓ-power F -Hecke correspondences, the main result of [8] implies that Z_{1} contains a supersingular point z_{1}. This point z_{1} ∈ Z_{1}(𝔽) corresponds to a triple (A_{z_{1}}, ι_{z_{1}}, ϕ_{z_{1}}), where A_{z_{1}} is a supersingular abelian variety. Again by the Serre-Tate theorem, the formal completion (ℳ_{F}^{Λ})^{/z_{1}} of ℳ_{F}^{Λ} at z_{1} is canonically isomorphic to a product D_{℘_{1}}^{z_{1}} × ⋯ × D_{℘_{r}}^{z_{r}}. The formal completion Z_{1}^{/z_{1}} of Z_{1} at z_{1} is a closed formal subscheme of (ℳ_{F}^{Λ})^{/z_{1}}, and the first projection of the product structure sends Z_{1}^{/z_{1}} to the base point of D_{℘_{1}}^{z_{1}}. The argument of “inspecting at the supersingular point” in the proof of [3, Prop. 7] then shows that J = {2, …, r}: The stabilizer subgroup at z_{1} contains an open subgroup ∏_{i=1}^{r} U_{i} of the product ∏_{i=1}^{r} (B_{℘_{i}})_{1}^{×} of norm-one elements in quaternion algebras B_{℘_{i}}’s over F_{℘_{i}}. For each i, the action of elements of U_{i} changes the i-th coordinate with respect to the product structure of (ℳ_{F}^{Λ})^{/z_{1}}. This implies that i ∈ J unless the i-th projection of Z_{1}^{/z_{1}} is equal to the base point of D_{1}^{z_{1}}. On the other hand, if for some i ≥ 2 the i-th projection of Z_{1}^{/z_{1}} is equal to the base point of D_{1}^{z_{1}}, then the the p-rank of every point of Z_{1} is at most g − 2. This contradiction proves that dim(Z_{1}) = g − 1.

The conclusion that dim(Z_{1}) = g − 1 can be reformulated in terms of Hasse-invariants: The decomposition 𝒪_{F} ⊗_{ℤ}ℚ_{p} = 𝒪_{℘_{1}} × ⋯ × 𝒪_{℘_{r}} gives a decomposition of the Barsotti-Tate group attached to the universal abelian variety A over ℳ_{F}^{Λ} and also a decomposition of the locally free sheaf \underline{H}^{1}(A, 𝒪_{A}) of rank g on ℳ_{F}^{Λ}. So one gets r Hasse invariants from the relative Frobenius action on the r factors of \underline{H}^{1}(A, 𝒪_{A}). See [10] for documentation on Hasse invariants and stratification for ℳ_{F}^{Λ}. The Hasse invariant for the factor ℘_{1} is a section of an invertible sheaf on ℳ_{F}^{Λ}, and ℘_{1} is a degree-one place of F above p. Clearly the ℓ-power F -Hecke correspondences preserve the zero locus of each Hasse invariant. What we proved in this section is that for each degree-one place ℘ above p and each p-corank-one point x in the zero locus Z_{℘} of the Hasse invariant attached to ℘, the Zariski closure of the ℓ-power F -Hecke orbit of x is a union of irreducible components of the divisor Z_{℘} of ℳ_{F}^{Λ}.

We summarize our discussion so far as

Theorem 2. Let x_{1} ∈ ℳ_{F}^{Λ} be a point of p-corank one over an algebraically closed field k of characteristic p > 0. Then there exists a unique degree-one place ℘ of F above p such that x_{1} lies in the zero locus Z_{℘} ⊂ ℳ_{F}^{Λ} of the Hasse invariant attached to ℘. Moreover for any prime number ℓ ≠ p, the Zariski closure of the ℓ-power F -Hecke orbit of x_{1} in ℳ_{F}^{Λ} over k is a union of irreducible components of the divisor Z_{℘} on ℳ_{F}^{Λ}.

Now we take up the more general situation. Let F = F_{1} × ⋯ × F_{s}, where each F_{j} is a totally real number field. Let 𝒪_{F} = 𝒪_{F_{1}} × ⋯ × 𝒪_{F_{s}}. Let Λ be an invertible 𝒪_{F} -module with a notion of positivity. Then Λ decomposes naturally into a direct sum Λ = Λ_{1} ⊕ ⋯ ⊕ Λ_{s}, where each Λ_{j} is an invertible 𝒪_{F_{j}}-module with a notion of positivity. We define a moduli stack ℳ_{F}^{Λ} just as before. Clearly every triple (A, ι, ϕ) in ℳ_{F}^{Λ} decomposes as a product (A_{1} × ⋯ A_{s}, ι_{1} × ⋯ ι_{s}, ϕ_{1} × ⋯ ϕ_{s}), since 𝒪_{F} is a product. This shows that ℳ_{F}^{Λ} decomposes as ℳ_{F_{1}}^{Λ_{1}} × ⋯ × ℳ_{F_{s}}^{Λ_{s}}. Similarly, the reductive group attached to ℳ_{F}^{Λ} is equal to the product of the reductive groups attached to ℳ_{F_{j}}^{Λ_{j}} up to center. So it will be more convenient to use the derived groups. In other words one uses SL_{2}(F) = SL_{2}(F_{1}) × ⋯ × SL_{2}(F_{s}). The effect for Hecke orbits is that one replaces the ℓ-power Hecke orbit of a point x = [(A, ι, ϕ)] by something slightly smaller: the set of all points [(A′, ι′, ϕ′)] such that there exists an 𝒪_{A}-linear ℚ_{ℓ}-quasi-isogeny between A′ and A which preserves the polarizations. We call this set the reduced ℓ-power F -Hecke orbit of x. One verifies easily that the statements of Theorem 2 and [3, Thm. 1] remain true if one replaces the ℓ-power F -Hecke orbit of x_{1} by the reduced ℓ-power F -Hecke orbit of x_{1}; the same proofs work while the statements are strengthened a little bit. One advantage of using the reduced ℓ-power Hecke orbit is that the reduced ℓ-power F -Hecke orbit of a point (A_{1} × ⋯ A_{s}, ι_{1} × ⋯ ι_{s}, ϕ_{1} × ⋯ ϕ_{s}) in ℳ_{F}^{Λ} is simply the product of the reduced ℓ-power F -Hecke orbits of the (A_{j}, ι_{j}, ϕ_{j})’s. Combining Thm. 2 with [3, Thm. 1], we obtain the following generalization of Thm. 2 using the reduced F -Hecke orbits.

Theorem 3. Let F = F_{1} ×⋯×F_{s} be a product of totally real number fields, 𝒪_{F} = 𝒪_{F_{1}} × ⋯ × 𝒪_{F_{s}}, and let Λ = Λ_{1} × ⋯ × Λ_{s} be an invertible 𝒪_{F} -module with a notion of positivity as above. Let x_{1} ∈ ℳ_{F}^{Λ} be a point of p-corank one over an algebraically closed field k of characteristic p > 0. Then there exists a unique degree-one place ℘ of F above p such that x_{1} lies in the zero locus Z_{℘} ⊂ ℳ_{F}^{Λ} of the Hasse invariant attached to ℘. Moreover for any prime number ℓ ≠ p, the Zariski closure of the reduced ℓ-power F -Hecke orbit of x_{1} in ℳ_{F}^{Λ} over k is a union of irreducible components of the divisor Z_{℘} on ℳ_{F}^{Λ}.

4 End of Proof

Let (A_{0}, λ_{0}) be a principally polarized abelian variety over 𝔽 of dimension g such that the p-rank f(A_{0}) of A_{0} is equal to g − 1. Consider the semi-simple algebra End(A_{0})^{∘} := End_{𝔽}(A_{0})⊗_{ℤ}ℚ over ℚ with the positive definite involution ∗ induced by the principal polarization λ_{0}. Then End(A_{0})^{∘} decomposes into a product of End(A_{0})^{∘} = E_{1} × ⋯ E_{m} of semisimple algebras, such that each E_{i} is isotypic and Hom(E_{i}, E_{j}) = (0) if i ≠ j. The positive definite involution ∗ stabilizes each E_{i} and induces a positive definite involution ∗_{i} on E_{i}. Let C_{i} be the center of E_{i}. Write [E_{i} : ℚ] as [E_{i} : ℚ] = [C_{i} : ℚ] · d_{i}^{2}, and let g_{i} = \frac{1}{2}[C_{i} : ℚ]d_{i}. One knows that each g_{i} is an integer, and g_{1} + ⋯ + g_{m} = g.

It is well-known that for each i, there exists a commutative semisimple subalgebra S_{i} ⊂ E_{i}, such that elements of S_{i} are fixed under ∗_{i} and [S_{i} : ℚ] = g_{i}. Since ∗_{i} is a positive definite involution on E_{i}, S_{i} is a product of totally real number fields. Actually one can choose S_{i} to be a totally real field; we will not need this though.

Let F be the product of the S_{i}’s, so F is a product of totally real number fields F_{1} × ⋯ × F_{s} embedded in End(A_{0})^{∘} and fixed under the Rosati involution ∗. Then there exist an invertible 𝒪_{F} -module Λ = Λ_{1} × ⋯ × Λ_{s} with a notion of positivity, a triple (A_{1}, ι_{1}, ϕ_{1}) ∈ ℳ_{F}^{Λ}(𝔽) and an isogeny β : A_{1} → A_{0}, such that β ∈ Hom(A_{1}, A_{0})^{∘} is F -linear and the pull-back of the principal polarization λ_{0} on A_{0} is the image of a totally positive element λ_{1} of Λ under ϕ_{1}. Let d^{2} be the degree of the polarization λ_{1} on A_{1}, that is the symmetric homomorphism λ_{1} : A_{1} → A_{1}^{t} has degree d^{2}. Using the element λ_{1} ∈ Λ to polarize the universal abelian variety over ℳ_{F}^{Λ}, we obtain a morphism i from ℳ_{F}^{Λ} to the moduli stack 𝒜_{g,d} of g-dimensional abelian varieties with polarization of degree d^{2}. It is known that i is a finite morphism which sends (A_{1}, ι_{1}, ϕ_{1}) ∈ ℳ_{F}^{Λ}(𝔽) to (A_{1}, λ_{1}) ∈ 𝒜_{g,d}(𝔽). For any prime number ℓ ≠ p, i maps the reduced ℓ-power F -Hecke orbit of (A_{1}, ι_{1}, ϕ_{1}) into the ℓ-power Hecke orbit of (A_{1}, λ_{1}). Moreover we have an isogeny correspondence κ between 𝒜_{g,d} and 𝒜_{g} which contains β; κ is finite over the open subset of 𝒜_{g,d} and 𝒜_{g} where the p-corank is at most one. The ℓ-power Hecke orbit of (A_{1}, λ_{1}) and the ℓ-power Hecke orbit of (A_{0}, λ_{0}) correspond under κ; especially their Zariski closure have the same dimension.

Let Z_{1} be the Zariski closure of the reduced ℓ-power F -Hecke orbit of the point x_{1} = [(A_{1}, ι_{1}, ϕ_{1})] in ℳ_{F}^{Λ}. By Thm. 3, there exists a unique degree-one place ℘_{1} of F above p such that x_{1} lies in the zero locus Z_{℘_{1}} ⊂ ℳ_{F}^{Λ} of the Hasse invariant attached to ℘_{1}, and Z_{1} is a union of irreducible components of the divisor Z_{℘_{1}} on ℳ_{F}^{Λ}.

As in the proof of [3, Prop. 6], the main result of [8] implies that Z_{1} contains a supersingular point z_{1}. Let y = i(z_{1}) ∈ 𝒜_{g,d}(𝔽). Let Y_{1} be the Zariski closure of the ℓ-power Hecke orbit of (A_{1}, λ_{1}) in 𝒜_{g,d}, so i(Z_{1}) ⊂ Y_{1}. Write y = (A_{y}, λ_{y}), and let E_{y} := End_{𝔽}(A_{y})^{∘}. The Rosati involution for λ_{1} gives a positive definite involution ∗ = ∗_{y} on E = E_{y}. The action of 𝒪_{F} on A_{1} gives us an embedding j : F ↪ E of F which is fixed under ∗. Since Y_{1} is stable under all ℓ-power Hecke correspondences, the formal completion Y_{1}^{/y} of Y_{1} at y is stable under the natural action of an open subgroup K_{1} ⊂ U(E, ∗)(ℚ_{p}). We are now exactly in the situation of §2, and Corollary 2.1 implies that there exists an element u ∈ K_{1}, a commutative finite dimensional semisimple algebra F′ over ℚ which splits into a product F′ = F′_{1}×F′_{2} with F′_{1} ≅ ℚ, an embedding j′ : F′ ↪ E fixed under ∗, such that u · j(F ⊗_{ℚ}ℚ_{p}) · u^{−1} = j′(F′) ⊗_{ℚ}ℚ_{p} and u · j(F_{℘_{1}} ⊗_{ℚ}ℚ_{p}) · u^{−1} = j′(F′_{1}) ⊗_{ℚ}ℚ_{p}. Since Y_{1}^{/y} is stable under the action of K_{1} and Y_{1} contains i(Z_{1}), Y_{1}^{/y} contains the image u · i(Z_{1})^{/y} of the formal completion i(Z_{1})^{/y} of i(Z_{1}) at y under the action of u ∈ K_{1}. Clearly the universal abelian scheme over the coordinate ring of i(Z_{1})^{/y} has endomorphisms by j(𝒪_{F}) ⊆ End_{𝔽}(A_{y}), and its Barsotti-Tate group has endomorphisms by j(𝒪_{F}) ⊗_{ℤ} ℤ_{p} ⊆ End_{𝔽}(A_{y}) ⊗_{ℤ} ℤ_{p}. So the Barsotti-Tate group of the universal abelian scheme over the coordinate ring of u · i(Z_{1})^{/y} has endomorphism by

u · j(𝒪_{F}) ⊗_{ℤ} ℤ_{p} · u^{−1} = j′(𝒪_{F′}) ⊗_{ℤ} ℤ_{p} ⊆ End_{𝔽}(A_{y}) ⊗_{ℤ} ℤ_{p} .

By the Serre-Tate theorem, the universal abelian scheme over the coordinate ring of u · i(Z_{1})^{/y} has endomorphisms by j′(𝒪_{F′}) ∩ End_{𝔽}(A_{y}), an order of 𝒪_{F′} which happens to be maximal at all primes above p. It follows that Y_{1} contains a (g − 1)-dimensional subvariety Z′_{1}, the image of a finite morphism i′ from a Hilbert-Blumenthal modular stack ℳ_{F′}^{Λ′} to 𝒜_{g}, such that the the restriction A_{/Z′_{1}} of the universal abelian variety to Z′_{1} have endomorphisms by an order of 𝒪_{F′}. In particular Z′_{1} contains an element x′_{2} = (A′_{2}, λ′_{2}) ∈ 𝒜_{g,d}(𝔽) of p-corank one such that End_{𝔽}(A′_{2})^{∘} contains a commutative semisimple algebra isomorphic to F′, such that the factor F′_{1} of F′ cuts out a one-dimensional supersingular abelian subvariety in A′_{2} up to isogeny. This completes the step 3) of the proof as outlined in §1.

The argument used above actually proves a little more. As the statement may be useful in other situations, a short digression seems warranted. The proof is omitted because it is entirely similar to that in the previous paragraph.

Proposition 4. Let (A, λ) be a principally polarized abelian variety over 𝔽 of dimension g. Let Z be the Zariski closure of the ℓ-power Hecke orbit of [(A, λ)] ∈ 𝒜_{g}(𝔽) in 𝒜_{g} over 𝔽, where ℓ is a prime number different from p. Then there exists a point (A′, λ′) ∈ Z(𝔽) such that

• A′ has the same slopes as A, with the same multiplicities,

• A′ is isogenous to a product of abelian varieties B_{1} × ⋯ × B_{a},

• each B_{i} has at most two distinct slopes, i = 1, …, a, • B_{i} and B_{j} do not have common slopes if i ≠ j, 1 ≤ i, j ≤ a.

We resume the proof of main result. From the isogeny correspondence κ between 𝒜_{g,d} and 𝒜_{g}, we conclude that the Zariski closure Z of the ℓ-power Hecke orbit of x_{0} = [(A_{0}, λ_{0})] ∈ 𝒜_{g}(𝔽) contains a point x_{2} = [(A_{2}, λ_{2})] ∈ 𝒜_{g}(𝔽) such that A_{2} is isogenous to the product of a supersingular elliptic curve and a g − 1-dimensional ordinary abelian variety of dimension g − 1. Hence there exists an isogeny correspondence κ′ between 𝒜_{g} and 𝒜_{g,d′} for some positive integer d′ such that there exists a point x_{3} = [(E_{3} × X_{3}, λ_{3})] ∈ 𝒜_{g,d′}(𝔽) which corresponds with x_{2} under κ′, where E_{3} is a supersingular elliptic curve over 𝔽 and X_{3} is an ordinary abelian variety of dimension g − 1 over 𝔽. This forces the polarization λ_{3} to be the product of a polarization λ_{E_{3}} on E_{3} and a polarization λ_{X_{3}} on X_{3}. So 𝒜_{g,d′} contains a “diagonally embedded” copy of 𝒜_{1} × 𝒜_{g−1,d_{1}} passing through x_{3}, where d_{1} is the degree of the induced polarization on X_{3}. By [3, Thm. 2], the Zariski closure of the ℓ-power Hecke orbit of [(X_{3}, λ_{X_{3}})] in 𝒜_{g−1,d_{1}} contains the irreducible component 𝒜_{g−1,δ} of (𝒜_{g−1,d_{1}})_{red} which contains [(X_{3}, λ_{X_{3}})]. Here δ denotes (δ_{1}, …, δ_{g−1}) for suitable positive integers δ_{1}, …, δ_{g−1} with δ_{1}| ⋯ δ_{g−1}. Since the ℓ-power Hecke orbit of x_{3} and the ℓ-power Hecke orbit of x_{2} correspond under κ′, dim(Z) = dim(Z_{3}), where Z_{3} is denotes the Zariski closure of x_{3} in 𝒜_{g,d′}. Now we use another isogeny correspondence. There exist a point [(X_{4}, λ_{X_{4}})] ∈ 𝒜_{g−1,δ}(𝔽), a point [(X_{5}, λ_{X_{5}})] ∈ 𝒜_{g}(𝔽), and an isogeny α : X_{4} → X_{5} such that λ_{X_{4}} = α^{∗}(λ_{X_{5}}). Similarly for (E_{3}, λ_{E_{3}}). So we can find

• a point x_{4} = [(A_{4}, λ_{4})] ∈ Z_{3}(𝔽) of p-corank one

• a point

x_{5} = [(E_{ss}, λ_{ss}) ⊕ (E_{or}, λ_{or})^{⊕(g−1)}] ∈ 𝒜_{g}(𝔽)

with (E_{ss}, λ_{ss}) a principally polarized supersingular elliptic curve over 𝔽 and (E_{or}, λ_{or}) an ordinary principally polarized elliptic curve over 𝔽

• an isogeny correspondence κ′′ between 𝒜_{g,d′} and 𝒜_{g}

such that x_{4} and x_{5} correspond under κ′′. As before the dimension of Z_{3} is equal to the dimension of the Zariski closure Z_{5} of the ℓ-power Hecke orbit of x_{5} in 𝒜_{g}. Moreover Z_{5} contains a copy of ‘diagonally embedded’ 𝒜_{g−1} passing through x_{5}.

By [4, Thm. 4.4, Prop. 4.7], the formal completion V^{/x_{5}} of the p-corank-one locus V in 𝒜_{g} at x_{5} has a natural fibration structure; it is a torsor under a formal torus T of dimension \frac{g(g−1)}{2} over a smooth formal group G of dimension g − 1. The smooth formal group G is canonically isomorphic to N ⊗_{ℤ_{p}} E_{ss}, where N is the p-adic Tate module of the maximal étale quotient of E_{or}^{⊕(g−1)}. The fact that Z_{5} contains a copy of A_{g−1} means that V^{/x_{5}} contains the fiber T over the base point of G. The stabilizer subgroup at x_{5} is quite big even integrally, for it contains the stabilizer subgroup for the point (E_{or}, λ_{or})^{⊕(g−1)} in 𝒜_{g−1}, which is equal to GL(N) ≅ GL_{g−1}(ℤ_{p}); see [3, p. 443].

It is easy to see that dim(Z_{5}) > \frac{g(g−1)}{2}, for instance by considering the action of the stabilizer subgroup at a supersingular point of Z_{5}. If g − 1 = 1, then dim(G) = 1 and dim(Z_{5}) = \frac{g(g+1)}{2} − 1 trivially, so we may assume that g > 2. As shown in [8], V is irreducible. So if we can show that the tangent space of Z_{5}^{/x_{5}} is equal to the tangent space of V^{/x_{5}}, then Z_{5} has dimension \frac{g(g+1)}{2} − 1 and therefore equal to V.

Recall that Z_{5} is smooth at x_{5}, as explained on [3, p. 443], so Z_{5}^{/x_{5}} is formally smooth. We already know that Z_{5}^{/x_{5}} contains T. The tangent space of Z_{5}^{/x_{5}} is stable under the action of GL(N). From the definition of G one sees that the action of GL(N) on the tangent space of G, identified with the quotient of the tangent space of V^{/x_{5}} by the tangent space of T, is given by the composition of the canonical map GL(N) → GL(N ⊗_{ℤ_{p}} 𝔽_{p}) and the standard representation of GL(N ⊗_{ℤ_{p}} 𝔽_{p}) on N ⊗_{ℤ_{p}} 𝔽_{p}. Since the tangent space of Z_{5}^{/x_{5}} properly contains the tangent space of T, the tangent space of Z_{5}^{/x_{5}} must be equal to the tangent space of V^{/x_{5}}. This shows that dim(Z) = dim(Z_{5}) = dim(V), so Z_{5} = V = Z. We have proved the first statement of

Theorem 5. Let k be an algebraically closed field of characteristic p > 0, let ℓ be a prime number different from p. let g ≥ 2 and d be two positive integers.

1. Suppose that x_{0} = (A_{x_{0}}, λ_{x_{0}}) ∈ 𝒜_{g}(k) is a point of 𝒜_{g}, then the ℓ-power Hecke orbit of x_{0} in 𝒜_{g} is dense in the zero locus of the Hasse invariant on 𝒜_{g} → Spec k.

2. Suppose that y_{0} = (A_{y_{0}}, λ_{y_{0}}) ∈ 𝒜_{g,d}(k) is a point in 𝒜_{g,d}, then the Zariski closure of the ℓ-power Hecke orbit of y_{0} in 𝒜_{g,d} has dimension \frac{g(g+1)}{2} − 1. It is an irreducible component of the zero locus of the Hasse invariant on 𝒜_{g,d} → Spec k if (ℓ, d) = 1; otherwise it is a union of irreducible components of the zero locus of the Hasse invariant.

Proof. Let Z_{y_{0}} be the Zariski closure of the ℓ-power Hecke orbit of y_{0}. Using a suitable isogeny correspondence, we deduce that dim(Z_{y_{0}}) = \frac{g(g+1)}{2} −1 from the principally polarized case. It remains only to prove the last assertion of part 2 of the theorem. By standard argument, we may and do assume that d = p^{m}, a power of p. We know that each irreducible component of the zero locus 𝒱_{g−1} of the Hasse invariant has dimension \frac{g(g+1)}{2} − 1 according to [14, Thm. 4.1]. From the description of the irreducible components of 𝒱_{g−1} by discrete invariants given in [5, Prop. 3.3], it follows that Z_{y_{0}} is equal to the irreducible component of 𝒱_{g−1} determined by the type of the étale part (or equivalently the multiplicative part) of Ker(λ_{y_{0}}). More precisely, let δ̄_{1}, ⋯, δ̄_{g−1} be the elementary divisor of the étale part of Ker(λ_{y_{0}}), with δ̄_{1}| ⋯ |δ̄_{g−1}. Using the notation as in [5], it is clear that Z_{y_{0}} is contained in the subvariety 𝒵(p^{m}, δ̄)^{c} of 𝒱_{g−1}. Since 𝒵(p^{m}, δ̄)^{c} has codimension one in 𝒜_{g} and is irreducible according to [5, Prop. 3.3], it is equal to Z_{y_{0}}.

Remark. Near the end of the proof, we produced a ‘very symmetric’ point x_{5} and explored the action of its stabilizer subgroup. It is possible to use a point which is only isogenous to a product like E_{ss} ⊕ E_{or}^{⊕(g+1)}, but the proof would be more complicated. It is also possible to use a boundary point in the closure of the Hecke orbit in the minimal compactification of 𝒜_{g} which lies in the one-dimensional cusp. Then one explores the action of the stabilizer subgroup at this boundary point just as in [3, §2], where the zero-dimensional cusp was used. Again the proof would be more involved.

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