Dedicated to the memory of Armand Borel

Monodromy of Hecke-invariant subvarieties Ching-Li Chai^{1}

§1. Introduction

The main result of this note is that for any primes ℓ ≠ p, the ℓ-adic monodromy group of the Zariski closure \overline{H(x_{0})} of the ℓ-adic Hecke orbit of a non-supersingular point x_{0} in A_{g} over \overline{𝔽_{p}} is equal to the full symplectic group; see Prop. 4.1. Here A_{g} denotes the moduli space of g-dimensional principally polarized abelian varieties. The proof consists mostly of group theory. It does use some nontrivial information from algebraic geometry, namely Grothendieck’s semisimplicity theorem for pure ℚ_{ℓ}-sheaves and the Riemann hypothesis for abelian varieties over finite fields.

Some immediate consequences of the main results are recorded as corollaries in §4. A generalization from the case of one prime ℓ to the set of all primes not equal to p is formulated in Prop. 4.5.4. The method used in this note applies to the reduction of other Shimura varieties as well, with the reductive group in the Shimura input data replaced by the simply connected cover of its derived group.

The group-theoretic argument has an unexpected geometric consequence. Suppose that Z is a smooth, locally closed subset Z of A_{g} over \overline{𝔽_{p}} stable under all ℓ-adic Hecke correspondences, such that the Hecke correspondences operate transitively on the set of irreducible components of Z, and the maximal points of Z are not supersingular. Then Z is irreducible; see Prop. 4.4. This result is useful for proving the irreducibility of certain subschemes of A_{g} defined by p-adic invariants of the Barsotti-Tate group of the universal abelian scheme; see 4.7 and 4.8.

It is a pleasure to thank E. P. van den Ban and A. Borel for their help with Lemma 3.3, B. Moonen for pertinent and helpful comments, and F. Oort for stimulating discussions on applications of Prop. 4.4. The author would like to thank the referee for a very careful reading and for pointing out an error in §2. This paper was written during a visit to Utrecht University in November 2002, and I would like to the Mathematisch Instituut of Utrecht University for hospitality. Finacial support from the NSF and the NWO are gratefully acknowledged.

§2. Notations and preliminaries

(2.1) Throughout this note, we work over an algebraically closed field k of characteristic p > 0. Let ℓ be a prime number, ℓ ≠ p, and let N ≥ 3 be a natural number relatively prime to pℓ. Let Z be a smooth locally closed subvariety of A_{g,N} over k. Here A_{g,N} denotes the moduli space of g-dimensional principally polarized abelian varieties with principal symplectic level-N structure. Assume that Z is stable under all ℓ-adic Hecke correspondences coming from Sp_{2g}(ℚ_{ℓ}), and that the ℓ-adic Hecke correspondences operate transitively on the set of connected components of Z. Let A → Z be the restriction to Z of the universal abelian scheme. Let Z^{0} be an irreducible component of Z, let η = η_{Z} be the generic point of Z^{0}, and let η̄ be a geometric point of Z above η.

1Partially supported by grants DMS01-00441 and DMS04-00482 from the National Science Foundation, and a grant from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek

(2.2) Denote by T = T_{ℓ} = A[ℓ^{∞}]η̄ the ℓ-adic Tate module of A_{η}, and let V = V_{ℓ} = T_{ℓ} ⊗ ℚ_{ℓ}. Let ⟨ , ⟩_{ℓ} : T_{ℓ} × T_{ℓ} → ℤ_{ℓ}(1) be the Weil pairing. Let G = Sp(V, ⟨ , ⟩_{ℓ}), considered as an algebraic group over ℚ_{ℓ}. The semisimple group G has a natural ℤ_{ℓ}-model whose ℤ_{ℓ}-points is Sp(T_{ℓ}, ⟨ , ⟩_{ℓ}). We choose and fix an isomorphism between (T_{ℓ}, ⟨ , ⟩_{ℓ}) and the standard symplectic pairing on ℤ_{ℓ}^{2g}, and use it to identify G with Sp_{2g,ℚ_{ℓ}}.

(2.3) Let ρ_{A,ℓ} : π_{1}(Z^{0}, η̄) → Sp(T_{ℓ}, ⟨ , ⟩_{ℓ}) be the ℓ-adic representation attached to the ℓ-adic Tate module of A → Z^{0}. For each n ∈ ℕ, denote by ⟨ , ⟩_{n} : (T_{ℓ}/ℓ^{n}T_{ℓ}) × (T_{ℓ}/ℓ^{n}T_{ℓ}) → (ℤ/ℓ^{n}ℤ)(1) the restriction of the Weil pairing to the ℓ^{n}-torsion points. Let ρ_{n} : π_{1}(Z^{0}, \overline{η}) → Sp(T_{ℓ}/ℓ^{n}T_{ℓ}, ⟨ , ⟩_{ℓ}) be the monodromy representation attached to A[ℓ^{n}] → Z^{0}, the subgroup of ℓ^{n}-torsion points of A → Z^{0}. Let M_{n} = ρ_{n}(π_{1}(Z, \overline{η})), the image of ρ_{n}. The finite groups M_{n} form a projective system whose limit M is the monodromy group ρ(π_{1}(Z^{0}, \overline{η})). The group M is a closed subgroup of G(ℚ_{ℓ}).

(2.4) For each n ∈ ℕ, “forgetting the ℓ-part of the level structure” defines a morphism A_{g,ℓ^{n}N} → A_{g,N} between moduli spaces; this morphism is a principal étale Sp_{2g}(ℤ/ℓ^{n}ℤ)- cover. Moreover we have natural maps A_{g,ℓ^{n+1}N} → A_{g,ℓ^{n}N}. Let A_{g,ℓ^{∞}N} be the projective limit of the A_{g,ℓ^{n}N}’s. It is a principal pro-étale Sp_{2g}(ℤ_{ℓ})-cover of A_{g,N}. There is a natural action of Sp_{2g}(ℚ_{ℓ}) on A_{g,ℓ^{∞}N}, extending the action of Sp_{2g}(ℤ_{ℓ}). This Sp_{2g}(ℚ_{ℓ})-action on A_{g,ℓ^{∞}N} induces the ℓ-adic Hecke correspondences on A_{g,N}.

(2.5) For each n ∈ ℕ, let Z(n) = A_{g,ℓ^{n}N} ×_{A_{g,N}} Z and let Z^{0}(n) be the restriction of Z(n) → Z to Z^{0} ⊂ Z. The natural map Z(n) → Z is a principal étale Sp_{2g}(ℤ/ℓ^{n}ℤ)-cover Z(n) → Z. Moreover the contraction product

Z(n) ×^{Sp_{2g}(ℤ/ℓ^{n}ℤ),std} (ℤ/ℓ^{n}ℤ)^{2g}

of the principal cover Z(n) → Z with the standard representation of Sp_{2g}(ℤ/ℓ^{n}ℤ) on (ℤ/ℓ^{n}ℤ)^{2g} is isomorphic to the group scheme A[ℓ^{n}] → Z of ℓ^{n}-torsion points of A → Z. As n varies the principal covers Z(n) form a projective system; denote by Z̃ be the projective limit of the the Z(n)’s; it is a pro-étale principal Sp_{2g}(ℤ_{ℓ})-cover of Z. Similarly denote by Z̃^{0} the projective limit of the Z^{0}(n)’s; it is the restriction to Z^{0} of the pro-étale principal Sp_{2g}(ℤ_{ℓ})-cover Z̃ → Z to Z^{0}. Of course Z̃ is canonically isomorphic to A_{g,ℓ^{∞}N} ×_{A_{g,N}} Z.

The assumption that Z is stable under the ℓ-adic Hecke correspondences means that the closed subscheme Z̃ of A_{g,ℓ^{∞}N} is stable under the action of Sp_{2g}(ℚ_{ℓ}) on A_{g,ℓ^{∞}N}. So the action of Sp_{2g}(ℤ_{ℓ}) on Z̃ extends to an action of Sp_{2g}(ℚ_{ℓ}) on Z̃. This action of Sp_{2g}(ℚ_{ℓ}) on Z̃ induces the ℓ-adic Hecke correspondences on the base scheme Z.

(2.6) Choose a point η̃ ∈ Z̃(η̄) above η; one can write η̃ as the limit of a compatible system of points η̃_{n} ∈ Z(n)(η̄). Let Y_{n} be the connected component of Z(n) containing η_{n}. The projective limit Y of the Y_{n}’s is a connected component of Z̃ and is also a connected component of Z̃^{0}. According to general Galois theory, the stabilizer in Sp_{2g}(ℤ/ℓ^{n}ℤ) of the connected component Y_{n} is equal to M_{n}, and the stabilizer of Y in Sp_{2g}(ℤ_{ℓ}) is M. Also we know from Galois theory that Aut(Y/Z^{0}) = M and Y ×^{M} Sp_{2g}(ℤ_{ℓ}) = Z̃^{0}.

The set π_{0}(Z̃^{0}) of connected components of Z̃^{0} is a pro-finite set, endowed with the pro-finite topology: π_{0}(Z̃^{0}) = \underleftarrow{lim} π_{0}(Z(n)). The same holds for the set π_{0}(Z̃) of connected components of Z̃. The assumption that the ℓ-adic Hecke correspondences operate transitively on π_{0}(Z) means that Sp_{2g}(ℚ_{ℓ}) operates transitively on the set π_{0}(Z̃) of connected components of Z̃. Clearly π_{0}(Z̃^{0}) is a subset of π_{0}(Z̃), and π_{0}(Z̃) is non-canonically homeomorphic to the disjoint union of a finite number of copies of π_{0}(Z̃^{0}) because Sp_{2g}(ℚ_{ℓ}) operates transitively on π_{0}(Z̃). The base point [Y] ∈ π_{0}(Z̃^{0}) gives us a continuous bijection f_{1} : M\ Sp_{2g}(ℤ_{ℓ}) ⟶^{∼} π_{0}(Z̃^{0}) which is necessarily a homeomorphism since the source and the target are both compact.

(2.7) Let Q be the the stabilizer subgroup of Y in Sp_{2g}(ℚ_{ℓ}), consisting of all elements of Sp_{2g}(ℚ_{ℓ}) which send Y to itself; Q is a closed subgroup of Sp_{2g}(ℚ_{ℓ}) because the action of Sp_{2g}(ℚ_{ℓ}) on π_{0}(Z̃) is continuous. By definition, each element of Q operates as an automorphism of Y; Q ∩ Sp_{2g}(ℤ_{ℓ}) = M because Aut(Y/Z^{0}) = M. The base point [Y] ∈ π_{0}(Z̃) gives us a continuous bijection f_{2} from Q\ Sp_{2g}(ℚ_{ℓ}) to π_{0}(Z̃).

(2.8) Lemma The bijection f_{2} above is a homeomorphism.

Proof. The source of f_{2} is locally compact, and is a countable union of compact-open subsets. The target of f_{2} is a profinite topological space, hence is a complete metric space. By Baire’s category theorem applied to the complete metric space π_{0}(Z̃), there exists an compact open subset U in Q\ Sp_{2g}(ℚ_{ℓ}) such that f_{2}(U) contains an open subset W of π_{0}(Z̃). Shrinking W, we see that there exists a compact open subset U_{1} of Q\ Sp_{2g}(ℚ_{ℓ}) contained in U such that f_{2} induces a homeomorphism from U_{1} to a compact open subset W_{1} of π_{0}(Z̃). Because f_{2} is compatible with the continuous action of Sp_{2g}(ℚ_{ℓ}) on its source and its target, we conclude that the continuous bijection f_{2} is open, hence it is a homeomorphism.

(2.9) Remark Infinite Galois coverings with action by a large non-compact ℓ-adic Lie group related to Hecke correspondences have been considered in [16], [9], [7] and [6].

§3. Lemmas

(3.1) Lemma Let L be a subgroup of finite index of Sp_{2g}(ℚ_{ℓ}). Then L = Sp_{2g}(ℚ_{ℓ}).

Proof. This Lemma is surely well-known; here is a proof. Let T be a split maximal torus of G = Sp_{2g} over ℚ_{ℓ}. For every root α of G, denote by U_{α} the corresponding root subgroup. Denote by χ_{α} : T → 𝔾_{m} the character of T attached to α. By assumption L ∩ U_{α}(ℚ_{ℓ}) is an open subgroup of finite index in U_{α}(ℚ_{ℓ}). The p-adic group U_{α}(ℚ_{ℓ}) is divisible as an abelian group, being the abelian group underlying a vector space over ℚ_{ℓ}. So the subgroup L ∩ U_{α}(ℚ_{ℓ}) of finite index in U_{α}(ℚ_{ℓ}) is equal to U_{α}(ℚ_{ℓ}) itself. Since Sp_{2g}(ℚ_{ℓ}) is generated by the root subgroups U_{α}(ℚ_{ℓ}), we have proved that L = Sp_{2g}(ℚ_{ℓ}).

(3.2) Lemma Let P be a connected closed subgroup of a connected reductive G over ℚ_{ℓ}. Then the coset space P (ℚ_{ℓ})\G(ℚ_{ℓ}) is compact if and only if P is a parabolic subgroup of G.

Proof. This is a special case of [2, Prop. 9.3].

(3.3) Lemma Let H be a connected semisimple subgroup of G = Sp_{2g} over a field E of characteristic 0. Then the neutral component N_{G}(H)^{0} of the normalizer of H in G is reductive.

Proof. We offer two proofs. The first one uses compact groups. We may and do assume that E = ℂ. Choose a maximal compact subgroup K_{H} of H(ℂ), and choose a maximal compact subgroup K_{G} of G(ℂ) containing K_{H}. From the compact subgroups K_{H} and K_{G} we get compact real forms H_{ℝ} and G_{ℝ} of H and G respectively, such that H_{ℝ}(ℝ) = K_{H} and G_{ℝ}(ℝ) = K_{G} as subgroups of G(ℂ). Therefore N_{G}(H)^{0} has a compact real form, namely N_{G_{ℝ}}(H_{ℝ})^{0}, the neutral component of the normalizer of H_{ℝ} in G_{ℝ}. Hence N_{G}(H)^{0} is reductive.

Here is an algebraic proof, due to A. Borel. We may and do assume that G is semisimple. Let C = Z_{G}(H)^{0} be the neutral component of the centralizer subgroup of H, and let U be the unipotent radical of C. The group H being reductive, we have natural orthogonal decompositions

Lie(G) = Lie(C) ⊕ V = Lie(U) ⊕ Lie(C/U) ⊕ V

with respect to the Killing form B on Lie(G), and each direct summand is stable under the action of C · H = N_{G}(H)^{0}. Since U is unipotent, there is a separated decreasing filtration Fil^{•} of Lie(G) such that ad(u) · Fil^{i} ⊂ Fil^{i+1} for all u ∈ Lie(U) and all i. This implies that the restriction of the Killing form B to Lie(U) vanishes identically. Therefore Lie(U) = (0) since B is nondegenerate. Hence C and N_{G}(H)^{0} = C · H are both reductive.

(3.4) Lemma Assumptions as in §2. If Aη̄ is not supersingular, then the abelian scheme A ×_{Z^{0}} Z′ → Z′ is not isogenous to a constant abelian scheme for any finite surjective morphism Z′ → Z . Conversely if Aη̄ is supersingular, then A → Z^{0} is isogenous to a constant abelian scheme.

Proof. Suppose that Aη̄ is not supersingular. Let \overline{Z} be the closure of Z in A_{g,N}. If A → Z^{0} is isogenous to an isotrivial abelian scheme after a finite surjective base change Z′ → Z, then \overline{Z} is proper. Clearly \overline{Z} is stable under all ℓ-power Hecke correspondences since Z is. The argument of [3, Prop. 6], which uses [14, Thm. 4.1] and [3, Prop. 1], shows that \overline{Z} contains a supersingular point. This is a contradiction. The converse statement is well-known.

(3.5) Corollary Assumptions as in §2. Assume moreover that Aη̄ is not supersingular. Then the image ρ_{ℓ}(π_{1}(Z_{x}, \overline{η})) of the ℓ-adic monodromy representation attached to A → Z^{0} is not finite. Conversely, if Aη̄ is supersingular, then ρ_{ℓ}(π_{1}(Z_{x}, \overline{η})) is finite.

Proof. Suppose that ρ_{ℓ}(π_{1}(Z_{x}, \overline{η})) is finite. By [12, Thm. 2.1], there exists a finite surjective base change map Z′ → Z^{0} such that A×_{Z^{0}} Z′ → Z is isogenous to a constant abelian scheme. This contradicts Lemma 3.4. The converse holds because the geometric ℓ-adic monodromy group of any abelian scheme isogenous to a constant one is finite.

§4. Results

(4.1) Proposition Assumptions as in §2. Assume moreover that the image of the ℓ-adic monodromy representation ρ_{A,ℓ} attached to A → Z^{0} is not finite. Then ρ_{A,ℓ}(π_{1}(Z^{0}, \overline{η})) contains an open subgroup of Sp(T_{ℓ}, ⟨ , ⟩_{ℓ})

Proof. Recall that G = Sp(V, ⟨ , ⟩_{ℓ}), identified with Sp_{2g,ℚ_{ℓ}} as an algebraic group over ℚ_{ℓ}. Let M the image of π_{1}(Z^{0}, \overline{η}) in G(ℚ_{ℓ}).

Let H be the neutral component of the Zariski closure of M in G; H is not the trivial subgroup by assumption. The closed subgroup M of Sp_{2g}(ℤ_{ℓ}) is an ℓ-adic Lie group; denote its Lie algebra by 𝔪. By [1, §7, Cor. 7.9], the commutator of the Lie algebra of H is contained in the Lie algebra of M. Hence it suffices to show that the subgroup H of G is equal to G.

Since Z is stable under all ℓ-power Hecke correspondences, the group Sp_{2g}(ℚ_{ℓ}) operates naturally on Z̃. This gives us an action of Sp_{2g}(ℚ_{ℓ}) on Z̃, extending the action of Sp_{2g}(ℤ_{ℓ}) on Z̃. Notice that this action is not compatible with the projection map Z̃ → Z.

Recall that Q is the stabilizer subgroup in Sp_{2g}(ℚ_{ℓ}) of the connected component Y of Z̃. Although M is not stable under the conjugation action of Q, this turns out to hold on the level of Lie algebras. More precisely, for any element γ ∈ Q, there exists an open subgroup U ⊂ Sp_{2g}(ℤ_{ℓ}) such that γ·U ·γ^{−1} ⊂ Sp_{2g}(ℤ_{ℓ}). Therefore γ·(M ∩U)·γ^{−1} ⊂ Q∩Sp_{2g}(ℤ_{ℓ}) = M. Passing to the Lie algebras of the ℓ-adic Lie groups M ∩U and M yields the desired conclusion that Ad(γ)(𝔪) ⊆ 𝔪 for all γ ∈ Q.

The fact that the Lie algebra 𝔪 of M is stable under Ad(γ) for all γ ∈ Q implies that the neutral component H of the Zariski closure of M is stable under conjugation by any element γ ∈ Q. In other words Q is contained in N_{G}(H)(ℚ_{ℓ}). Since Q\G(ℚ_{ℓ}) is compact, we conclude that coset space N_{G}(H)^{0}(ℚ_{ℓ})\G(ℚ_{ℓ}) is compact, where N_{G}(H)^{0} denotes the neutral component of the algebraic group N_{G}(H). By Lemma 3.2, N_{G}(H)^{0} is a parabolic subgroup of G.

Now we use a theorem of Grothendieck and a theorem of Deligne in [8, Cor. 1.3.9] and [8, Thm. 3.4.1]. They imply that the neutral component of the Zariski closure of the geometric monodromy group of any pure ℚ_{ℓ}-sheaf over a variety defined over a finite field is semisimple. In our situation Grothendieck’s theorem tells us that H is semisimple. Notice that the purity condition comes from the Riemann hypothesis for abelian varieties over finite field, proved by Weil. Since H is semisimple, its normalizer subgroup in G is reductive by Lemma 3.3.

The subgroup N_{G}(H)^{0} of G is a parabolic subgroup of G and it is also reductive, hence N_{G}(H)^{0} = G. So H is a non-trivial normal subgroup of G. Therefore H is equal to G since G = Sp_{2g} is a simple algebraic group over ℚ_{ℓ}.

(4.1.1) Remark According to Cor. 3.5, the assumption that ρ_{A,ℓ}(π_{1}(Z^{0}, \overline{η})) is not finite is equivalent to the condition that Aη̄ is not supersingular.

(4.2) Corollary Assumptions as in §2. Assume moreover that Aη̄} is not supersingular. Then End(Aη̄) = ℤ.

Proof. Obviously the action of End(Aη̄) ⊗_{ℤ} ℚ_{ℓ} on V = T_{ℓ}(Aη̄) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} commutes with the monodromy representation, hence End(Aη̄) ⊗_{ℤ} ℚ_{ℓ} ⊆ ℚ_{ℓ} · Id_{V}.

(4.3) Corollary Notation and assumption as in §2. Suppose that ground field k is \overline{𝔽_{p}}. Then there exists a point y ∈ Z(\overline{𝔽_{p}}) such that End\overline{𝔽_{p}}(A_{y}) ⊗_{ℤ} ℚ is a number field E with [E : ℚ] = g, and there is exactly one place v in E above ℓ.

Proof. Use Chebotarev and Prop. 4.1.

(4.4) Proposition Notation and assumption as in §2. Assume that Aη̄ is not supersingular. Then the image ρ_{A,ℓ}(π_{1}(Z^{0}, \overline{η})) of the ℓ-adic monodromy representation of A → Z^{0} is equal to Sp(T_{ℓ}, ⟨ , ⟩_{ℓ}) ≅ Sp_{2g}(ℤ_{ℓ}). Moreover Z = Z^{0}, i.e. Z is irreducible.

Proof. We use the notation in the proof of Prop. 4.1. Cor. 3.5 shows that the ℓ-adic monodromy representation of A → Z^{0} is not finite. Prop. 4.1 tells us that the set π_{0}(Z̃^{0}) ≅ M\ Sp_{2g}(ℤ_{ℓ}) of connected components of Z̃^{0} is finite, and so is the set of connected components π_{0}(Z̃) ≅ Q\ Sp_{2g}(ℚ_{ℓ}). Lemma 3.1 implies that Q = Sp_{2g}(ℚ_{ℓ}). Hence Z̃ is connected. Since the map Z̃ → Z is faithfully flat, we deduce that Z is connected.

(4.4.1) Remark We have assumed that the level N is prime to ℓp for simplicity. Our argument still applies if we assume only that the level N ≥ 3 is prime to p. Then Prop. 4.1 remains true, while Prop. 4.4 should be changed to state that the image ρ_{ℓ}(π_{1}(Z^{0}, η̄)) of ρ is equal to the principal congruence subgroup of level M = (N, ℓ^{∞}) in Sp_{2g}(ℤ_{ℓ}).

(4.5) Let S be a set of prime numbers different from p. Then one can consider S-adic Hecke correspondences coming from elements of the restricted product ∏′_{ℓ∈S} Sp_{2g}(ℚ_{ℓ}) =: Sp_{2g}(𝔸_{ℚ,S}). The method used in the proof of Prop. 4.1, 4.4 applies as well and yield analogous statements. Here we formulate only the case when S is the set of all prime numbers different from p. The corresponding Hecke correspondences are usually referred to as the set of all prime-to-p Hecke correspondences.

(4.5.1) Let W be a smooth locally closed subset of A_{g,N} over \overline{𝔽_{p}}, such that W is stable under the all prime-to-p Hecke correspondences and the prime-to-p Hecke correspondences operate transitively on π_{0}(W).

(4.5.2) Denote by Γ(N) the subgroup of Sp_{2g}(ℤ̂^{(p)}) = ∏_{ℓ≠p} Sp_{2g}(ℤ_{ℓ}), where ℓ runs through all prime numbers different from p, consisting of all elements of Sp_{2g}(ℤ̂^{(p)}) which are congruent to Id modulo N.

(4.5.3) Let W^{0} be a connected component of W, let η be the generic point of W^{0}, and let η̄ be a geometric point above η. Denote by

ρ_{A,ℤ̂^{(p)}} : π_{1}(W^{0}, η̄) → Γ(N) ⊂ Sp_{2g}(ℤ̂^{(p)})

the monodromy representation attached to the prime-to-p torsion points of A → W^{0}. The following proposition can be proved by the same argument as before; the proof is omitted.

(4.5.4) Proposition Notation as above. Assume that the abelian variety Aη̄ is not supersingular. Then the image of the monodromy representation ρ_{A,ℤ̂^{(p)}} is equal to Γ(N). Moreover W = W^{0}, i.e. W is irreducible.

Remark We shall prove in Prop. 4.6 a sort of “independence of ℓ” result, to the effect that every subvariety Z in A_{g,N} which is stable under all ℓ-adic Hecke correspondences is actually stable under the action of all prime-to-p Hecke correspondences. Therefore Prop. 4.5.4 applies, giving the maximality of the Galois representation attached to the prime-to-p torsion points of A → Z.

(4.6) Proposition Notation and assumptions as above. In particular, Z is a smooth irreducible subvariety of A_{g,N} stable under all ℓ-adic Hecke correspondences, and Aη̄ is not supersingular. Then Z is stable under the action of all λ-adic Hecke correspondences for any prime number λ ≠ p.

Proof. The idea of the proof already appeared on page 477 of [3]; here we explain the use of the strong approximation theorem in more detail, correcting some inaccuracies in loc. cit. We may and do assume that g ≥ 2. As in the proof of Lemma 3.4, the Zariski closure \overline{Z} of Z contains a supersingular point s. Denote by U (resp. G) the unitary group (resp. the special unitary group) over ℚ attached to the semisimple algebra End(A_{s}) ⊗_{ℤ} ℚ with the Rosati involution. The group G is an inner form of Sp_{2g} such that G(ℝ) is compact, while G(ℚ_{r}) is non-compact for every prime number r, including r = p. For every prime number r ≠ p, the groups G is split over ℚ_{r}. Moreover, if r ≠ p, then the ℤ_{r}-lattice End(A_{s}) ⊗_{ℤ} ℤ_{r} in End(A_{s}) ⊗_{ℤ} ℚ_{r} defines a hyperspecial maximal compact subgroups G(ℤ_{r}) of G(ℚ_{r}), isomorphic to Sp_{2g}(ℤ_{r}). The isomorphism is unique up to conjugation by elements of Sp_{2g}(ℤ_{r}).

Let U(ℤ_{p}) := U(ℚ_{p}) ∩ (End(A_{s}) ⊗_{ℤ} ℤ_{p})^{×}, G(ℤ_{p}) := G(ℚ_{p}) ∩ (End(A_{s}) ⊗_{ℤ} ℤ_{p})^{×}. We note that End(A_{s})⊗_{ℤ}ℤ_{p} ⟶^{∼} End(A_{s}[p^{∞}]) by Tate’s theorem, because there exists a finite extension 𝔽_{q} of 𝔽_{p} with q = p^{2a}, a ∈ ℕ, such that the principally polarized abelian variety A_{s} is defined over 𝔽_{q}, and the Frobenius morphism Fr_{q} is equal to p^{a} · Id. So we can identify U(ℤ_{p}) with Aut(End(A_{s}[p^{∞}], ∗)). The latter group operates naturally on the local deformation space A_{g,N}^{/s}, and is sometimes referred to as the local automorphism group of the deformation space.

For each prime number r ≠ p, let Γ_{r} = G(ℤ_{r}) if (r, N) = 1, and let Γ_{r} be the principal congruence subgroup of level r^{e} if r^{e}||N, e ≥ 1. According to the strong approximation theorem, the image of

G(ℚ) ∩ (Γ_{p} × ∏_{r≠p,ℓ} Γ_{r})

in G(ℤ_{p}) is dense in G(ℤ_{p}). This implies that the formal completion \overline{Z}^{/s} of \overline{Z} at s, as a closed formal subscheme of A_{g,N}^{/s}, is stable under the action of an G(ℤ_{p}). This is an example of a general principle underlying the methods used in [3]; it can be seen as follows. For any element γ_{p} in G(ℤ_{p}) and any compact open subgroup K_{m} of G(ℤ_{p}), there exists an element u ∈ G(ℚ) such that u_{r} ∈ Γ_{r} for every prime number r ≠ p, ℓ, and u_{p} · γ_{p} ∈ K_{m} in G(ℤ_{p}). Here u_{r} and u_{p} denote the image of u in G(ℚ_{r}) and G(ℚ_{p}) respectively. Let u_{ℓ} be the image of u in G(ℚ_{ℓ}). Then the action on A_{g,N}^{/s} of the Hecke correspondence H(u_{ℓ}) attached to u_{ℓ} is equal to the natural action of u_{p}^{−1} on the local deformation space A_{g,N}^{/s}. For every natural number m ≥ 1, one can choose K_{m} sufficiently small so that the actions of γ_{p} and u_{p}^{−1} on A_{g,N}^{/s} coincide on the m-th infinitesimal neighborhood Spec(𝒪_{s,A_{g,N}} /𝔪_{s,A_{g,N}}^{m}) of s in A_{g,N}. Since \overline{Z} is stable under the ℓ-adic Hecke correspondence H(u_{ℓ}), the m-th infinitesimal neighborhood Spec(𝒪_{\overline{Z}_{s}}/𝔪_{s,\overline{Z}}^{m}) of s in \overline{Z}, as a closed subscheme Spec(𝒪_{s,A_{g,N}} /𝔪_{s,A_{g,N}}^{m}), is stable under the

action of γ_{p}, for every natural number m ≥ 1. Therefore the closed formal subscheme \overline{Z}^{/s} of A_{g,N}^{/s} is stable under the action of γ_{p}.

We may and do assume that λ ≠ ℓ. Let γ_{λ} be an element of G(ℚ_{λ}). Let W be the image of \overline{Z} under the Hecke correspondence H(γ_{λ}) attached to γ_{λ}. According to the strong approximation theorem, the image of G(ℚ) in the restricted product ∏′_{r≠λ} G(ℤ_{r}) is dense. So there exists an element δ in G(ℚ), such that

• δ_{r} ∈ Γ_{r} for all prime number r ≠ ℓ, λ,

• δ_{λ} ∈ (γ_{λ} ·Γ_{λ}) ∩ (Γ_{λ} ·γ_{λ}), where δ_{r} and δ_{λ} denote the image of δ in G(ℚ_{r}) and G(ℚ_{λ}) respectively.

Then the effect on A_{g,N}^{/s} of the Hecke correspondence H(γ_{λ}) attached to γ_{λ} is equal to the composition of the action δ_{p}^{−1} and the Hecke correspondence H(δ_{ℓ}^{−1}) attached to δ_{ℓ}^{−1}. Since \overline{Z}^{/s} is invariant under the action of G(ℤ_{p}) and Z is stable under all ℓ-adic Hecke correspondences, we deduce that W contains s and the formal completion W^{/s} coincide with Z^{/s}. Notice that the same statement holds if s is replaced by any supersingular point s_{1} in \overline{Z}.

The prime-to-p Hecke orbit H_{non−p} · s of s in A_{g,N}, induced by elements of the symplectic group, is in natural bijection with the finite set (G(ℚ) ∩ G(ℤ_{p})) \ ∏′_{r≠p} G(ℚ_{r})/Γ(N), where Γ(N) is the principle congruence subgroup of ∏′_{r≠p} G(ℚ_{r}) of level N. By the strong approximation theorem, the ℓ-adic Hecke correspondences induced by elements of Sp_{2g}(ℚ_{ℓ}) ≅ G(ℚ_{ℓ}) operates transitively on H_{non−p} ·s. It is clear that the subvariety W of A_{g,N}, being the image of the irreducible subvariety \overline{Z} under the Hecke correspondence H(γ_{λ}), has the property that each of its irreducible component contains a point of H_{non−p} · s. The conclusion of the previous paragraph, applied to points of the Hecke orbit H_{non−p} · s of s, implies that W is contained in and hence equal to \overline{Z}. We have shown that \overline{Z} is stable under the Hecke correspondences attached to elements of Sp_{2g}(ℚ_{λ}).

(4.7) Remark The irreducibility results in 4.4 and 4.5.4 is handy for proving the irreducibility of certain subsets of A_{g,N} defined by geometric invariants which are invariant under prime-to-p isogenies. An example is a result of Oort on the set of irreducibility of any non-supersingular Newton polygon stratum in A_{g,N}. We sketch the argument here; see [15] for more details. Let W_{ξ} be a Newton polygon stratum in A_{g,N} which is not supersingular. It is known that W_{ξ} is equi-dimensional, and the closure of every irreducible component of W_{ξ} contains an irreducible component of the supersingular locus of A_{g,N}; see [11], [10]. One can verify that the prime-to-p Hecke correspondences operate transitively on the irreducible components of the supersingular locus, using the description of the supersingular locus in [11]. Furthermore local information is available from deformation theory at points of a dense open subset of the supersingular locus, see [13]. A standard degeneration method in algebraic geometry enables one to deduce from adequate local information that prime-to-p Hecke correspondences operate transitively on the irreducible components of W_{ξ}. Then one can apply Prop. 4.5.4 to conclude that W_{ξ} is irreducible.

(4.8) Remark One can deduce the irreducibility of non-supersingular central leaves from the irreducibility of non-supersingular Newton polygon strata, using the notation of hypersymmetric points in [5] and Prop. 4.4. As the last example, C.-F. Yu has proved the irreducibility of non-supersingular central leaves in a Hilbert modular variety using Prop. 4.4.

References

[1] A. Borel. Linear Algebraic Groups. (Notes taken by Hyman Bass.) Benjamin, 1969.

[2] A. Borel and J. Tits. Groupes Réductifs. Publ. Math. IHES., 27:55-150, 1965.

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

[4] C.-L. Chai. Hecke orbits on Siegel modular varieties. To appear in Geometric Methods in Algebra and Number Theory. Birkhäuser, 2004. 35 pages. Available from

   http://www.math.upenn.edu/^{∼}chai

[5] C.-L. Chai. Hypersymmetric abelian varieties. Preprint, 6 pages. Available from

   http://www.math.upenn.edu/^{∼}chai

[6] I. V. Cerednik. Uniformization of algebraic curves by discrete arithmetic subgroups of P GL_{2}(k_{w}) with compact quotient spaces. Math. USSR-Sb., 29:55–76, 1976.

[7] P. Deligne. Traveaux de Shimura. Sém. Bourbaki, 1970/71, no. 389. In LNM 244, Springer-Verlag, pp. 123–165, 1971.

[8] P. Deligne. La conjecture de Weil. II. Publ. Math. IHES., 58:138–252, 1980.

[9] Y. Ihara. On Congruence Monodromy Problems, vol. I, II. Lecture Notes no. 1, 2, Department of Mathematics, University of Tokyo, 1968, 1969.

[10] A. J. de Jong and F. Oort. Purity and the stratification by Newton polygons. J. Amer. Math. Soc., 13:209–241, 2000.

[11] K. Z. Li and F. Oort. Moduli of Supersingular Abelian Varieties. LNM 1680, Springer-Verlag, 1998.

[12] F. Oort. Subvarieties of moduli spaces. Invent. Math., 24:95–119, 1974.

[13] F. Oort. Newton polygon and formal groups: conjectures by Manin and Grothendieck. Ann. Math., 152:183–206, 2000.

[14] F. Oort. A stratification of a moduli space of polarized abelian varieties. In Moduli of Abelian Varieties, Progress in Math. 195, Birkhäuser, pp. 345–416, 2001.

[15] F. Oort. Monodromy, Hecke orbits and Newton polygon strata. Available from URL http://www.math.uu.nl/people/oort/ Informal notes of a lecture given in Bonn on Feb. 24, 2003.

[16] I. I. Pjateckii-Shapiro and I. R. Shafarevich. Galois theory of transcendental extensions and uniformization. Izv. Akad. Nauk SSSR Ser. Math., 30:671–704, 1966.

Ching-Li Chai Department of Mathematics University of Pennsylvania Philadelphia, PA 19003 U. S. A.

email: chai@math.upenn.edu
home page: http://www.math.upenn.edu/^{∼}chai