SUBGROUPS OF INERTIA GROUPS ARISING FROM ABELIAN VARIETIES

A. SILVERBERG AND YU. G. ZARHIN

Abstract. Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction. We study this subgroup, and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction.

1. Introduction

Suppose X is an abelian variety over a field F, and v is a discrete valuation on F. Fix an extension v̄ of v to a separable closure F^{s} of F, and write I_{v} for the inertia subgroup in Gal(F^{s}/F) for v̄. In [6] (see pp. 354–355), Grothendieck defined a subgroup I′ of I_{v} with the property that X has semistable reduction at the restriction w of v̄ to a finite separable extension of F if and only if I_{w} ⊆ I′. In particular, if F_{v}^{nr} denotes the maximal unramified extension of the completion of F at v, then I′ cuts out the smallest Galois extension of F_{v}^{nr} over which X has semistable reduction. We denote the group I′ by I_{v,X} because of its dependence on X and v.

In §4 we give some properties of the group I_{v,X}. We show that the Zariski closure of its image under the ℓ-adic representation (for ℓ different from the residue characteristic) coincides with the identity connected component of the Zariski closure of the image of I_{v}. The proofs of the results in §4 are in the spirit of [18], where we dealt with connectedness questions for Zariski closures of images of ℓ-adic representations.

In §5 we show that the finite group G_{v,X} = I_{v}/I_{v,X} injects into GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z_{ℓ}) for all but finitely many primes ℓ, where t_{v} and a_{v} (respectively, t and a) are the toric and abelian ranks of the special fiber of the Néron model of X at v (respectively, at an extension of v over which X has semistable reduction). Here, the projection onto the first factor is independent of ℓ, and the characteristic polynomial of the projection onto the second factor has integer coefficients independent of ℓ. The group G_{v,X} was introduced by Serre in the case of elliptic curves in §5.6 of [10] (where it was called Φ_{p}).

In §6 we obtain divisibility bounds on the order of G_{v,X}, and in §7 we deduce results on semistable reduction of abelian varieties. Bounds on the prime divisors and the exponent of #G_{v,X} were obtained by Lorenzini (see Proposition 3.1 of [7]). In particular, the bound on Q_{v,X} in Corollary 6.3 was essentially obtained by Lorenzini.

The paper continues our earlier work on semistable reduction of abelian varieties (see [17] and [19]). The proofs are heavily influenced by the fundamental results of Grothendieck and Serre.

The authors would like to thank Karl Rubin for helpful conversations, and NSA and NSF for financial support.

2. Notation and preliminaries

If K is a field, write K^{s} for a separable closure. If G is an algebraic group, let G^{0} denote its identity connected component. Let φ denote the Euler φ-function, let ζ_{M} denote a primitive M-th root of unity, and let F_{ℓ} denote the finite field with ℓ elements.

If Y is a commutative algebraic group over a field K (e.g., an abelian variety or an algebraic torus), let Y_{n} denote the kernel of multiplication by n in Y (K^{s}), let T_{ℓ}(Y) = lim_{←}Y_{ℓ^{n}}, and let V_{ℓ}(Y) = T_{ℓ}(Y) ⊗_{Z_{ℓ}} Q_{ℓ}. For example, if G_{m} is the multiplicative group, then Z_{ℓ}(1) := T_{ℓ}(G_{m}) is a free Z_{ℓ}-module of rank 1.

Throughout this paper, X is a d-dimensional abelian variety over a field F, v is a discrete valuation on F of residue characteristic p ≥ 0, and ℓ is a prime not equal to p. Fix an extension v̄ of v to F^{s}. If w is the restriction of v̄ to a finite separable extension L of F, let I_{w} denote the inertia subgroup in Gal(F^{s}/L) for v̄, let X_{w} denote the special fiber of the Néron model of X at w, and let T_{w} denote the maximal subtorus of X_{w}. Let

ρ_{ℓ} : Gal(F^{s}/F) → GL(T_{ℓ}(X))

denote the ℓ-adic representation. Let 𝔊 denote the Zariski closure of ρ_{ℓ}(I_{v}) in GL(V_{ℓ}(X)). We will make repeated use of the following result.

Theorem 2.1 (Galois Criterion for Semistable Reduction). X has semistable reduction at v if and only if I_{v} acts unipotently on V_{ℓ}(X).

Proof. See Proposition 3.5 and Corollaire 3.8 of [6] and Theorem 6 on p. 184 of [1].

Theorem 2.2. The Zariski closure 𝔊 of ρ_{ℓ}(I_{v}) is connected if and only if X has semistable reduction at v.

Proof. See Theorem 5.2 of [18]; see also Remarque 1 on p. 396 of [13].

Suppose λ is a polarization on X. Then λ gives rise to a non-degenerate, alternating, Gal(F^{s}/F)-equivariant, Z_{ℓ}(1)-valued pairing on T_{ℓ}(X) (see §§1.0 and 2.5 of [6]). Since I_{v} acts trivially on Z_{ℓ}(1), we obtain a non-degenerate, alternating, I_{v}-invariant pairing

E_{λ} : T_{ℓ}(X) × T_{ℓ}(X) → Z_{ℓ}.

Then E_{λ} is perfect if and only if deg(λ) is not divisible by ℓ (see (2.5.1) and §1.0 of [6]). Let ⊥ denote the orthogonal complement with respect to E_{λ}.

Proposition 2.3. (i) We can identify

T_{ℓ}(X_{v}) = T_{ℓ}(X)^{I_{v}} and V_{ℓ}(X_{v}) = V_{ℓ}(X)^{I_{v}}.

(ii) T_{ℓ}(T_{v}) = T_{ℓ}(X)^{I_{v}} ∩ (T_{ℓ}(X)^{I_{v}})^{⊥} = T_{ℓ}(X_{v}) ∩ T_{ℓ}(X_{v})^{⊥}. (iii) T_{ℓ}(X_{v})/T_{ℓ}(T_{v}) is a free Z_{ℓ}-module.

Proof. For (i), see Lemma 2 on p. 495 of [15]. For (ii), use (i) and Grothendieck’s Orthogonality Theorem (see (2.5.2) of [6]). For (iii), see (2.1.6) of [6].

Proposition 2.4. Suppose L is a finite separable extension of F, w is the restriction of v̄ to L, and X has semistable reduction at w. Then

T_{ℓ}(T_{w}) = T_{ℓ}(X_{w})^{⊥} = (T_{ℓ}(X)^{I_{w}})^{⊥} ⊆ T_{ℓ}(X)^{I_{w}} = T_{ℓ}(X_{w}).

Proof. See Proposition 3.5 of [6].

3. Linear Algebra

We will use the following linear algebra facts in Theorem 5.2 below.

Definition 3.1. Suppose R is a principal ideal domain, and M_{1} and M_{2} are free R-modules. A bilinear form e : M_{1} × M_{2} → R is called a perfect pairing if the natural homomorphisms M_{1} → Hom(M_{2}, R) and M_{2} → Hom(M_{1}, R) are bijective. If L is a submodule of M_{1} (resp., M_{2}), we write L^{⊥} for the orthogonal complement of L with respect to e in M_{2} (resp., M_{1}).

Remark 3.2. Suppose R is a principal ideal domain, M is a free R-module of rank 2n, and e : M × M → R is an alternating bilinear form. If e is perfect, then Aut(M, e) ≅ Sp_{2n}(R), where Sp_{2n}(R) denotes the group of 2n × 2n symplectic matrices over R (see §5 of [2]).

Proposition 3.3. Suppose ℓ is a prime number, G is a finite group whose order is not divisible by ℓ, V is a finite-dimensional Q_{ℓ}-vector space with a linear G-action, and e : V ×V → Q_{ℓ} is a G-invariant non-degenerate alternating (resp., symmetric) bilinear form. Suppose M is a G-stable Z_{ℓ}-lattice in V (there always exist such). Then there exists a perfect G-invariant alternating (resp., symmetric) bilinear form e′ : M × M → Z_{ℓ}.

Proof. First note that the result is true when e is symmetric and dim(V) = 1 and when e is alternating and dim(V) = 2, by scaling e.

We may assume that e(M, M) = Z_{ℓ} (by replacing e by ℓ^{i}e for a suitable i, if necessary). Then e induces a non-zero alternating (resp., symmetric) bilinear form ē on M/ℓM. By Nakayama’s Lemma, if ē is non-degenerate then e is perfect and we are done. Assume ē is degenerate, and let M̄_{1} = ker(ē). Then M̄_{1} is a proper subset of M/ℓM, since e(M, M) = Z_{ℓ}. Since #G is not divisible by ℓ, there is a G-invariant splitting M/ℓM = M̄_{1} ⊕ M̄_{2}, where M̄_{2} is a non-zero G-invariant subspace of M/ℓM, which can be lifted to a G-invariant splitting of Z_{ℓ}- lattices M = M_{1} ⊕ M_{2} with M_{1}/ℓM_{1} = M̄_{1} and M_{2}/ℓM_{2} = M̄_{2} (see §15.5 and Corollary 1 of §14.4 of [11]). Denote by e_{2} the restriction of e to V_{2} := M_{2} ⊗_{Z_{ℓ}} Q_{ℓ}, and let V_{1} denote the orthogonal complement of V_{2} in V with respect to e. The restriction of e to V_{1} is non-degenerate, and the restriction of e to M_{2} is perfect. We obtain a G-invariant orthogonal splitting V = V_{1} ⊕ V_{2}. Replace M_{1} by its (isomorphic) image in V_{1} under the projection map from V_{1} ⊕ V_{2} to V_{1}. Since dim(V_{1}) < dim(V), we obtain inductively a perfect G-invariant alternating (resp., symmetric) form e_{1} on M_{1}. Let M = M_{1} ⊕ M_{2} and e′ = e_{1} ⊕ e_{2}.

Lemma 3.4. Suppose R is a principal ideal domain, M_{1} and M_{2} are free Rmodules, L is a submodule of M_{1}, M_{1}/L is torsion-free, and e : M_{1} × M_{2} → R is a perfect pairing. Then:

(i) (L^{⊥})^{⊥} = L, (ii) the natural map Hom(M_{1}, R) → Hom(L, R) is surjective, (iii) the induced form L × M_{2}/L^{⊥} → R is a perfect pairing.

Proof. Clearly, L ⊆ (L^{⊥})^{⊥}. Let K denote the fraction field of R. By dimension arguments we have L ⊗ K = ((L ⊗ K)^{⊥})^{⊥} = (L^{⊥})^{⊥} ⊗ K. Therefore, (L^{⊥})^{⊥}/L is torsion. Since M_{1}/L is torsion-free, we obtain (i). Further, since M_{1}/L is torsion-free, L is a direct summand of M_{1}, and therefore we have (ii).

Since e is perfect, we can identify Hom(M_{2}, R) with M_{1}. Under this identification, Hom(M_{2}/L^{⊥}, R) = (L^{⊥})^{⊥} = L, by (i). Since e is perfect we can identify M_{2} with Hom(M_{1}, R). The natural injection M_{2}/L^{⊥} ↪ Hom(L, R) is surjective since M_{2} = Hom(M_{1}, R) → Hom(L, R) is surjective by (ii).

Proposition 3.5. Suppose R is a principal ideal domain, M is a free R-module of finite rank, e : M × M → R is an alternating (resp., symmetric) perfect pairing, N is a submodule of M, and M/N is torsion-free. Assume that N^{⊥} ⊆ N. Then

ẽ(a + N^{⊥}, b + N^{⊥}) = e(a, b)

defines an alternating (resp., symmetric) perfect pairing ẽ: N /N^{⊥} × N /N^{⊥} → R.

Proof. Applying Lemma 3.4iii with M_{1} = M_{2} = M and L = N gives an alternating (resp., symmetric) perfect pairing e′ : N × M/N^{⊥} → R. Now applying Lemma 3.4iii to e′ with M_{1} = M/N^{⊥}, M_{2} = N, and L = N /N^{⊥}, we obtain the desired result.

4. Properties of I_{v,X}

We will give a different definition for I_{v,X} than Grothendieck did, and will then show that the two definitions are equivalent. Grothendieck’s definition coincides with (ii) of Theorem 4.2 below.

Definition 4.1. Define I_{v,X} to be the kernel of the natural surjective homomorphism I_{v} → 𝔊/𝔊^{0}. Define G_{v,X} = I_{v}/I_{v,X}.

Theorem 4.2. I_{v,X} is an open normal subgroup of I_{v} which enjoys the following properties:

(i) I_{v,X} is the largest open subgroup of I_{v} such that the Zariski closure of its image under ρ_{ℓ} is 𝔊^{0}.

(ii) I_{v,X} = {σ ∈ I_{v} : σ acts unipotently on V_{ℓ}(X)}. (iii) If L is a finite separable extension of F and w is the restriction of v̄ to L,

then the following are equivalent:

(a) X has semistable reduction at w,

(b) I_{w} ⊆ I_{v,X},

(c) the Zariski closure of ρ_{ℓ}(I_{w}) is 𝔊^{0}. (iv) I_{v,X} = I_{v} if and only if X has semistable reduction at v.

(v) I_{v,X} is independent of the choice of ℓ.

Proof. By the definition of I_{v,X}, it is an open normal subgroup of I_{v}, and it is the largest open subgroup of I_{v} such that the Zariski closure of its image under ρ_{ℓ} is 𝔊^{0}.

Suppose L is a finite separable extension of F, and w is the restriction of v̄ to L. Let 𝔊_{w} denote the Zariski closure of ρ_{ℓ}(I_{w}), and let V = V_{ℓ}(X)^{I_{w}}.

Suppose first that X has semistable reduction at w. Then 𝔊_{w} is connected, by Theorem 2.2. Therefore, I_{w} ⊆ I_{v,X} and 𝔊_{w} = 𝔊^{0}. Since 𝔊^{0} (= 𝔊_{w}) is a normal subgroup of 𝔊, it follows that V is stable under I_{v}. Next we will show that

I_{v,X} = {σ ∈ I_{v} : σ acts unipotently on V_{ℓ}(X)}

= {σ ∈ I_{v} : σ acts unipotently on V }.

Since 𝔊^{0} = 𝔊_{w}, 𝔊^{0} acts as the identity on V and on V_{ℓ}(X)/V. Therefore, every element of I_{v,X} acts unipotently on V_{ℓ}(X), and therefore on V. To show the reverse inclusions, suppose g ∈ I_{v} and g acts unipotently on V. By Proposition 2.4 (and after tensoring with Q_{ℓ}), V^{⊥} ⊆ V. Since V^{⊥} is the dual of V_{ℓ}(X)/V, it follows that g acts unipotently on V_{ℓ}(X)/V, and therefore acts unipotently on V_{ℓ}(X). By Proposition 2.5 of [18], g ∈ I_{v,X}. We therefore obtain the desired equalities.

By (i), if 𝔊^{0} = 𝔊_{w}, then I_{w} ⊆ I_{v,X}. By (ii) and Theorem 2.1, if I_{w} ⊆ I_{v,X} then X has semistable reduction at w. We therefore have (iii). We easily deduce (iv) from (iii).

By Théorème 4.3 of [6], if σ ∈ I_{v} then the characteristic polynomial of ρ_{ℓ}(σ) is independent of ℓ. By (ii),

I_{v,X} = {σ ∈ I_{v} : the characteristic polynomial of ρ_{ℓ}(σ) is (x − 1)^{2d}}.

Therefore, I_{v,X} is independent of ℓ.

Proposition 4.3. If L is a finite separable extension of F, and X has semistable reduction at the restriction w of v̄ to L, then

(i) I_{v,X} = {σ ∈ I_{v} : σ acts unipotently on V_{ℓ}(X)^{I_{w}}}, (ii) G_{v,X} acts faithfully on T_{ℓ}(X)^{I_{w}}/T_{ℓ}(X)^{I_{v}}, (iii) T_{ℓ}(X)^{I_{w}} = T_{ℓ}(X)^{I_{v,X}}.

Proof. The proof of Theorem 4.2 included a proof of (i), and easily implies (ii). For (iii), let 𝔊_{w} denote the Zariski closure of ρ_{ℓ}(I_{w}), and note that T_{ℓ}(X)^{I_{v,X}} = T_{ℓ}(X)^{𝔊^{0}} = T_{ℓ}(X)^{𝔊_{w}} = T_{ℓ}(X)^{I_{w}}, by Theorem 4.2.

5. Restrictions on G_{v,X}

Remark 5.1. Note that G_{v,X} is isomorphic to the group of connected components of 𝔊. If p = 0 then G_{v,X} is cyclic, and if p > 0 then G_{v,X} is an extension of a cyclic group of order prime to p by a p-group (as can be seen by replacing F by the maximal unramified extension of the completion of F at v, looking at the extension cut out by I_{v,X}, taking its maximal tamely ramified subextension, and applying §8 of [5]). In particular, G_{v,X} is solvable, and has a unique Sylow-p-subgroup if p > 0.

The group G_{v,X} does not change if we replace F by an extension unramified at v. Therefore, in this section we can and do replace F by the maximal unramified extension of the completion of F at v. Then I_{v,X} cuts out the smallest Galois extension L of F over which X has semistable reduction. Let w denote the restriction of v̄ to L. Then I_{v,X} = I_{w}. Therefore, I_{w} is an open normal subgroup of I_{v} of finite index, and T_{ℓ}(X_{w}) = T_{ℓ}(X)^{I_{w}} is I_{v}-stable. By Proposition 2.3i we have

T_{ℓ}(X_{v}) = T_{ℓ}(X)^{I_{v}} ⊆ T_{ℓ}(X)^{I_{w}} = T_{ℓ}(X_{w})

as I_{v}-modules.

Over an algebraic closure of the residue field, there are exact sequences

0 → T_{w} → X_{w}^{0} → B_{w} → 0, 0 → U_{v} × T_{v} → X_{v}^{0} → B_{v} → 0,

where B_{w} and B_{v} are abelian varieties and U_{v} is a unipotent group (see §2.1 of [6]). Base change for Néron models induces a homomorphism ι : X_{v} → X_{w} such that if n is a positive integer not divisible by p, then the restriction of ι to the n-torsion (X_{v})_{n} is injective (see Lemma 2 of [15]; see also (3.1.1) of [6]). Here, X_{v} and X_{w} are viewed as commutative algebraic groups over an algebraic closure of the residue field at w. The map ι induces homomorphisms T_{v} → T_{w} and B_{v} → B_{w} whose kernels are finite group schemes killed by appropriate powers of p. The image of T_{v} (resp., B_{v}) is a subtorus (resp., abelian subvariety) in T_{w} (resp., B_{w}), and we let T (resp., B) denote the corresponding quotient. Let a and t (respectively, a_{v} and t_{v}) denote the abelian and toric ranks of X_{w} (respectively, X_{v}). Note that a and t are independent of the valuation w above v at which X has semistable reduction. We have rk(T_{ℓ}(X_{w})) = 2a + t and rk(T_{ℓ}(X_{v})) = 2a_{v} + t_{v}. By the functoriality of Néron models, G_{v,X} acts on X_{w} (see §4.2 of [6]), and therefore acts on T_{w} and on B_{w}. One may easily check that this action is trivial on the image of X_{v} → X_{w}. It follows that G_{v,X} acts on T and on B.

Fix a polarization λ on X. Let W_{ℓ} (respectively, S_{ℓ}) denote the orthogonal complement of V_{ℓ}(X_{v})/V_{ℓ}(T_{v}) (respectively, T_{ℓ}(X_{v})/T_{ℓ}(T_{v})) with respect to the pairing e_{λ} on V_{ℓ}(X_{w})/V_{ℓ}(T_{w}) (respectively, T_{ℓ}(X_{w})/T_{ℓ}(T_{w})) induced by E_{λ}. Then W_{ℓ} is a G_{v,X}-stable Q_{ℓ}-vector space of dimension 2a − 2a_{v} and S_{ℓ} is a G_{v,X}-stable Z_{ℓ}-sublattice of rank 2a − 2a_{v}.

Recall that ℓ is always a prime not equal to p.

Theorem 5.2. (i) The form e_{λ} : W_{ℓ} × W_{ℓ} → Q_{ℓ} is non-degenerate, alternating, and G_{v,X}-invariant. The vector space W_{ℓ} and the lattice S_{ℓ} do not depend on the choice of polarization λ. The natural actions of G_{v,X} on T, W_{ℓ}, and e_{λ} induce an injection

G_{v,X} ↪ Aut(T) × Aut(W_{ℓ}, e_{λ}) ≅ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Q_{ℓ})

such that the projection onto the first factor is independent of ℓ, and the characteristic polynomial of the projection onto the second factor has integer coefficients independent of ℓ.

(ii) If ℓ does not divide deg(λ)#G_{v,X}, then e_{λ} : S_{ℓ} × S_{ℓ} → Z_{ℓ} is perfect and the above injection takes values in

Aut(T) × Aut(S_{ℓ}, e_{λ}) ≅ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z_{ℓ}).

(iii) Suppose ℓ does not divide #G_{v,X}. Then for every G_{v,X}-stable Z_{ℓ}-lattice M in W_{ℓ}, there exists a perfect Z_{ℓ}-valued G_{v,X}-invariant alternating pairing e′ on M. I.e., there is an injection

G_{v,X} ↪ Aut(T) × Aut(M, e′) ≅ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z_{ℓ}).

Proof. By Proposition 4.3ii, G_{v,X} acts faithfully on T_{ℓ}(X_{w}). The natural homomorphism

φ_{ℓ} : G_{v,X} → Aut(T_{ℓ}(T_{w})) × Aut(T_{ℓ}(X_{w})/T_{ℓ}(T_{w}))

is injective, since its kernel consists of unipotent operators of finite order in characteristic zero. The map φ_{ℓ} factors through Aut(T) × Aut(T_{ℓ}(X_{w})/T_{ℓ}(T_{w})), since G_{v,X} acts trivially on T_{v}. By Theorem 2.3ii and Proposition 2.4,

T_{ℓ}(T_{v}) ⊆ T_{ℓ}(X_{v}) ∩ T_{ℓ}(T_{w}) = T_{ℓ}(X_{v}) ∩ T_{ℓ}(X_{w})^{⊥} ⊆ T_{ℓ}(X_{v}) ∩ T_{ℓ}(X_{v})^{⊥} = T_{ℓ}(T_{v}).

Therefore,

T_{ℓ}(T_{v}) = T_{ℓ}(X_{v}) ∩ T_{ℓ}(T_{w}) = T_{ℓ}(X_{v}) ∩ T_{ℓ}(X_{v})^{⊥}.

By Proposition 2.4, T_{ℓ}(T_{w}) = T_{ℓ}(X_{w})^{⊥}. It follows that e_{λ} is non-degenerate on T_{ℓ}(X_{w})/T_{ℓ}(T_{w}) and on T_{ℓ}(X_{v})/T_{ℓ}(T_{v}), and therefore on V_{ℓ}(X_{w})/V_{ℓ}(T_{w}) and on V_{ℓ}(X_{v})/V_{ℓ}(T_{v}). Further, e_{λ} is G_{v,X}-invariant.

Let r = #G_{v,X} and let u = \frac{1}{r} ∑_{g∈G_{v,X}} g ∈ Q_{ℓ}[G_{v,X}]. Then

u(V_{ℓ}(X_{w})) = V_{ℓ}(X_{w})^{G_{v,X}} = V_{ℓ}(X_{v}), u(V_{ℓ}(T_{w})) = V_{ℓ}(T_{w})^{G_{v,X}} = V_{ℓ}(T_{v}),

W_{ℓ} = (1 − u)(V_{ℓ}(X_{w})/V_{ℓ}(T_{w})), and S_{ℓ} = W_{ℓ} ∩ (T_{ℓ}(X_{w})/T_{ℓ}(T_{w})).

Therefore W_{ℓ} and S_{ℓ} are independent of the choice of λ, and we have a G_{v,X}- invariant e_{λ}-orthogonal splitting

V_{ℓ}(X_{w})/V_{ℓ}(T_{w}) ≅ V_{ℓ}(X_{v})/V_{ℓ}(T_{v}) ⊕ W_{ℓ}.

Since G_{v,X} acts trivially on V_{ℓ}(X_{v})/V_{ℓ}(T_{v}), the map φ_{ℓ} factors through Aut(T) × Aut(W_{ℓ}). Further, since e_{λ} is non-degenerate, alternating, and G_{v,X}-invariant, φ_{ℓ} factors through Aut(T) × Aut(W_{ℓ}, e_{λ}) ≅ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Q_{ℓ}). See p. 360 of [6] for the integrality and ℓ-independence.

If r is not divisible by ℓ, then u ∈ Z_{ℓ}[G_{v,X}],

u(T_{ℓ}(X_{w})) = T_{ℓ}(X_{w})^{G_{v,X}} = T_{ℓ}(X_{v}), u(T_{ℓ}(T_{w})) = T_{ℓ}(T_{w})^{G_{v,X}} = T_{ℓ}(T_{v}),

S_{ℓ} = (1 − u)(T_{ℓ}(X_{w})/T_{ℓ}(T_{w})),

and we have a G_{v,X}-invariant e_{λ}-orthogonal splitting

T_{ℓ}(X_{w})/T_{ℓ}(T_{w}) ≅ T_{ℓ}(X_{v})/T_{ℓ}(T_{v}) ⊕ S_{ℓ}.

To derive (iii), apply Proposition 3.3 and Remark 3.2. To derive (ii), suppose deg(λ) is not divisible by ℓ. Then E_{λ} is perfect. Applying Proposition 3.5 with M = T_{ℓ}(X), N = T_{ℓ}(X_{w}), R = Z_{ℓ}, and e = E_{λ}, we deduce that e_{λ} is perfect, giving (ii).

Next we give a variation of Theorem 5.2, whose proof and statement are of independent interest. Retain the notation from the proof of Theorem 5.2. Since G_{v,X} is finite, there exists a G_{v,X}-invariant polarization δ on the abelian variety B.

Theorem 5.3. If ℓ does not divide deg(δ), then the action of G_{v,X} on T and on B induces injections

G_{v,X} ↪ Aut(T) × Aut(B, δ) ↪ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z_{ℓ})

where the projections onto the first factors are independent of ℓ, and the characteristic polynomials of the projections onto the second factors have integer coefficients independent of ℓ. In particular, the conclusion holds for all ℓ ≠ p, whenever a − a_{v} = 0 or 1 (e.g., if X or B is an elliptic curve).

Proof. As I_{v}-modules, T_{ℓ}(B_{w}) ≅ T_{ℓ}(X_{w})/T_{ℓ}(T_{w}), T_{ℓ}(B_{v}) ≅ T_{ℓ}(X_{v})/T_{ℓ}(T_{v}),

T_{ℓ}(B) ≅ T_{ℓ}(B_{w})/T_{ℓ}(B_{v}), and T_{ℓ}(T) ≅ T_{ℓ}(T_{w})/T_{ℓ}(T_{v}),

and we have dim(T) = t − t_{v} and dim(B) = a − a_{v}. By considering the pairing E_{δ} on T_{ℓ}(B) induced by δ, it follows that the image of G_{v,X} in Aut(T_{ℓ}(B)) lies in Aut(T_{ℓ}(B), E_{δ}). Since ℓ does not divide deg(δ), then E_{δ} is perfect. The actions of G_{v,X} on T and B therefore induce a homomorphism

η_{ℓ} : G_{v,X} → Aut(T) × Aut(T_{ℓ}(B), E_{δ}) ≅ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z_{ℓ}). By Proposition 4.3ii, G_{v,X} acts faithfully on T_{ℓ}(X_{w}). The kernel of η_{ℓ} consists of elements in G_{v,X} ⊂ Aut(T_{ℓ}(X_{w})) which act as the identity on T_{ℓ}(T_{w}) and on T_{ℓ}(X_{w})/T_{ℓ}(T_{w}). Therefore, these elements act as unipotent operators on T_{ℓ}(X_{w}). Since they also have finite order, it follows that η_{ℓ} is injective. As before, see p. 360 of [6] for integrality and ℓ-independence. When dim(B) = 0 or 1, we may suppose that deg(δ) = 1.

Remark 5.4. If ℓ ≠ 2, then reducing the second factor modulo ℓ in Theorem 5.2iii or 5.3 gives

G_{v,X} ↪ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(F_{ℓ}).

If either p#G_{v,X} or p deg(δ) is odd, then G_{v,X} ↪ GL_{t−t_{v}}(Z) × Sp_{2(a−a_{v})}(Z/4Z).

Remark 5.5. It follows from Corollary 6.3 that the statement in Theorem 5.2iii holds true whenever p ≠ ℓ > 1 + max{t − t_{v}, 2(a − a_{v})} or p ≠ ℓ > 2d + 1.

Remark 5.6. Theorems 5.2 and 5.3 and Remark 5.4 have led us to the question of when a finite interia group embedded in Sp_{2D}(Q_{ℓ}) can also be embedded in Sp_{2D}(Z_{ℓ}) or in Sp_{2D}(F_{ℓ}) in such a way that the characteristic polynomials are preserved. Suppose p is a prime number, and G is a finite group with a normal Sylow-p subgroup P such that G/P is cyclic. Suppose ℓ is a prime number, ℓ ≠ p, V is a 2D-dimensional Q_{ℓ}-vector space, e : V × V → Q_{ℓ} is a non-degenerate alternating bilinear form, and φ : G ↪ Aut(V, e) is an injection. Does there always exist a G-stable Z_{ℓ}-lattice S in V with a perfect G-invariant alternating Z_{ℓ}-valued pairing? The answer is no. However, the answer is yes if ℓ > D + 1, and this bound is sharp. Does there always exist an injection η : G ↪ Sp_{2D}(F_{ℓ}) such that for all g ∈ G, the characteristic polynomial of η(g) is equal to the characteristic polynomial of φ(g) mod ℓ? The answer is no, but is yes if ℓ > 3, and this bound is sharp. Proofs and counterexamples (including counterexamples with inertia groups of the form G_{v,X} for abelian surfaces X) will appear in a later paper.

Remark 5.7. Note that if #G_{v,X} is not divisible by p, then X acquires semistable reduction over a tamely ramified extension; see [3] for a study of Néron models in this important case.

6. Bounds on the order of G_{v,X}

One can use Theorem 5.2, Remark 5.1, and group theory to obtain more precise information about the finite group G_{v,X}. In Corollary 6.3 below we give one such result. The next two results, along with Theorem 5.2, will be used to prove Corollary 6.3.

Lemma 6.1. If ℓ is a prime number, ℓ ≡ 5 (mod 8), r and m are positive integers, and Sp_{2m}(F_{ℓ}) has an element of (exact) order 2^{r}, then 2^{r−1} ≤ 2m.

Proof. We may assume r ≥ 3. Let ζ be a primitive 2^{r}-th root of unity in F̄_{ℓ}. It is easy to check that the condition ℓ ≡ 5 (mod 8) implies that [F_{ℓ}(ζ) : F_{ℓ}] = 2^{r−2} and that ζ^{−1} is not a conjugate of ζ. It follows that in F_{ℓ}[x] we have Φ_{2^{r}} = fg, where Φ_{2^{r}} is the 2^{r}-th cyclotomic polynomial, f and g are irreducible polynomials in F_{ℓ}[x] of degree 2^{r−2}, and the roots of g are the inverses of the roots of f. Let γ denote an element of order 2^{r} in Sp_{2m}(F_{ℓ}). If α is an eigenvalue of γ, then so is α^{−1}. It follows that the characteristic polynomial of γ is divisible by Φ_{2^{r}}. Therefore, 2^{r−1} ≤ 2m.

Proposition 6.2. If q is a prime number, r and m are positive integers, and for all prime numbers ℓ in a set of density 1 the group Sp_{2m}(F_{ℓ}) contains an element of (exact) order q^{r}, then φ(q^{r}) ≤ 2m.

Proof. If q = 2 the result follows from Lemma 6.1. Suppose q is odd. Then (Z/q^{r}Z)^{×} is cyclic, and by the Chebotarev density theorem there is a set of primes ℓ of positive density such that Gal(F_{ℓ}(ζ_{q^{r}})/F_{ℓ}) ≅ (Z/q^{r}Z)^{×} ≅ Gal(Q(ζ_{q^{r}})/Q). If q ≠ ℓ, then GL_{n}(F_{ℓ}) contains an element of order q^{r} if and only if [F_{ℓ}(ζ_{q^{r}}) : F_{ℓ}] ≤ n (see [20]). Therefore, φ(q^{r}) ≤ 2m.

An explicit description of all possible orders of elements of general linear groups over arbitrary fields is given in [20].

Let [ ] denote the greatest integer function, let s(n, q) = ∑_{j=0}^{∞} [\frac{n}{q^{j}(q−1)}], and let J(n) = ∏ q^{s(n,q)}, where q runs over the prime numbers. Note that the prime divisors of J(n) are the primes q ≤ n + 1. For example, J(0) = 1, J(1) = 2, and J(2) = 24. For n ≥ 1, Theorem 3.2 of [16] shows that J(n) < (4.462n)^{n} if n is even and J(n) < \sqrt{2}(4.462n)^{n} if n is odd. The method of Minkowski and Serre ([8] and pp. 119–121 of [12]; see also Formula 3.1 of [16]) shows that, for all N ≥ 3, J(2m) is equal to the greatest common divisor of the orders of the groups Sp_{2m}(F_{ℓ}), for primes ℓ ≥ N. Further ([14]), J(n) is the least common multiple of the orders of the finite subgroups of GL_{n}(Q) (or equivalently, of GL_{n}(Z)). While J(n) is optimal from the point of view of divisibility, there are sharper upper bounds on the orders of finite subgroups of GL_{n}(Q). The determination of the finite subgroups of maximum order for general linear groups over Q and over cyclotomic fields is given in [4].

Let

r_{p} = s(t − t_{v}, p) + s(2(a − a_{v}), p), M_{v,X} = max{t − t_{v}, 2(a − a_{v})}, and for all primes q such that p ≠ q ≤ M_{v,X} + 1 let

r_{q} = 1 + [log_{q} (\frac{M_{v,X}}{q − 1} )] .

If X has semistable reduction at v let N_{v,X} = 1, and otherwise let N_{v,X} = ∏ q^{r_{q}}, where the product runs over all prime numbers q ≤ M_{v,X} + 1 (this might include q = p). Let Q_{v,X} denote the largest prime divisor of #G_{v,X} (let Q_{v,X} = 1 if I_{v} = I_{v,X}).

Corollary 6.3. The order of G_{v,X} divides N_{v,X}. In particular, Q_{v,X} ≤ M_{v,X}+1 ≤ 2d + 1, and #G_{v,X} divides J(t − t_{v})J(2(a − a_{v})) and divides J(2d).

Proof. By Theorem 5.2ii, #G_{v,X} divides J(t−t_{v})J(2(a−a_{v})), and therefore Q_{v,X} ≤ M_{v,X} + 1. Note that J(t − t_{v})J(2(a − a_{v})) divides J(2d), since t − t_{v} + 2(a − a_{v}) ≤ t + 2a = 2d − t ≤ 2d. As noted in Remark 5.1, the prime-to-p part of G_{v,X} is cyclic. Suppose q is a prime divisor of #G_{v,X}, and q ≠ p. Then GL_{n}(Q) contains an element of order q^{r} if and only if φ(q^{r}) ≤ n, i.e., if and only if r ≤ 1+[log_{q} (\frac{n}{q−1})]. The result now follows from Proposition 6.2.

7. Applications

Retain the notation of the previous sections. The next result follows immediately from Corollary 6.3 and Theorem 4.2iii,iv.

Corollary 7.1. Suppose L is a finite separable extension of F, and w is the restriction of v̄ to L. Suppose X has semistable reduction at w but not at v. Then [I_{v} : I_{w}] has a prime divisor q such that q ≤ Q_{v,X} ≤ M_{v,X} + 1 ≤ 2d + 1.

Remark 7.2. Suppose L is a finite separable extension of F, and w is the restriction of v̄ to L. Let k_{w} and k_{v} denote the residue fields and let e(w/v) = [w(L^{×}) : v(F^{×})] (= the ramification degree). Then [I_{v} : I_{w}] = e(w/v)[k_{w} : k_{v}]_{i}, where the subscript i denotes the inseparable degree (see Proposition 21 on p. 32 of [9] for the case where L/F is Galois. In the non-Galois case, take a Galois extension L′ of F which contains L, and apply the result to L′/L and L′/F, to obtain the result for L/F). Taking completions, then [L_{w} : F_{v}] = e(w/v)[k_{w} : k_{v}] = [I_{v} : I_{w}][k_{w} : k_{v}]_{s}, where the subscript s denotes the separable degree. Therefore, [I_{v} : I_{w}] divides [L_{w} : F_{v}].

Corollary 7.3. Suppose L is a finite separable extension of F. Suppose in addition that either F is complete with respect to v, or L/F is Galois. Suppose X has semistable reduction at the restriction w of v̄ to L, but does not have semistable reduction at v. Then [L : F] has a prime divisor q such that q ≤ Q_{v,X}. In particular, if [L : F] is a power of a prime q, then q ≤ Q_{v,X}.

Proof. Under our assumptions on L/F, its degree is divisible by [I_{v} : I_{w}]. The result now follows from Corollary 7.1.

Recall that there exists a finite Galois extension L of F such that X has semistable reduction at the extensions of v to L (see Proposition 3.6 of [6]).

Corollary 7.4. Let r = #G_{v,X} and let ζ_{r} denote a primitive r-th root of unity. Suppose that r is not divisible by p. Then there is a cyclic degree r extension L of F (ζ_{r}) such that X acquires semistable reduction over every extension of v to L. If either F = F (ζ_{r}) or p > Q_{v,X}, then there exists a finite Galois extension L of F of degree prime to p such that X acquires semistable reduction over every extension of v to L.

Proof. Let L be the field obtained by adjoining an r-th root of a uniformizing parameter to F (ζ_{r}). Then L/F is Galois, F (ζ_{r})/F is unramified above v, L/F (ζ_{r}) is totally ramified, and Gal(L/F (ζ_{r})) ≅ G_{v,X} ≅ Z/rZ. Let w be the restriction of v̄ to L. By the construction of L, we have I_{w} ≅ I_{v,X}. By Theorem 4.2iii, X has semistable reduction at w. Since L/F is Galois, X has semistable reduction at every extension of v to L by Theorem 2.1. If p > Q_{v,X}, then p does not divide φ(r) by Corollary 6.3, and therefore p does not divide [F (ζ_{r}) : F].

By Corollary 6.3, Q_{v,X} can be replaced by 2d + 1 (or by M_{v,X} + 1) in Corollaries 7.3 and 7.4.

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Department of Mathematics, Ohio State University, 231 W. 18 Avenue, Columbus, Ohio 43210–1174, USA

E-mail address: silver@math.ohio-state.edu

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

E-mail address: zarhin@math.psu.edu