REDUCTION OF ABELIAN VARIETIES

A. SILVERBERG AND YU. G. ZARHIN

1. Introduction

In this paper we study the reduction of abelian varieties. We assume F is a field with a discrete valuation v, X is an abelian variety over F, and n is an integer not divisible by the residue characteristic.

In Part 1 we give criteria for semistable reduction. Suppose n ≥ 5. In Theorem 4.4 we show that X has semistable reduction if and only if (σ − 1)^{2} = 0 on the n-torsion in X, for every σ in the absolute inertia group. In Theorem 4.5 we show (using Theorem 4.4) that X has semistable reduction if and only if there exists a subgroup of n-torsion points such that the absolute inertia group acts trivially on both it and its orthogonal complement with respect to the e_{n}-pairing. We deduce as special cases both Raynaud’s criterion (that the abelian variety have full level n structure for n ≥ 3; see Theorem 4.2) and the criterion of [7] (that the abelian variety have partial level n structure for n ≥ 5; see Theorem 4.6). We also obtain a (near) converse to the criterion of [7]. The proofs are based on the fundamental results of Grothendieck on semistable reduction of abelian varieties (see [3]). In §5 we allow n < 5. In §6 we give a measure of potentially good reduction. We discuss other measures of potentially good reduction in Part 2.

In Part 2 we study Néron models of abelian varieties with potentially good reduction and torsion points of small order. Suppose that the valuation ring is henselian and the residue field is algebraically closed. If X has good reduction, then X_{n} ⊆ X(F) (this is an immediate corollary of the existence of Néron models; see Lemma 7.4 below). On the other hand, if X_{n} ⊆ X(F) and n ≥ 3, then by virtue of Raynaud’s criterion for semistable reduction, X has good reduction. Notice that the failure of X to have good reduction is measured by the dimension u of the unipotent radical of the special fiber of the Néron model of X. In particular, u = 0 if and only if X has good reduction. In general, 0 ≤ u ≤ dim(X). The equality u = dim(X) says that X has purely additive reduction. Another measure of the deviation from good reduction is the (finite) group of connected components Φ of the special fiber of the Néron model. If X has good reduction then Φ = {0}, but the converse statement is not true in general.

The aim of §8 is to connect explicitly the invariants u and Φ with the failure of X(F) to contain all the n-torsion points. This failure can be measured by the index [X_{n} : X_{n}(F)]. We assume that at least “half” of the n-torsion points are rational over F. More precisely, we assume that there exists an F -rational polarization λ on X and a maximal isotropic (with respect to the pairing e_{λ,n} induced from the Weil e_{n}-pairing by λ) subgroup of X_{n} consisting of F -rational points. If in addition n ≥ 5, then X has good reduction (see Theorem 7.4 of [7]), and therefore u = 0, Φ = {0}, and X_{n} = X_{n}(F). Therefore, we have to investigate only the cases n = 2, 3, and 4. Let Φ′ denote the prime-to-p part of Φ, where p is the residue characteristic (with Φ′ = Φ if p = 0).

We show that if n = 2 then Φ′ is an elementary abelian 2-group and [X_{2} : X_{2}(F)]#Φ′ = 4^{u}, if n = 3 then [X_{3} : X_{3}(F)] = 3^{u} and Φ′ ≅ (Z/3Z)^{u}, and if n = 4 then X_{2} ⊆ X(F), [X_{4} : X_{4}(F)] = 4^{u}, and Φ′ ≅ (Z/2Z)^{2u}. If instead of assuming partial level n structure we assume that all the points of order 2 on X are defined over F, then [X_{4} : X_{4}(F)] = 4^{u} and Φ′ ≅ (Z/2Z)^{2u}.

Earlier work on abelian varieties with potentially good reduction and on groups of connected components of Néron models has been done by Serre and Tate [6], Silverman [13], Lenstra and Oort [4], Lorenzini [5], and Edixhoven [2].

Silverberg would like to thank the IHES and the Bunting Institute for their hospitality, and the NSA and the Science Scholars Fellowship Program at the Bunting Institute for financial support. Zarhin would like to thank the NSF for financial support. He also would like to thank the organizers of the NATO/CRM 1998 Summer School on the Arithmetic and Geometry of Algebraic Cycles for twelve wonderful days in Banff.

2. Notation and definitions

If F is a field, let F^{s} denote a separable closure. Suppose that X is an abelian variety defined over F, and n is a positive integer not divisible by the characteristic of F. Let X^{∗} denote the dual abelian variety of X, let X_{n} denote the kernel of multiplication by n in X(F^{s}), let X_{n}^{∗} denote the kernel of multiplication by n in X^{∗}(F^{s}), and let μ_{n} denote the Gal(F^{s}/F)-module of n-th roots of unity in F^{s}. The e_{n}-pairing

e_{n} : X_{n} × X_{n}^{∗} → μ_{n}

is a Gal(F^{s}/F)-equivariant nondegenerate pairing (see §74 of [14]). If S is a subgroup of X_{n}, let

S^{⊥_{n}} = {y ∈ X_{n}^{∗} : e_{n}(x, y) = 1 for every x ∈ S} ⊆ X_{n}^{∗}.

If λ is a polarization on X, define

e_{λ,n} : X_{n} × X_{n} → μ_{n}

by e_{λ,n}(x, y) = e_{n}(x, λ(y)) (see §75 of [14]). Then

σ(e_{λ,n}(x_{1}, x_{2})) = e_{σ(λ),n}(σ(x_{1}), σ(x_{2}))

for every σ ∈ Gal(F^{s}/F) and x_{1}, x_{2} ∈ X_{n}. If n is relatively prime to the degree of the polarization λ, then the pairing e_{λ,n} is nondegenerate. If ℓ is a prime not equal to the characteristic of F, and d = dim(X), let

ρ_{ℓ,X} : Gal(F^{s}/F) → Aut(T_{ℓ}(X)) ≅ M_{2d}(Z_{ℓ})

denote the ℓ-adic representation on the Tate module T_{ℓ}(X) of X, and let V_{ℓ}(X) = T_{ℓ}(X) ⊗_{Z_{ℓ}} Q_{ℓ}. Let I denote the identity matrix in M_{2d}(Z_{ℓ}).

If L is a Galois extension of F, v is a discrete valuation on F, and w is an extension of v to L, let I(w/v) denote the inertia subgroup at w of Gal(L/F). If X is an abelian variety over F, let X_{v} denote the special fiber of the Néron model of X at v and let X_{v}^{0} denote its identity connected component. Let a, u, and t denote, respectively, the abelian, unipotent, and toric ranks of X_{v}. Then a + u + t = dim(X). If p ( ≥ 0) is the residue characteristic of v, let Φ′ denote the prime-to-p part of the group of connected components of the special fiber of the Néron model of X at v (with Φ′ the full group of components if p = 0).

Definition 2.1. If v is a discrete valuation on a field F, we say the valuation ring is strictly henselian if the valuation ring is henselian and the residue field is algebraically closed.

Definition 2.2. Suppose L/F is an extension of fields, w is a discrete valuation on L, and v is the restriction of w to F. We say that w/v is unramified if a uniformizing element of the valuation ring for v induces a uniformizing element of the valuation ring for w and the residue field extension is separable (see Definition 1 on page 78 of [1]).

Remark 2.3 (Remark 5.3 of [7]). Suppose v is a discrete valuation on a field F, and m is a positive integer not divisible by the residue characteristic. Then every degree m Galois extension of F totally ramified at v is cyclic. If F (ζ_{m}) = F, then F has a cyclic extension of degree m which is totally ramified at v. In particular, if the residue characteristic is not 2 then F has a quadratic extension which is (totally and tamely) ramified at v. If the valuation ring is henselian and the residue field is separably closed, then F = F (ζ_{m}) and therefore F has a cyclic totally ramified extension of degree m. (See Remark 5.3 of [7].) Note also that F has no non-trivial unramified extensions if and only if the valuation ring is henselian and the residue field is separably closed.

Part 1. Semistable reduction of abelian varieties

3. Preliminaries

Definition 3.1. If k is a positive integer, define a finite set of prime powers N(k) by

N(k) = {prime powers ℓ^{m} : 0 ≤ m(ℓ − 1) ≤ k}.

For example,

N(1) = {1, 2}, N(2) = {1, 2, 3, 4},
N(3) = {1, 2, 3, 4, 8}, N(4) = {1, 2, 3, 4, 5, 8, 9, 16}.

Theorem 3.2. Suppose n and k are positive integers, O is an integral domain of characteristic zero such that no rational prime which divides n is a unit in O, α ∈ O, α has finite multiplicative order, and (α − 1)^{k} ∈ nO. If n ∉ N(k), then α = 1. In particular, if (α − 1)^{2} ∈ nO and n ≥ 5, then α = 1.

Proof. See Corollary 3.3 of [9].

Lemma 3.3 (Lemma 5.2 of [7]). Suppose that d and n are positive integers, and for each prime ℓ which divides n we have a matrix A_{ℓ} ∈ M_{2d}(Z_{ℓ}) such that the characteristic polynomials of the A_{ℓ} have integral coefficients independent of ℓ, and such that (A_{ℓ} − I)^{2} ∈ nM_{2d}(Z_{ℓ}). Then for every eigenvalue α of A_{ℓ}, (α − 1)/\sqrt{n} satisfies a monic polynomial with integer coefficients.

Lemma 3.4 (Lemma 4.2 of [10]). Suppose v is a discrete valuation on a field F with residue characteristic p ≥ 0, m is a positive integer, ℓ is a prime, p does not divide mℓ, K is a degree m extension of F which is totally ramified above v, and v̄ is an extension of v to a separable closure K^{s} of K. Suppose that X is an abelian variety over F, and for every σ ∈ I(v̄/v), all the eigenvalues of ρ_{ℓ,X}(σ) are m-th roots of unity. Then X has semistable reduction at the extension of v to K.

4. Criteria for semistable reduction

Theorem 4.1 (Galois Criterion for Semistable Reduction). Suppose X is an abelian variety over a field F, v is a discrete valuation on F, ℓ is a prime not equal to the residue characteristic of v, v̄ is an extension of v to F^{s}, and I = I(v̄/v). Then the following are equivalent:

(i) X has semistable reduction at v, (ii) I acts unipotently on T_{ℓ}(X); i.e., all the eigenvalues of ρ_{ℓ,X}(σ) are 1, for

every σ ∈ I, (iii) for every σ ∈ I, (ρ_{ℓ,X}(σ) − I)^{2} = 0.

Proof. See Proposition 3.5 and Corollaire 3.8 of [3] and Theorem 6 on p. 184 of [1].

Theorem 4.2 (Raynaud Criterion for Semistable Reduction). Suppose X is an abelian variety over a field F with a discrete valuation v, m is a positive integer not divisible by the residue characteristic of v, and the points of X_{m} are defined over an extension of F which is unramified over v. If m ≥ 3, then X has semistable reduction at v.

Proof. See Proposition 4.7 of [3].

Proposition 4.3. Suppose X is an abelian variety over a field F, v is a discrete valuation on F, n is an integer not divisible by the residue characteristic of v, v̄ is an extension of v to F^{s}, and I = I(v̄/v). Let S = X_{n}^{I}, the elements of X_{n} on which I acts as the identity. If X has semistable reduction at v, then

(i) (σ − 1)^{2}X_{n} = 0 for every σ ∈ I, and (ii) I acts as the identity on S^{⊥_{n}}.

Proof. Suppose X has semistable reduction at v. By Theorem 4.1, we have (i). It follows that σ^{n} = 1 on X_{n}. Since n is not divisible by the residue characteristic, X_{n} is tamely ramified over F. Let J denote the first ramification group. Then the action of I on X_{n} factors through I/J. Let τ denote a lift to I of a topological generator of the pro-cyclic group I/J. Since

e_{n}((τ − 1)X_{n}, (X_{n}^{∗})^{I}) = 1,

we have

#((X_{n}^{∗})^{I})#((τ − 1)X_{n}) ≤ #X_{n}^{∗}.

The map from X_{n} to (τ − 1)X_{n} defined by y ↦ (τ − 1)y defines a short exact sequence

0 → S → X_{n} → (τ − 1)X_{n} → 0.

Therefore,

#S#((τ − 1)X_{n}) = #X_{n} = #S#S^{⊥_{n}}.

Similarly,

#((X_{n}^{∗})^{I})#((τ − 1)X_{n}^{∗}) = #X_{n}^{∗}.

Therefore,

#S^{⊥_{n}} = #((τ − 1)X_{n}) ≤ #((τ − 1)X_{n}^{∗}).

Since (τ − 1)X_{n}^{∗} ⊆ S^{⊥_{n}}, we conclude that

S^{⊥_{n}} = (τ − 1)X_{n}^{∗}.

By (i), we have (τ − 1)^{2}X_{n} = 0. It follows from the natural Gal(F^{s}/F)-equivariant isomorphism X_{n}^{∗} ≅ Hom(X_{n}, μ_{n}) that (τ − 1)^{2}X_{n}^{∗} = 0, and therefore I acts as the identity on S^{⊥_{n}}.

Theorem 4.4. Suppose X is an abelian variety over a field F, v is a discrete valuation on F, n is an integer not divisible by the residue characteristic of v, n ≥ 5, v̄ is an extension of v to F^{s}, and I = I(v̄/v). Then X has semistable reduction at v if and only if (σ − 1)^{2}X_{n} = 0 for every σ ∈ I.

Proof. If X has semistable reduction at v then for every σ ∈ I we have (σ−1)^{2}X_{n} = 0, by Proposition 4.3i.

Conversely, suppose n ≥ 5 and (σ − 1)^{2}X_{n} = 0 for every σ ∈ I. Let I′ ⊆ I be the inertia group for the prime below v̄ in a finite Galois extension of F over which X has semistable reduction. Take σ ∈ I. Then σ^{m} ∈ I′ for some m. Let ℓ be a prime divisor of n. Theorem 4.1 implies that (ρ_{ℓ,X}(σ)^{m} − I)^{2} = 0. Let α be an eigenvalue of ρ_{ℓ,X}(σ). Then (α^{m} − 1)^{2} = 0. Therefore, α^{m} = 1. By our hypothesis,

(ρ_{ℓ,X}(σ) − I)^{2} ∈ nM_{2d}(Z_{ℓ}),

where d = dim(X). By Theorem 4.3 on p. 359 of [3], the characteristic polynomial of ρ_{ℓ,X}(σ) has integer coefficients which are independent of ℓ. By Lemma 3.3, (α − 1)^{2} ∈ nZ̄, where Z̄ denotes the ring of algebraic integers. Since n ≥ 5, by Theorem 3.2 we have α = 1 (i.e., I acts unipotently on T_{ℓ}(X)). By Theorem 4.1, X has semistable reduction at v.

Theorem 4.5. Suppose X is an abelian variety over a field F, v is a discrete valuation on F, n is an integer not divisible by the residue characteristic of v, n ≥ 5, v̄ is an extension of v to F^{s}, and I = I(v̄/v). Then X has semistable reduction at v if and only if there exists a subgroup S of X_{n} such that I acts as the identity on S and on S^{⊥_{n}}.

Proof. Suppose there exists a subgroup S as in the statement of the theorem. The map x ↦ (y ↦ e_{n}(x, y)) induces a Gal(F^{s}/F)-equivariant isomorphism from X_{n}/S onto Hom(S^{⊥_{n}}, μ_{n}). Suppose σ ∈ I. Then σ = 1 on S^{⊥_{n}} and on μ_{n}. Therefore, σ = 1 on X_{n}/S. Thus, (σ − 1)^{2}X_{n} ⊆ (σ − 1)S = 0. By Theorem 4.4, X has semistable reduction at v.

Conversely, suppose X has semistable reduction at v. Let S = X_{n}^{I}, and apply Proposition 4.3ii.

Theorem 4.6. Suppose X is an abelian variety over a field F, v is a discrete valuation on F, λ is a polarization on X defined over an extension of F which is unramified over v, n is a positive integer not divisible by the residue characteristic of v, and n ≥ 5.

(i) If X̃_{n} is a maximal isotropic subgroup of X_{n} with respect to e_{λ,n}, and the points of X̃_{n} are defined over an extension of F which is unramified over v, then X has semistable reduction at v.

(ii) Conversely, if X has semistable reduction at v, and the degree of the polarization λ is relatively prime to n, then there exists a maximal isotropic subgroup of X_{n} with respect to e_{λ,n}, all of whose points are defined over an extension of F which is unramified over v.

Proof. Under the hypotheses in (i), let S = X̃_{n}. Then S^{⊥_{n}} = λ(S), and X has semistable reduction at v by applying Theorem 4.5.

Conversely, suppose X has semistable reduction at v. Let v̄ be an extension of v to F^{s} and let I = I(v̄/v). Let S = X_{n}^{I}. If G is a subgroup of X_{n}, let

G^{⊥_{λ,n}} = {y ∈ X_{n} : e_{λ,n}(x, y) = 1 for every x ∈ G}.

Since the degree of λ is relatively prime to n, λ induces an isomorphism between S^{⊥_{λ,n}} and S^{⊥_{n}}. Since λ is defined over an unramified extension, I acts as the identity on S^{⊥_{λ,n}} by Proposition 4.3ii. Therefore, S^{⊥_{λ,n}} ⊆ S = X_{n}^{I}. The pairing e_{λ,n} induces a nondegenerate pairing on S/S^{⊥_{λ,n}}. Let H be the inverse image in S (under the natural projection) of a maximal isotropic subgroup of S/S^{⊥_{λ,n}}. It is easy to check that H is a maximal isotropic subgroup of X_{n} with respect to e_{λ,n}, proving (ii).

Remarks 4.7. Raynaud’s criterion (Theorem 4.2) follows from Theorem 4.5 by letting n = m^{2} and S = X_{m} ⊂ X_{n} (since then S^{⊥_{n}} = X_{m}^{∗}, the dual Galois module of X_{m}, and n ≥ 5 whenever m ≥ 3). The converse of Raynaud’s criterion is clearly false, i.e., semistable reduction does not imply that the n-torsion points are unramified (for n ≥ 3 and n not divisible by the residue characteristic), as can be seen, for example, by comparing Raynaud’s criterion with the Néron-Ogg-Shafarevich criterion for good reduction, and considering an abelian variety with semistable but not good reduction.

Theorem 4.6i is Theorem 6.2 of [7]. Similarly, the other results of [7] and of §3 of [8] can readily be generalized to the setting of Theorem 4.5. Theorem 4.6ii shows that the sufficient condition for semistability given in Theorem 6.2 of [7] comes close to being a necessary condition. Note that Theorem 4.6ii would be false if the condition on the degree of the polarization were omitted.

Definition 4.8. Suppose v is a discrete valuation on F of residue characteristic p. We say v satisfies (*) if at least one of the following conditions is satisfied:

(a) p ≠ 2,

(b) the valuation ring is henselian and the residue field is separably closed.

The techniques of the above proofs can be extended to prove the following result. The proof will appear in [12].

Theorem 4.9. Suppose X is an abelian variety over a field F, and v is a discrete valuation on F of residue characteristic p ≥ 0. Suppose k ∈ Z, and 0 < k < 2dim(X).

(i) If either X has semistable reduction at v, or k is even and X has purely additive reduction at v which becomes semistable over a quadratic extension of F, then

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z_{ℓ}) = 0

for every σ ∈ I and every prime ℓ ≠ p, and

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z/nZ) = 0

for every σ ∈ I and every positive integer n not divisible by p. (ii) Suppose n is a positive integer not divisible by p, and

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z/nZ) = 0

for every σ ∈ I. Suppose L is a degree R(k + 1, n) extension of F which is totally ramified above v, and let w be the extension of v to L. If k is odd, then X has semistable reduction at w. If k is even and v satisfies (*), then either X has semistable reduction at w, or X has purely additive reduction at w which becomes semistable over a quadratic extension of L.

If we restrict to the case where n ∉ N(k + 1), we obtain the following result. This result gives necessary and sufficient conditions for semistable reduction, and also necessary and sufficient conditions for X to have either semistable reduction or purely additive reduction which becomes semistable over a quadratic extension.

Corollary 4.10. Suppose X is an abelian variety over a field F, v is a discrete valuation on F of residue characteristic p ≥ 0, k and n are positive integers, ℓ is a prime number, k < 2dim(X), n and ℓ are not divisible by p, and n ∉ N(k + 1).

(i) Suppose k is odd. Then the following are equivalent: (a) X has semistable reduction at v,

(b) for every σ ∈ I,

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z_{ℓ}) = 0,

(c) for every σ ∈ I,

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z/nZ) = 0.

(ii) Suppose k is even and v satisfies (*). Then the following are equivalent:

(a) either X has semistable reduction at v, or X has purely additive reduction at v which becomes semistable over a quadratic extension of F,

(b) for every σ ∈ I,

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z_{ℓ}) = 0,

(c) for every σ ∈ I,

(σ − 1)^{k+1}H_{ét}^{k}(X ×_{F} F^{s}, Z/nZ) = 0.

5. Exceptional n

In this section we discuss briefly the “exceptional” cases n = 2, 3, 4. For the proofs, and for examples which show the results are sharp, we refer the reader to [10].

First, let us state the following “one-way” generalization of Theorem 4.5.

Theorem 5.1. Suppose X is an abelian variety over a field F, v is a discrete valuation on F, and n is an integer greater than 1 which is not divisible by the residue characteristic of v. Suppose there exists a subgroup S of X_{n} such that I acts as the identity on S and on S^{⊥_{n}}. Then X has semistable reduction over every degree R(n) extension of F totally ramified above v.

It turns out that the converse statement is not true. However, the following result gives an “approximate converse”.

Theorem 5.2. Suppose n = 2, 3, or 4, respectively. Suppose X is an abelian variety over a field F, and v is a discrete valuation on F whose residue characteristic does not divide n. Suppose L is an extension of F of degree 4, 3, or 2, respectively, which is totally ramified above v. Then the following are equivalent:

(i) X has semistable reduction over L above v, (ii) there exist an abelian variety Y over a finite extension K of F unramified

above v, a separable K-isogeny φ : X → Y, and a subgroup S of Y_{n} such that I acts as the identity on S and on S^{⊥_{n}}.

Further, φ can be taken so that its kernel is killed by 8, 9, or 4, respectively. If X has potentially good reduction at v, then φ can be taken so that its kernel is killed by 2, 3, or 2, respectively.

In the case of low-dimensional X this result may be improved as follows.

Theorem 5.3. In Theorem 5.2, with d = dim(X), φ can be taken so that its kernel is killed by 4 if d = 3 and n = 2, by 3 if d = 2 and n = 3, and by 2 if d = n = 2. If d = 1, then we can take Y = X and φ the identity map.

In the case of elliptic curves this implies the following statement.

Corollary 5.4. Suppose X is an elliptic curve over a field F, and v is a discrete valuation on F of residue characteristic p ≥ 0.

(a) If p ≠ 2, then X has semistable reduction above v over a totally ramified quartic extension of F if and only if X has an I-invariant point of order 2.

(b) If p ≠ 3, then X has semistable reduction above v over a totally ramified cubic extension of F if and only if X has an I-invariant point of order 3.

(c) If p ≠ 2, then X has semistable reduction above v over a quadratic extension of F if and only if either X has an I-invariant point of order 4, or all the points of order 2 on X are I-invariant.

(d) If p ≠ 2 and X has bad but potentially good reduction at v, then X has good reduction above v over a quadratic extension of F if and only if X has no I-invariant point of order 4 and all its points of order 2 are I-invariant.

(e) Suppose p is not 2 or 3. Then the following are equivalent:

(i) X has no I-invariant points of order 2 or 3, (ii) there does not exist a finite separable extension L of F of degree less than

6 such that X has semistable reduction at the restriction of v̄ to L.

(f) Suppose p is not 2 or 3. Then the following are equivalent:

(i) X has no I-invariant points of order 4 or 3 and not all the points of order 2 are I-invariant,

(ii) there does not exist a finite separable extension L of F of degree less than 4 such that X has semistable reduction at the restriction of v̄ to L.

In the case of potentially good reduction the following statement holds true.

Theorem 5.5. Suppose X is an abelian variety over a field F, v is a discrete valuation on F of residue characteristic p ≥ 0, and X has purely additive and potentially good reduction at v.

(a) If p ≠ 2, then X has good reduction above v over a quadratic extension of F if and only if there exists a subgroup S of X_{4} such that I acts as the identity on S and on S^{⊥_{4}}.

(b) If p ≠ 3, then X has good reduction above v over a totally ramified cubic extension of F if and only if there exists a subgroup S of X_{3} such that I acts as the identity on S and on S^{⊥_{3}}.

(c) Suppose p ≠ 2, and L/F is a degree 4 extension, totally ramified above v, which has a quadratic subextension over which X has purely additive reduction. Then X has good reduction above v over L if and only if there exists a subgroup S of X_{2} such that I acts as the identity on S and on S^{⊥_{2}}.

6. A measure of potentially good reduction

Suppose v is a discrete valuation on a field F, and X is an abelian variety over F which has potentially good reduction at v. Let F_{v}^{nr} denote the maximal unramified extension of the completion of F at v, let L denote the smallest extension of F_{v}^{nr} over which X has good reduction, and let

G_{v,X} = Gal(L/F_{v}^{nr}).

Then G_{v,X} can also be characterized as the inertia group of the extension F (X_{n})/F, where n is any integer greater than 2 and not divisible by the residue characteristic of v (see Corollary 2 on p. 497 of [6]). Clearly, X has good reduction at v if and only if G_{v,X} = 1. The finite group G_{v,X} is a measure of how far X is from having good reduction at v.

If A is a matrix, let P_{A} denote its characteristic polynomial. The following result gives constraints on the group G_{v,X}.

Theorem 6.1. Suppose v is a discrete valuation on a field F, and X is a ddimensional abelian variety over F which has potentially good reduction at v. Let G = G_{v,X}. Suppose ℓ is a prime number not equal to the residue characteristic of v. Then the action of Gal(F^{s}/F) on the ℓ-adic Tate module V_{ℓ}(X) induces an embedding

f : G ↪ Sp_{2d}(Q_{ℓ})

which satisfies the following properties.

(i) For every σ ∈ G, the coefficients of P_{f(σ)} are integers which are independent of ℓ. If X has an F -polarization of degree not divisible by ℓ, then one may choose f so that its image lies in Sp_{2d}(Z_{ℓ}).

(ii) If either (ℓ, #G) = 1 or ℓ > d + 1, then there exists an embedding

g : G ↪ Sp_{2d}(Z_{ℓ})

such that P_{g(σ)} = P_{f(σ)} for every σ ∈ G. (iii) If ℓ ≥ 5 then there exists an embedding

h : G ↪ Sp_{2d}(F_{ℓ})

such that P_{h(σ)} ≡ P_{f(σ)} (mod ℓ) for every σ ∈ G.

Further, if ℓ ≥ 5 then there exists an embedding

G ↪ Sp_{2d}(Z_{ℓ})

(which does not necessarily “preserve” the characteristic polynomials obtained from the embedding f).

See [6] for (i), and see [11] for the case (ℓ, #G) = 1 of (ii). The remainder of Theorem 6.1 follows from results whose proofs will appear elsewhere (along with examples which show that the results are sharp). Those results apply more generally to measure how far an abelian variety (not necessarily with potentially good reduction) is from having semistable reduction. In some cases, these results apply to more general finite groups than those obtained as G_{v,X}’s.

Part 2. Néron models of abelian varieties with potentially good reduction

7. Preliminaries

In [9], the following result was obtained as a corollary of Theorem 3.2 above.

Proposition 7.1 (Theorem 6.10a of [9]). Suppose ℓ is a prime, m and r are positive integers, O is an integral domain of characteristic zero with no non-zero infinitely ℓ-divisible elements, ℓO is a maximal ideal of O, M is a free O-module of finite rank, and A is an endomorphism of M of finite multiplicative order such that (A−1)^{m(ℓ−1)ℓ^{r−1}} ∈ ℓ^{m}End(M). If r > 1, then the torsion subgroup of M/(A−1)M is killed by ℓ^{r−1}.

Proposition 7.2 (see Proposition 6.1i and Corollary 7.1 of [7]). Suppose X is a ddimensional abelian variety over a field F, v is a discrete valuation on F with residue characteristic not equal to 2, λ is a polarization on X, X̃_{2} is a maximal isotropic subgroup of X_{2} with respect to e_{λ,2}, λ and the points of X̃_{2} are defined over an extension of F which is unramified over v, v̄ is an extension of v to a separable closure of F, and σ ∈ I(v̄/v). Then (ρ_{2,X}(σ) − I)^{2} ∈ 2M_{2d}(Z_{2}), and X has semistable reduction above v over every totally ramified Galois (necessarily cyclic) extension of F of degree 4.

Recall that u denotes the unipotent rank of X_{v}, a denotes the abelian rank, and Φ′ denotes the prime-to-p part of the group of connected components of the special fiber of the Néron model of X at v, where p is the residue characteristic of the discrete valuation v. If X has potentially good reduction, then dim(X) = a + u.

Theorem 7.3 (Theorem 7.5 of [7]). Suppose v is a discrete valuation on a field F with strictly henselian valuation ring, X is an abelian variety over F which has potentially good reduction at v, and either

(a) n = 2 and the points of X_{2} are defined over F, or

(b) n = 3 or 4, λ is a polarization on X defined over F, and the points of a maximal isotropic subgroup of X_{n} with respect to e_{λ,n} are defined over F.

Suppose the residue characteristic p ( ≥ 0) of v does not divide n. Then Φ′ ≅ (Z/2Z)^{2u} if n = 2 or 4, and Φ′ ≅ (Z/3Z)^{u} if n = 3.

Lemma 7.4. Suppose v is a discrete valuation on a field F such that the valuation ring is strictly henselian. Suppose X is an abelian variety over F which has potentially good reduction at v, and suppose n is a positive integer not divisible by the residue characteristic of v. Let Φ_{n} denote the subgroup of X_{v}/X_{v}^{0} of points of order dividing n. Then:

(i) (X_{v})_{n} ≅ X_{n}(F), (ii) (X_{v}^{0})_{n} ≅ (Z/nZ)^{2a}, (iii) Φ_{n} ≅ (X_{v})_{n}/(X_{v}^{0})_{n}, and (iv) if X_{n}(F) ≅ (Z/nZ)^{b}, then Φ_{n} ≅ (Z/nZ)^{b−2a}.

Proof. By Lemma 2 of [6], the reduction map defines an isomorphism of X_{n}^{I} onto (X_{v})_{n}, where I = I(v̄/v) for some extension v̄ of v to F^{s}. Under our hypotheses on v, we have X_{n}^{I} ≅ X_{n}(F). Therefore, (X_{v})_{n} ≅ X_{n}(F). As shown in the proof of Lemma 1 of [6], (X_{v}^{0})_{n} ≅ (Z/nZ)^{2a+t}, where t denotes the toric rank of X_{v}. Since X has potentially good reduction at v, t = 0. Since X_{v}^{0} is n-divisible, we have Φ_{n} ≅ (X_{v})_{n}/(X_{v}^{0})_{n}. Part (iv) follows easily from (i), (ii), and (iii).

8. Néron models

In Theorem 8.1 we generalize Theorem 7.3 to the case of partial level 2 structure. We can recover Theorem 7.3a as a special case. Recall that u denotes the unipotent rank of X_{v}, a denotes the abelian rank, and Φ′ denotes the prime-to-p part of the group of connected components of the special fiber of the Néron model of X at v, where p is the residue characteristic of v (with Φ′ the full group of components if p = 0).

Theorem 8.1. Suppose v is a discrete valuation on a field F, suppose the valuation ring is strictly henselian, and suppose the residue field has characteristic p ≠ 2. Suppose (X, λ) is a d-dimensional polarized abelian variety over F, X has potentially good reduction at v, and the points of a maximal isotropic subgroup of X_{2} with respect to e_{λ,2} are defined over F. Then:

(i) Φ′ ≅ (Z/2Z)^{b−2a} = (Z/2Z)^{b+2u−2d}, where b is defined by X_{2}(F) ≅ (Z/2Z)^{b}, (ii) [X_{2} : X_{2}(F)]#Φ′ = 2^{2u}, and (iii) X has good reduction at v if and only if Φ′ = {0} and X_{2} ⊆ X(F).

Proof. Let v̄ be an extension of v to a separable closure of F, let I = I(v̄/v), let k be the residue field of v, and let J be the first ramification group (i.e., J is trivial if p = 0 and J is the pro-p-Sylow subgroup of I if p > 0). Suppose q is a prime not equal to p, and let Φ_{q} denote the q-part of the group of connected components of the special fiber of the Néron model of X. Since X has potentially good reduction at v, ρ_{q,X}(σ) has finite multiplicative order for every σ ∈ I. Let τ be a lift to I of a generator of the pro-cyclic group I/J. By §11 of [3] (see Lemma 2.1 of [5]),

Φ_{q} is isomorphic to the torsion subgroup of T_{q}(X)^{J} /(τ − 1)T_{q}(X)^{J} .

By Proposition 7.2 and Remark 2.3, X has semistable reduction (and therefore good reduction) above v over a totally ramified cyclic Galois extension of F of degree 4. Therefore I acts on T_{q}(X) through a cyclic quotient of order 4, so ρ_{q,X}(σ)^{4} = I for every σ ∈ I. Since p ≠ 2, we have ρ_{q,X}(σ) = I for every σ ∈ J. Therefore, T_{q}(X)^{J} = T_{q}(X). If q ≠ 2, then T_{q}(X)/(ρ_{q,X}(τ) − I)T_{q}(X) is torsion-free, so Φ_{q} is trivial. Further,

Φ_{2} is isomorphic to the torsion subgroup of T_{2}(X)/(τ − 1)T_{2}(X).

We have (ρ_{2,X}(τ) − I)^{2} ∈ 2M_{2d}(Z_{2}), by Proposition 7.2. By Proposition 7.1 with ℓ = 2, r = 2, m = 1, and O = Z_{2}, Φ_{2} is annihilated by 2. Therefore, Φ′ is an elementary abelian 2-group. By Lemma 7.4, Φ′ ≅ (Z/2Z)^{b−2a}. Parts (ii) and (iii) follow immediately. Note that Theorem 7.3a is a special case of Theorem 8.1.

Theorem 8.2. Suppose v is a discrete valuation on a field F, suppose the valuation ring is strictly henselian, and suppose the residue field has characteristic p ≠ 3. Suppose (X, λ) is a d-dimensional polarized abelian variety over F, X has potentially good reduction at v, and the points of a maximal isotropic subgroup of X_{3} with respect to e_{λ,3} are defined over F. Then:

(i) X_{3}(F) ≅ (Z/3Z)^{2d−u}, (ii) X has good reduction at v if and only if X_{3}(F) = X_{3}, and (iii) X has purely additive reduction at v if and only if X_{3}(F) ≅ (Z/3Z)^{d}.

Proof. By Theorem 7.3, Φ′ ≅ (Z/3Z)^{u}. Write X_{3}(F) ≅ (Z/3Z)^{b}. By Lemma 7.4, Φ′ ≅ (Z/3Z)^{b−2d+2u}. Therefore, b = 2d − u, and we obtain the desired result.

Theorem 8.3. Suppose v is a discrete valuation on a field F with strictly henselian valuation ring, X is an abelian variety over F which has potentially good reduction at v, the residue field has characteristic p ≠ 2, and either

(a) the points of X_{2} are defined over F, or

(b) λ is a polarization on X defined over F, and the points of a maximal isotropic subgroup of X_{4} with respect to e_{λ,4} are defined over F.

Then

X_{4}(F) ≅ (Z/4Z)^{2a} × (Z/2Z)^{2u}.

In particular:

(i) X_{2} ⊆ X_{4}(F) ⊆ X_{4}, [X_{4} : X_{4}(F)] = 2^{2u}, [X_{4}(F) : X_{2}] = 2^{2a}, (ii) X has good reduction at v if and only if X_{4}(F) = X_{4}, and (iii) X has purely additive reduction at v if and only if X_{4}(F) = X_{2}.

Proof. By Theorem 7.3, we have Φ′ ≅ (Z/2Z)^{2u}. By Lemma 7.4, we have a short exact sequence

0 → (Z/4Z)^{2a} → X_{4}(F) → (Z/2Z)^{2u} → 0.

Let d = dim(X). Since X_{4}(F) ⊆ X_{4} ≅ (Z/4Z)^{2d}, we conclude that X_{4}(F) ≅ (Z/4Z)^{2a} × (Z/2Z)^{2u}. Note that X_{2} ≅ (Z/2Z)^{2d} = (Z/2Z)^{2a+2u}. The rest of the result follows immediately.

As an example, let X be the elliptic curve defined by the equation y^{2} = x^{3} − 9x, and let F be the maximal unramified extension of Q_{3}. Then X_{2}(F) = X_{2} = X_{4}(F), X has additive and potentially good reduction, and Φ′ ≅ (Z/2Z)^{2}.

Remarks 8.4. If X has a polarization λ of odd degree, then X_{2} is a maximal isotropic subgroup of X_{4} with respect to e_{λ,4}.

As stated in the Introduction, Theorems 8.2ii and 8.3ii are immediate corollaries of Raynaud’s criterion for semistable reduction.

If X has purely additive reduction, then X_{n}(F) ≅ Φ_{n} (see [5]).

Suppose v is a discrete valuation on a field F, X is an abelian variety over F with potentially good reduction at v, the valuation ring is strictly henselian, ℓ = 2 or 3, and the residue characteristic is not equal to ℓ. Then Theorem 6.1 of [2] implies that if Φ′ is an elementary abelian ℓ-group, then Φ′ is a subgroup of (Z/2Z)^{2u} if ℓ = 2 or of (Z/3Z)^{u} if ℓ = 3.

For simplicity of exposition, we do not generalize the results of §8 (or the prerequisite results from [7], or related results in §3 of [8]) to the setting of Theorem 4.5, but leave such generalizations as a straightforward exercise for the reader.

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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