SELF-DUAL ℓ-ADIC REPRESENTATIONS OF FINITE GROUPS

A. SILVERBERG AND YU. G. ZARHIN

1. Introduction

It is well-known that every finite subgroup of GL_{d}(Q_{ℓ}) is conjugate to a subgroup of GL_{d}(Z_{ℓ}) (see Part II, Chapter IV, Appendix 1 of [15]). However, this does not remain true if we replace general linear groups by symplectic groups.

Throughout this paper, p is a fixed prime number. Suppose G is a finite group. We will say that G is a group of inertia type if G has a normal Sylow-p-subgroup with cyclic quotient. Such groups are exactly the inertia groups of finite Galois extensions of discrete valuation fields of residue characteristic p (see Corollaire 4 and the remark which follows in Chapitre IV, §2 of [13]).

Suppose that ℓ is a prime number different from p. Symplectic representations of groups of inertia type arise naturally from Galois actions on ℓ-adic Tate modules of polarized abelian varieties with non-semistable reduction over discrete valuation fields of residue characteristic p (see [17], [19]).

It is known (see Proposition 3.3 of [19]) that every finite ℓ′-subgroup of the symplectic group Sp_{2d}(Q_{ℓ}) is conjugate in GL_{2d}(Q_{ℓ}) to a subgroup of Sp_{2d}(Z_{ℓ}). Note that every finite subgroup of Sp_{2d}(Q_{ℓ}) is conjugate in Sp_{2d}(F) to a subgroup of Sp_{2d}(O_{F}), for some totally ramified extension F of Q_{ℓ} with valuation ring O_{F} (see Proposition 8 of [16]).

In Proposition 6.2 below we give examples of subgroups of Sp_{2d}(Q_{ℓ}) of inertia type which are not conjugate in GL_{2d}(Q_{ℓ}) to a subgroup of Sp_{2d}(Z_{ℓ}). In these examples, ℓ ≤ d + 1. However, we show that if ℓ > d + 1, and G is a subgroup of Sp_{2d}(Q_{ℓ}) of inertia type, then G is conjugate in GL_{2d}(Q_{ℓ}) to a subgroup of Sp_{2d}(Z_{ℓ}) (see Theorem 4.3).

Despite the fact that G can fail to be conjugate in GL_{2d}(Q_{ℓ}) to a subgroup of Sp_{2d}(Z_{ℓ}), we prove (Theorem 5.1) that it can nevertheless be embedded in Sp_{2d}(F_{ℓ}) in such a way that the characteristic polynomials are preserved (mod ℓ), as long as ℓ > 3. These results hold for arbitrary finite groups, even if they are not of inertia type. The same statement holds true for orthogonal groups. Proposition 6.5 gives examples which show that the bound ℓ > 3 is sharp.

In §7 we give an application to abelian varieties. For ℓ an odd prime number, we obtain isogeny classes of abelian varieties all of whose polarizations have degree divisible by ℓ.

We refer to [2] and [3] for a concrete description of classical groups over local fields (see also Chapters 19 and 20 in [6]).

Silverberg would like to thank NSA, NSF, and the Science Scholars Fellowship Program at the Bunting Institute for financial support. Zarhin would like to thank NSF for financial support.

2. Definitions and notation

Suppose ℓ is a prime number, and K is a discrete valuation field of characteristic zero and residue characteristic ℓ. We will let v : K^{×} → Z denote the valuation map, normalized so that v(K^{×}) = Z. Let O_{K} denote the valuation ring, and let e(K) denote the (absolute) ramification degree, i.e., e(K) = v(ℓ). Let ζ_{n} denote a primitive n-th root of unity in an algebraic closure of K, and let μ_{ℓ} denote the multiplicative subgroup generated by ζ_{ℓ}. If K is a complete discrete valuation field, then K contains Q_{ℓ}. If F is a field and Q_{ℓ} ⊆ F ⊆ K, let e(K/F) denote the (relative) ramification degree of K over F.

Definition 2.1. A semisimple algebra is quasisplit if it is a direct sum of matrix algebras over (commutative) fields.

The terminology “split semisimple” is used instead of “quasisplit” in [4] (Vol. I, Defn. 3.35).

Recall that p is a fixed prime number.

Definition 2.2. We say G is a group of inertia type if G is a finite group, and there exists a normal p-subgroup H of G such that the quotient G/H is cyclic of order prime to p.

Definition 2.3. We say G satisfies (D_{ℓ}) if ℓ is a prime number, G is a finite group, and there is a normal subgroup N of G of order prime to ℓ such that the quotient G/N is a cyclic ℓ-group.

Definition 2.4. If K is a field, V is a K-vector space, G is a group, and V is a G-module, then we say V is a self-dual K[G]-module if there exists a nondegenerate G-invariant bilinear form f : V × V → K. We say V is a symplectic (respectively, orthogonal) K[G]-module if f is alternating (respectively, symmetric).

Definition 2.5. Suppose R is a principal ideal domain, and T_{1} and T_{2} are free Rmodules. A bilinear form f : T_{1} × T_{2} → R is perfect if the natural homomorphisms T_{1} → Hom(T_{2}, R) and T_{2} → Hom(T_{1}, R) are bijective.

Remark 2.6. Suppose R is a principal ideal domain, T is a free R-module of rank 2n, and f : T × T → R is an alternating bilinear form. If f is perfect, then Aut(T, f) ≅ Sp_{2n}(R), where Sp_{2n}(R) denotes the group of 2n × 2n symplectic matrices over R.

Definition 2.7. The generalized quaternion group Q_{m} is the group of order 4m with the presentation:

Q_{m} = ⟨a, b : a^{2m} = 1, b^{2} = a^{m}, bab^{−1} = a^{−1}⟩

(see 1.24ii, Vol. I of [4]). Note that Q_{2} is the quaternion group of order 8.

3. Lemmas

Our main technical tool in §4 is the following theorem of Serre.

Theorem 3.1 (Serre [12]). If G is a group of inertia type, ℓ is a prime, and ℓ ≠ p, then Q_{ℓ}[G] is quasisplit.

Lemma 3.2. If G is a group of inertia type, then G satisfies (D_{ℓ}) for every prime ℓ ≠ p.

SELF-DUAL ℓ-ADIC REPRESENTATIONS 3

Proof. We use the notation of Definitions 2.2 and 2.3. Let C = G/H and let L be the quotient of C by its prime-to-ℓ part. Let N be the kernel of the composition G ↠ C ↠ L. Then N is a normal ℓ′-subgroup of G, G/N ≅ L, and L is a cyclic ℓ-group.

The next result follows immediately from the Schur-Zassenhaus Theorem.

Lemma 3.3. If G satisfies (D_{ℓ}), then there exists a cyclic ℓ-subgroup L of G such that G is the semidirect product of N by L.

The following result is an easy exercise.

Lemma 3.4. If G is a group of inertia type, and C is a normal subgroup of G, then G/C is a group of inertia type.

Lemma 3.5. Suppose K is a complete discrete valuation field of characteristic zero and finite residue characteristic, 𝔪 is the maximal ideal of the valuation ring O_{K}, and k = O_{K}/𝔪 is the residue field. Suppose N is a finite group whose order is not divisible by char(k). Suppose V is a finite-dimensional K-vector space and a simple K[N]-module, and T is an N-stable O_{K}-lattice in V. Then T/𝔪T is simple as a k[N]-module.

Proof. See §14.4 and §15.5 of [14].

Proposition 3.6. Suppose ℓ is a prime number, N is a finite group of order prime to ℓ, and K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ. Suppose V is a finite-dimensional K-vector space and a simple K[N]-module, suppose f : V × V → K is a K-bilinear nondegenerate N-invariant pairing, suppose T is an N-stable O_{K}-lattice in V, and suppose f(T, T) = O_{K}. Let 𝔪 denote the maximal ideal of O_{K}. Then the restriction f : T ×T → O_{K} is perfect, and the reduction

f̄: T/𝔪T × T/𝔪T → O_{K}/𝔪

is nondegenerate.

Proof. Since V is a simple K[N]-module, T/𝔪T is a simple (O_{K}/𝔪)[N]-module, by Lemma 3.5. Let W denote the (left) kernel of f̄. Since W is N-stable, and T/𝔪T is a simple N-module, we have W = 0. Therefore, f̄ is nondegenerate. By Nakayama’s Lemma, f : T × T → O_{K} is perfect.

Lemma 3.7. Suppose ℓ is a prime number, d is a positive integer, K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ, e = e(K), [K(ζ_{ℓ}) : K] = 2d, and ℓ = 2de + 1. Let T = O_{K}[ζ_{ℓ}]. Then there exists a perfect alternating μ_{ℓ} × {±1}-invariant pairing

f′ : T × T → O_{K}.

Proof. Note that e(Q_{ℓ}(ζ_{ℓ})) = ℓ − 1. Under our hypotheses, e(K(ζ_{ℓ})) = 2de = ℓ − 1, and K(ζ_{ℓ})/K is totally and tamely ramified. Let η = ζ_{ℓ} − ζ_{ℓ}^{−1}. Then η is a uniformizer for K(ζ_{ℓ}). Let y ↦ ȳ be the (unique) element of order two in Gal(K(ζ_{ℓ})/K), let η_{1} be a uniformizer for K, and define f′ by

f′(x, y) = η_{1}^{−1}tr_{K(ζ_{ℓ})/K}(xηȳ).

Since ℓ does not divide 2d = [K(ζ_{ℓ}) : K], we have tr_{K(ζ_{ℓ})/K}(T) = O_{K}. Therefore for i = 1, …, 2d we have

η_{1}O_{K} = tr_{K(ζ_{ℓ})/K}(η_{1}T) ⊆ tr_{K(ζ_{ℓ})/K}(η^{i}T) ⊆ η^{i}T ∩ O_{K} ⊆ η_{1}O_{K}.

If x ∈ T − η_{1}T, then x = η^{i}u with 0 ≤ i < 2d and u ∈ T^{×}. Therefore, f′(x, T) = O_{K}. It follows that f′ is perfect.

Proposition 3.8. Suppose G is a finite group, N is a normal subgroup of G, W is a finite-dimensional vector space over a field K, K[N] is quasisplit, and r is a positive integer. We can view Aut(W) as naturally contained in Aut(W^{r}). Suppose ρ : G → Aut(W^{r}) is a representation such that ρ(N) ⊂ Aut(W) ⊆ Aut(W^{r}) and such that the restriction of ρ to N is irreducible. Let Ad : G → Aut(End(W^{r})) denote the corresponding adjoint representation defined by

Ad(g)(u) = ρ(g)uρ(g)^{−1}

for u ∈ End(W^{r}) and g ∈ G. Let E = End_{N}(W) ⊆ End(W) ⊆ End(W^{r}) and let J denote the image of the natural map K[N] → End(W) ⊆ End(W^{r}). Then:

(a) J = End_{E}(W),

(b) E = End_{J}(W),

(c) E is a field,

(d) the center of J is E,

(e) J and E are stable under the action of Ad(g) for every g ∈ G,

(f) for every h ∈ N, the action of Ad(h) on E is trivial.

Proof. By the Jacobson density theorem we have (a) and (b). Since K[N] is quasisplit and W is a simple K[N]-module, we have that E is a field and E is the center of J. Suppose g ∈ G. Since N is normal in G, therefore J is stable under Ad(g). Since E is the center of J, it follows that E is stable under Ad(g). Since E commutes with ρ(N), the action of Ad(h) on E is trivial for h ∈ N.

Corollary 3.9. Suppose ℓ is a prime number, suppose N is a finite group of order prime to ℓ, suppose W is a finite-dimensional vector space over a complete discrete valuation field K of characteristic zero and residue characteristic ℓ, suppose W is a simple K[N]-module, and suppose K[N] is quasisplit. Let E = End_{N}(W). Then E is a field and E/K is an unramified extension.

Proof. Note that if A is a normal subgroup of N, then K[N/A] is also quasisplit. Therefore, replacing N by its image in Aut(W) we may assume that W is a faithful K[N]-module. By Proposition 3.8, E is a field. By Theorem 24.7 of [5] (see also Theorem 74.5ii, Vol. II of [4]), the field E is generated over K by the values of the character of the representation of N on the E-vector space W. Thus, E ⊆ K(ζ_{n}), where n = #N. Since n is not divisible by ℓ, the extension E/K is unramified.

Lemma 3.10. Suppose ℓ is a prime number, K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ, and e = e(K). Suppose A ∈ GL_{m}(K), suppose A is not a scalar, and suppose A^{ℓ} is a scalar. Then m ≥ (ℓ − 1)/e.

Proof. If A^{ℓ} is the scalar c ∈ K^{×}, let g(x) = x^{ℓ} − c. Either g has a root in K, or g is irreducible over K (see the end of p. 297 of [10]). If g is irreducible over K, then m ≥ ℓ since g(A) = 0. If g has a root γ ∈ K, then Aγ^{−1} is an element of GL_{m}(K) of exact order ℓ. Therefore m ≥ [K(ζ_{ℓ}) : K] ≥ (ℓ − 1)/e.

Lemma 3.11. Suppose ℓ is an odd prime number, K is an unramified extension of Q_{ℓ}, M = K(ζ_{ℓ}), δ ∈ M^{+} = K(ζ_{ℓ} + ζ_{ℓ}^{−1}), η = ζ_{ℓ} − ζ_{ℓ}^{−1}, and tr_{M/K}(δO_{M}) ⊆ O_{K}. Then

tr_{M/K}(δη^{ℓ−2}O_{M}) ⊆ ℓO_{K}.

Proof. Note that η is a uniformizer for M and η^{2} is a uniformizer for M^{+}. Since e(M) = ℓ − 1, the inverse different for the extension M/K is η^{2−ℓ}O_{M} (see Proposition 13 in Chapitre III, §6 of [13]). Therefore, δ ∈ η^{2−ℓ}O_{M} ∩ M^{+} = η^{3−ℓ}O_{M^{+}}. Thus,

tr_{M/K}(δη^{ℓ−2}O_{M}) ⊆ tr_{M/K}(ηO_{M}) = ℓO_{K}.

The following “rigidity” result is in the spirit of similar results by Minkowski and Serre (see for example Proposition 14, III.7.6 of [1]).

Proposition 3.12. Suppose ℓ is a prime number, and K is a discrete valuation field of characteristic zero and residue characteristic ℓ. Let 𝔪 denote the maximal ideal of O_{K}, and let e = e(K). Suppose S is a free O_{K}-module of finite rank, and A is an automorphism of S of finite order. If either

(a) 2e < ℓ − 1 and (A − 1)^{2} ∈ 𝔪End(S), or

(b) e < ℓ − 1 and A − 1 ∈ 𝔪End(S), then A = 1.

Proof. This follows directly from Theorem 6.2 of [18] for n = ℓ, with k = 2e in case (a), and with k = e in case (b).

Note that the hypothesis 2e < ℓ − 1 is satisfied if e = 1 and ℓ ≥ 5.

Lemma 3.13. Suppose L is a field of characteristic not equal to 2, and

f : L^{2} × L^{2} → L

is a nondegenerate symmetric pairing. Let g = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ SL_{2}(L). Then f is not

g-invariant.

Proof. Let {u, v} denote the standard basis of L^{2} over L. Suppose f is g-invariant. Then f(u, v) = f(gu, gv) = f(u, u+v), i.e., f(u, u) = 0. Also, f(v, v) = f(gv, gv) = f(u + v, u + v) = f(v, v) + 2f(u, v), since f is symmetric. Since char(L) ≠ 2, we have f(u, v) = 0. Therefore f(u, w) = 0 for every w ∈ L^{2}, contradicting the nondegeneracy of f.

4. Embeddings over Z_{ℓ}

Theorem 4.1. Suppose G satisfies (D_{ℓ}). Suppose K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ, and e = e(K). Suppose V is a K-vector space of even dimension 2d, suppose ℓ > de + 1, suppose V is a faithful simple self-dual K[G]-module, and suppose K[N] is quasisplit. Then either:

(a) V is simple as a K[N]-module, or

(b) ℓ = 2de + 1, V ≅ K(ζ_{ℓ}), and either G ≅ μ_{ℓ} or G ≅ μ_{ℓ} × {±1}, with the natural action on V.

Proof. By Lemma 3.3, G has a cyclic ℓ-subgroup L such that G is the semidirect product of N by L. We have

[K(ζ_{ℓ^{2}}) : K] ≥ e(K(ζ_{ℓ^{2}})/K) ≥ ℓ(ℓ − 1)/e > ℓd > 2d.

Since the characteristic polynomials of the elements of G acting on V have degree 2d, and [K(ζ_{ℓ^{2}}) : K] > 2d, therefore #L < ℓ^{2}. If #L = 1, then G = N and case (a) holds. Therefore we may assume #L = ℓ, and may identify L with μ_{ℓ}.

An isotypic K[N]-module is, by definition, a direct sum of isomorphic simple K[N]-modules. Let V = ⊕_{i=1}^{s}V_{i} be the canonical decomposition of the K[N]- module V into a direct sum of isotypic components (see 2.6 of [14]). The action of G on V induces an action of G on the set of V_{i}’s. Since V is a simple K[G]-module, the action of G on {V_{i}} is transitive. Since N is normal in G, this action factors through G/N. Since G/N has prime order ℓ, either s = 1 and V is isotypic, or s = ℓ and V = ⊕_{g∈L}(gV_{1}). In the latter case, 2d is divisible by ℓ, contradicting the hypothesis that ℓ > de + 1. Therefore V is isotypic, i.e., for some r ∈ Z^{+} we have V ≅ W^{r}, where W is a simple K[N]-module. The above proof followed the method of proof of Proposition 24 in §8.1 of [14]; see also Clifford’s Theorem [4], Vol. 1, 11.1.

Since V is a faithful K[G]-module, it is a faithful K[N]-module. Therefore, W is a faithful K[N]-module. If r = 1, then (a) holds. Thus, assume r > 1.

Suppose dim_{K}(W) = 1. Since W is a faithful K[N]-module, we have N ⊆ K^{×}. Since V is a self-dual K[N]-module, we conclude that N ⊆ {±1}. Therefore, G is isomorphic to μ_{ℓ} or to μ_{ℓ} × {±1}. Since V is a simple faithful K[G]-module, we have V ≅ K(ζ_{ℓ}). Therefore [K(ζ_{ℓ}) : K] = dim_{K}(V) = 2d, and ℓ − 1 divides 2de. Since ℓ − 1 > de, therefore ℓ − 1 = 2de and (b) holds.

Therefore, we may assume that dim_{K}(W) ≥ 2 and r ≥ 2. We will show that this leads to a contradiction.

Let E = End_{N}(W). We will show that E = K. By Corollary 3.9, E is a field and E/K is an unramified extension. Let J denote the image of K[N] → End(W). By Proposition 3.8, J = End_{E}(W), G acts on J and E by conjugation, and the induced action of N on E is trivial. If L acts nontrivially on E, then it acts faithfully, so [E : K] is divisible by ℓ. Since dim_{K}(W) is divisible by [E : K], rℓ divides dim_{K}(V) = 2d. This contradicts the fact that rℓ ≥ 2ℓ > 2(de + 1) > 2d. Therefore, G commutes with E. Let γ be a generator of L. Then γ is an E-linear automorphism of V of exact order ℓ, so dim_{E}(V) ≥ [E(ζ_{ℓ}) : E]. Since E/K is unramified,

dim_{E}(V) ≥ [E(ζ_{ℓ}) : E] ≥ e(K(ζ_{ℓ})/K) ≥ (ℓ − 1)/e > d.

If E ≠ K, then dim_{E}(V) ≤ d, a contradiction. Therefore, E = K.

Since J = End(W), therefore G acts on End(W) by “conjugation”. Since G acts trivially on E, by the Skolem-Noether Theorem there exists A ∈ Aut(W) ⊂ Aut(V) such that AuA^{−1} = γuγ^{−1} for every u ∈ End(W). Since γ^{ℓ} = 1, therefore A^{ℓ} commutes with all elements of End(W). Thus A^{ℓ} is a scalar.

Suppose A is a scalar. Then we may replace A by the identity. Since L ⊆ G ⊆ Aut(V), we can view γ as an element of Aut(V). Since N ⊂ End(W) ⊂ End(V), we have that γ is an element of Aut_{N}(V) = Aut_{N}(W^{r}) ≅ GL_{r}(K) of exact order ℓ. Therefore, r ≥ [K(ζ_{ℓ}) : K] > d. Thus, 2d = rdim_{K}(W) > 2d, a contradiction.

Therefore A is not a scalar. By Lemma 3.10, we have dim_{K}(W) ≥ (ℓ − 1)/e > d. Therefore 2d = rdim_{K}(W) > 2d, a contradiction.

Theorem 4.2. Suppose G satisfies (D_{ℓ}), suppose K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ, suppose e = e(K), suppose V is a 2d-dimensional K-vector space and is a faithful simple symplectic K[G]-module, and suppose ℓ > de+1. Suppose K[N] is quasisplit. Then there exist a G-stable O_{K}-lattice T in V and a perfect alternating G-invariant bilinear form

f′ : T × T → O_{K}.

Proof. By Theorem 4.1, we are in case (a) or (b). In case (b), apply Lemma 3.7. In case (a), let f be the associated alternating pairing on V, and let T be a G-stable O_{K}-lattice in V. By multiplying f by a suitable power of a uniformizer, we can assume that f(T, T) = O_{K}. Now apply Proposition 3.6.

Theorem 4.3. Suppose G is a group of inertia type, suppose ℓ is a prime number, and suppose ℓ ≠ p. Suppose K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ, suppose e = e(K), suppose V is a 2d-dimensional K-vector space and is a symplectic K[G]-module, and suppose ℓ > de+1. Then there exist a G-stable O_{K}-lattice T in V and a perfect alternating G-invariant bilinear form

f′ : T × T → O_{K}.

Proof. We can reduce to the case where V is a simple symplectic K[G]-module as follows. Let f be the associated alternating pairing on V. Write V as a direct sum of simple K[G]-modules

V = V_{1} ⊕ ⋯ ⊕ V_{k}.

If for some i the restriction f_{i} of f to V_{i} is not identically zero, then f_{i} is nondegenerate. Let V′_{i} be the orthogonal complement of V_{i} in V with respect to f. Clearly, the restriction of f to V′_{i} is also nondegenerate. Apply induction to the symplectic K[G]-modules V_{i} and V′_{i}, to obtain the desired result. We may therefore assume that f(V_{i}, V_{i}) = 0 for every i. Since f is nondegenerate, there exists a V_{j} such that f(V_{1}, V_{j}) ≠ 0. Since V_{1} and V_{j} are simple K[G]-modules, f : V_{1} × V_{j} → K is nondegenerate. Thus the restriction f̃ of f to W = V_{1} ⊕ V_{j} is nondegenerate. Let V′′ denote the orthogonal complement of W in V with respect to f. Again by induction, we obtain a G-stable lattice T′′ in V′′ and a G-invariant perfect O_{K}-valued alternating pairing f′′ on T′′. Let T_{1} be a G-stable O_{K}-lattice in V_{1}, let

T_{j} = {x ∈ V_{j} : f(T_{1}, x) ⊆ O_{K}},

let T = T_{1} ⊕ T_{j} ⊕ T′′, and let f′ = f̃⊕ f′′.

By Lemma 3.4, replacing G by its image in Aut(V), we may reduce to the case where V is a faithful simple symplectic K[G]-module. Suppose ℓ is a prime and ℓ ≠ p. By Lemma 3.2, G satisfies (D_{ℓ}), and we have a corresponding normal subgroup N. By Theorem 3.1, Q_{ℓ}[N] is quasisplit. Therefore, K[N] is quasisplit. The theorem now follows from Theorem 4.2.

Remark 4.4. If d = 1, then dim_{K}(V) = 2 and Aut(V, f) = SL(V). In this case the conclusion of Theorems 4.2 and 4.3 holds without the requirement that ℓ > de + 1. Indeed, one can let T be any G-stable O_{K}-lattice in V, and let f′ : T × T → O_{K} be any alternating perfect pairing.

5. Embeddings over F_{ℓ}

Theorem 5.1. Suppose ℓ is a prime number, and K is a discrete valuation field of characteristic zero and residue characteristic ℓ. Let 𝔪 denote the maximal ideal, let k = O_{K}/𝔪, let e = e(K), and suppose 2e < ℓ − 1. Suppose V is a K-vector space of finite dimension n, suppose f : V × V → K is a nondegenerate alternating (respectively, symmetric) K-bilinear form, suppose G is a finite group, and suppose

ρ : G ↪ Aut_{K}(V, f)

is a faithful representation of G on V that preserves the form f. Then there exist a nondegenerate alternating (respectively, symmetric) k-valued k-bilinear form f_{0} on k^{n}, and a faithful representation

ρ̄: G ↪ Aut_{k}(k^{n}, f_{0}),

such that for every g ∈ G, the characteristic polynomial of ρ̄(g) is the reduction modulo 𝔪 of the characteristic polynomial of ρ(g).

Proof. If S is a G-stable O_{K}-lattice in V, let

S^{∗} = {x ∈ V : f(x, S) ⊆ O_{K}}.

Fix a G-stable O_{K}-lattice S in V. Let η denote a uniformizer for O_{K}. Multiplying f by an integral power of η if necessary, we may assume that f(S, S) = O_{K}. Then S ⊆ S^{∗}. Let S_{0} = S and let

S_{i+1} = S_{i} + (η^{−1}S_{i} ∩ ηS_{i}^{∗}) for i ≥ 0.

Then S_{i} is a G-stable O_{K}-lattice in V, f(S_{i}, S_{i}) = O_{K}, and S_{i} ⊆ S_{i+1} ⊆ S_{i+1}^{∗} ⊆ S_{i}^{∗}. Note that S_{i+1} = S_{i} if and only if ηS_{i}^{∗} ⊆ S_{i}. We have

S = S_{0} ⊆ S_{1} ⊆ S_{2} ⊆ … ⊆ S^{∗}.

Since S^{∗}/S is finite, we have S_{j} = S_{j+1} for some j. Let T = S_{j}. Then T is a G-stable O_{K}-lattice in V such that f(T, T) = O_{K} and 𝔪T^{∗} = ηT^{∗} ⊆ T.

Let f̄ : T/𝔪T × T/𝔪T → k be the reduction of f modulo 𝔪. Then ker(f̄) = 𝔪T^{∗}/𝔪T ≅ T^{∗}/T. Clearly, f̄ is nondegenerate on (T/𝔪T)/ ker(f̄) ≅ T/𝔪T^{∗}. On T^{∗} × T^{∗}, the form ηf is O_{K}-valued. Let f̃ denote the reduction modulo 𝔪 of the restriction of ηf to T^{∗} ×T^{∗}. Since (T^{∗})^{∗} = T, we have ker(f̃) = T/𝔪T^{∗}. Therefore, f̃ is nondegenerate on (T^{∗}/𝔪T^{∗})/(T/𝔪T^{∗}) ≅ T^{∗}/T ≅ ker(f̄). We thus obtain a homomorphism

ψ : G → Aut_{k}((T/𝔪T)/ ker(f̄), f̄) × Aut_{k}(ker(f̄), f̃) ≅
Aut_{k}(T/𝔪T^{∗}, f̄) × Aut_{k}(T^{∗}/T, f̃) ↪ Aut_{k}(T^{∗}/𝔪T^{∗}, f̄× f̃) ≅ Aut_{k}(k^{n}, f_{0})

for an appropriate pairing f_{0}. All the elements σ ∈ ker(ψ) act as the identity on T/𝔪T^{∗} and on T^{∗}/T, and thus (σ − 1)^{2}T^{∗} ⊆ 𝔪T^{∗}. Proposition 3.12a implies ψ is injective. Let ρ̄ = ψ.

Remark 5.2. If one is concerned only with preserving the characteristic polynomials, and does not insist that ρ̄ be an embedding, then in the symplectic case the above result can instead be achieved with the aid of Proposition 8 of [16], rather than Proposition 3.12a above.

Theorem 5.3. Suppose ℓ is a prime number, and L is a discrete valuation field of characteristic zero and residue characteristic ℓ. Suppose K is a quadratic extension of L, let 𝔪 denote the maximal ideal, and let k = O_{K}/𝔪. Let e = e(K), and suppose 2e < ℓ−1. Suppose V is a K-vector space of finite dimension n, suppose f : V ×V → K is a nondegenerate pairing which is hermitian (respectively, skew-hermitian) with respect to the extension K/L, suppose G is a finite group, and suppose ρ : G ↪ Aut_{K}(V, f) is a faithful representation. Suppose K/L is unramified and thus k is a quadratic extension of the residue field k_{L} of L. Then there exist a nondegenerate k-valued pairing f_{0} on k^{n} which is hermitian (respectively, skew-hermitian) with respect to the extension k/k_{L}, and a faithful representation ρ̄: G ↪ Aut_{k}(k^{n}, f_{0}), such that for every g ∈ G, the characteristic polynomial of ρ̄(g) is the reduction modulo 𝔪 of the characteristic polynomial of ρ(g).

Proof. Let η denote a uniformizer for L. Then η is also a uniformizer for K, and the proof is a repetition of the proof of Theorem 5.1. The reductions f̄ and f̃ are now hermitian (respectively, skew-hermitian).

Remark 5.4. In the setting of Theorem 5.3, suppose now that K/L is ramified. Then k_{L} = k, and one can write K = L(\sqrt{D}) where D is a uniformizer for L and η = \sqrt{D} is a uniformizer for K. Let S be a G-stable O_{K}-lattice in V. Then η^{r}f(S, S) = O_{K} for some integer r. If r is even then η^{r}f is hermitian (respectively, skew-hermitian) and its reduction is symmetric (respectively, alternating). Further, η^{r+1}f is skew-hermitian (respectively, hermitian) and its reduction is alternating (respectively, symmetric). If r is odd, then η^{r}f is skew-hermitian (respectively, hermitian) and its reduction is alternating (respectively, symmetric). Further, η^{r+1}f is hermitian (respectively, skew-hermitian) and its reduction is symmetric (respectively, alternating). In all cases one can proceed as in Theorem 5.1 and obtain an embedding of G into a product of an orthogonal group O_{s}(k) and a symplectic group Sp_{n−s}(k), which “respects” the characteristic polynomials.

6. Sharpness

The next proposition shows that one cannot replace “symplectic” by “orthogonal” in the statements of Theorems 4.2 and 4.3. Here, e = 1 and ℓ − 1 = 2d = 2de > de.

Proposition 6.1. Suppose ℓ is an odd prime. Let G = μ_{ℓ}, let M = Q_{ℓ}(ζ_{ℓ}), and let V be M viewed as a Q_{ℓ}-vector space. Then V carries a natural structure of a faithful simple orthogonal Q_{ℓ}[G]-module, but there is no G-stable lattice T in V with a perfect symmetric G-invariant pairing f′ : T × T → Z_{ℓ}.

Proof. Define

f : V × V → Q_{ℓ} by f(x, y) = tr_{M/Q_{ℓ}}(xȳ),

where y ↦ ȳ is the automorphism of M over Q_{ℓ} which sends ζ_{ℓ} to its inverse. Then f is a nondegenerate symmetric G-invariant pairing. Let S = Z_{ℓ}[ζ_{ℓ}] and let M^{+} = Q_{ℓ}(ζ_{ℓ} + ζ_{ℓ}^{−1}). We have End_{G}(V) = M. Every nondegenerate G-invariant symmetric bilinear form on V is of the form f_{δ}(x, y) = f(δx, y) for some δ ∈ (M^{+})^{×}. Let η = ζ_{ℓ} − ζ_{ℓ}^{−1}. Then η is a uniformizer for M and η^{2} is a uniformizer for M^{+}. Clearly, every G-stable Z_{ℓ}-lattice in the Q_{ℓ}-vector space V is a Z_{ℓ}[ζ_{ℓ}]-lattice. By Exercise 15.3 of [14], every G-stable lattice in V is of the form η^{r}S where r ∈ Z. This implies that if T is a G-stable lattice in V and f′ : T × T → Z_{ℓ} is a Ginvariant perfect alternating pairing, then there exist r ∈ Z and δ ∈ M^{+} such that T = η^{r}S and f′ = f_{δ}. Replacing δ by δ(ηη̄)^{r}, we can reduce to the case where f_{δ} is a perfect Z_{ℓ}-valued pairing on S. Since f_{δ}(S, S) ⊆ Z_{ℓ}, therefore tr_{M/Q_{ℓ}}(δS) ⊆ Z_{ℓ}. By Lemma 3.11, f_{δ}(η^{ℓ−2}S, S) ⊆ ℓZ_{ℓ}. The perfectness of f_{δ} : S × S → Z_{ℓ} is now contradicted by the fact that η^{ℓ−2}S contains elements of S − ℓS.

The following results show that the bound ℓ > de + 1 in §4 is sharp.

Proposition 6.2. Suppose ℓ is an odd prime, N is a finite group of order not divisible by ℓ, K is a complete discrete valuation field of characteristic zero and residue characteristic ℓ which is an unramified extension of Q_{ℓ}, and W is a non-zero K-vector space which is also an absolutely simple faithful symplectic K[N]-module. Let G = N × μ_{ℓ}, and let V = W ⊗_{K} K(ζ_{ℓ}). Then V carries a natural structure of a faithful symplectic K[G]-module, but there is no G-stable O_{K}-lattice T in V with a perfect alternating G-invariant pairing f′ : T × T → O_{K}.

Proof. Let T_{1} be an N-stable Z_{ℓ}-lattice in W. Multiplying the given alternating Ninvariant pairing f_{1} on W by a suitable power of ℓ, we may assume that f_{1}(T_{1}, T_{1}) = O_{K}. Since #N is not divisible by ℓ, it follows from Proposition 3.6 that f_{1} : T_{1} × T_{1} → O_{K} is perfect.

Let M = K(ζ_{ℓ}), and let W_{2} be M viewed as a K-vector space. Define

f_{2} : W_{2} × W_{2} → K by f_{2}(α, β) = tr_{M/K}(αβ̄),

where β ↦ β̄ is the automorphism of M over K which sends ζ_{ℓ} to its inverse. Then f_{2} is a nondegenerate μ_{ℓ}-invariant symmetric pairing. Let f = f_{1} ⊗ f_{2}. Then f is a nondegenerate G-invariant alternating pairing, with respect to which V is a faithful symplectic K[G]-module. We have End_{N}(W) = K. Since End_{μ_{ℓ}}(W_{2}) = M, we have End_{G}(V) = M. It therefore follows from the definition of f that every nondegenerate G-invariant alternating bilinear form on V is of the form f_{δ}(x, y) = f(δx, y) for some δ ∈ (M^{+})^{×}, where M^{+} = K(ζ_{ℓ} +ζ_{ℓ}^{−1}). Let S = T_{1} ⊗_{O_{K}} O_{M} ⊂ V, and let k denote the residue field of K. By Lemma 3.5, S/(1 − ζ_{ℓ})S (= T_{1}/ℓT_{1}) is a simple k[N]-module, and therefore is a simple k[G]-module. Let η = ζ_{ℓ} − ζ_{ℓ}^{−1}. Every G-stable O_{K}-lattice in the K-vector space V is an O_{M}-lattice. By Exercise 15.3 of [14], it follows that every G-stable lattice in V is of the form η^{r}S with r ∈ Z. Thus if T is a G-stable lattice in V and f′ : T × T → Z_{ℓ} is a G-invariant perfect alternating pairing, then there exist r ∈ Z and δ ∈ M^{+} such that T = η^{r}S and f′ = f_{δ}. Replacing δ by δ(ηη̄)^{r}, we can reduce to the case where f_{δ} is a perfect O_{K}- valued pairing on S. Since f_{δ}(S, S) ⊆ O_{K}, therefore tr_{M/K}(δS) ⊆ O_{K}. By Lemma 3.11, f_{δ}(η^{ℓ−2}S, S) ⊆ ℓO_{K}. The perfectness of f_{δ} : S × S → O_{K} is contradicted by the fact that η^{ℓ−2}S contains elements of S − ℓS.

Applying Proposition 6.2 to the family of examples in the next result, we obtain examples which show that for all primes p and for all odd primes ℓ ≠ p, the bound ℓ > de + 1 in Theorem 4.3 is sharp. In these examples we have that e = 1 and ℓ − 1 = d = de, and the group G = N × μ_{ℓ} is a group of inertia type.

Proposition 6.3. Let q = p if p is odd, and let q = 4 if p = 2. Assume that the quadruple (ℓ, K, W, N) is as in one of the following two cases.

(a) Let ℓ be an odd prime different from p, let K = Q_{ℓ}(ζ_{q}), let W = K^{2}, and let

N be the subgroup of SL(W) generated by ( \begin{matrix}ζ_{q} & 0 \\ 0 & ζ_{q}^{−1} \end{matrix} ) and ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

(b) Let ℓ be an odd prime different from p such that ℓ ≡ −1 (mod q). Let K = Q_{ℓ}, and let W be the 2-dimensional Q_{ℓ}-vector space Q_{ℓ}(ζ_{q}). Let σ be the nontrivial automorphism of the quadratic extension Q_{ℓ}(ζ_{q})/Q_{ℓ}. Pick α ∈ Q_{ℓ}(ζ_{q}) such that Norm_{Q_{ℓ}(ζ_{q})/Q_{ℓ}}(α) = −1 (this can be done since Q_{ℓ}(ζ_{q})/Q_{ℓ} is unramified). Let N be the subgroup of SL(W) generated by ζ_{q} (with the natural action) and the element τ of order 4 defined by τ(x) = ασ(x) for x ∈ W.

Then ℓ, K, N, and W satisfy the hypotheses of Proposition 6.2, and G = N × μ_{ℓ} is a group of inertia type.

Proof. The group N is the generalized quaternion group Q_{p} of order 4p. Since N is non-abelian and W is a 2-dimensional faithful K[N]-module, it follows that W is an absolutely simple K[N]-module. If p is odd, then μ_{p} is a normal subgroup of N and G, and G/μ_{p} is cyclic of order 4ℓ. If p = 2, then N is a normal Sylow-2-subgroup of G, and G/N is cyclic of order ℓ. In each case, G is of inertia type.

Note that Q_{ℓ}(ζ_{q}) = Q_{ℓ} if and only if ℓ ≡ 1 (mod q). Therefore when p = 2 or 3, the preceding result yields examples which establish the sharpness of the bound ℓ > de + 1 in Theorem 4.3 with K = Q_{ℓ} and d = ℓ − 1, for all odd primes ℓ ≠ p.

When ℓ, K, N, and W are as in Proposition 6.3, and b is a positive integer, then the following result gives examples where e = 1 and de = d = ℓ − 1 + b, thereby providing examples which do not satisfy the conclusion of Theorem 4.3, for all odd primes ℓ ≤ de + 1 different from p.

Corollary 6.4. Suppose ℓ, N, K, W, V, and G are as in Proposition 6.2, and b ∈ Z^{+}. Let W_{0} be the K-vector space K^{2b} with trivial G-action, and let U be the K[G]-module V ⊕ W_{0}. Then U carries a natural structure of a faithful symplectic K[G]-module, but there is no G-stable O_{K}-lattice T in V with a perfect alternating G-invariant pairing f′ : T × T → O_{K}.

Proof. The natural symplectic structure on U arises from the sum of the natural alternating pairing on V and the standard alternating pairing on W_{0}. Let τ = \frac{1}{#N} ∑_{h∈N} h ∈ Z_{ℓ}[N]. Then τ^{2} = τ, τU = W_{0}, and (1 − τ)U = V. Suppose T is a G-stable O_{K}-lattice in U and f′ : T × T → O_{K} is a perfect alternating G-invariant pairing. Let T_{1} = (1−τ)T and let T_{2} = τT. Then T = T_{1}⊕T_{2}, and T_{1} (respectively, T_{2}) is a G-stable O_{K}-lattice in V (respectively, W_{0}). Since f′ is G-invariant and the G-action on W_{0} is trivial, we have f′(x, y) = f′(gx, gy) = f′(gx, y) for every x ∈ T_{1}, y ∈ T_{2}, and g ∈ G. Therefore, f′((1 − g)T_{1}, T_{2}) = 0 for every g ∈ G. Since V is a simple K[G]-module, therefore the (1 − g)T_{1}’s generate T_{1}. Thus, f′(T_{1}, T_{2}) = 0. Since f′ is perfect, it follows that the restriction of f′ to T_{1} × T_{1} is perfect, contradicting Proposition 6.2.

Propositions 6.5 and 6.6 below enable us to obtain examples with e = (ℓ − 1)/2. In particular, Proposition 6.5 gives examples which show that the bound ℓ > 2e + 1 in Theorem 5.1 is sharp. The proof of Proposition 6.6 uses Proposition 6.5.

Proposition 6.5. Suppose ℓ is an odd prime number, F is a complete discrete valuation field of characteristic zero and residue characteristic ℓ which is an unramified extension of Q_{ℓ}, and N is a finite group of order not divisible by ℓ. Let K = F (ζ_{ℓ} + ζ_{ℓ}^{−1}), let 𝔪 denote the maximal ideal of O_{K}, and let k = O_{K}/𝔪. Suppose W is a non-zero K-vector space which is also an absolutely simple faithful symplectic K[N]-module, and write dim_{K}(W) = 2t. Let V be the K-vector space W ⊗_{K} K(ζ_{ℓ}), and let G = N ×μ_{ℓ}. Then there are a natural nondegenerate alternating K-bilinear form f : V ×V → K and a natural faithful irreducible representation ρ : G ↪ Aut_{K}(V, f). If L is a field which contains k, and f_{0} : L^{4t} × L^{4t} → L is a nondegenerate L-bilinear alternating form, then there does not exist a faithful representation ρ̄ : G ↪ Aut_{L}(L^{4t}, f_{0}) having the property that for every g ∈ G, tr(ρ̄(g)) ≡ tr(ρ(g)) (mod 𝔪).

Proof. Let M = F (ζ_{ℓ}), let x ↦ x̄ denote the nontrivial automorphism of M over K, and let V_{2} be M viewed as a 2-dimensional K-vector space. By our assumptions, we have an absolutely irreducible, faithful representation ρ_{1} : N ↪ Aut_{K}(W, f_{1}) where f_{1} is a nondegenerate alternating K-valued pairing on W. The natural action of μ_{ℓ} on V_{2} respects the nondegenerate symmetric bilinear form f_{2} : V_{2} × V_{2} → K defined by f_{2}(x, y) = tr_{M/K}(xȳ). We thus obtain a faithful representation ρ_{2} : μ_{ℓ} ↪ Aut_{K}(V_{2}, f_{2}). Let ρ be the representation

ρ : G = N × μ_{ℓ} ↪ Aut_{K}(W, f_{1}) × Aut_{K}(V_{2}, f_{2}) ↪ Aut_{K}(V, f)

defined by ρ(a, b) = ρ_{1}(a) ⊗ ρ_{2}(b), where f = f_{1} ⊗ f_{2}.

Suppose L is a field which contains k, V_{0} is a 4t-dimensional L-vector space, f_{0} is a nondegenerate L-valued alternating pairing on V_{0}, and ρ̄ : G ↪ Aut_{L}(V_{0}, f_{0}) is a faithful representation. Since #N is not divisible by ℓ, V_{0} is a semisimple L[N]-module. Let T be an N-stable O_{K}-lattice T in W. By Proposition 3.6, for some non-zero multiple f′ of f_{1} we have ρ_{1}(N) ⊂ Aut(T, f′), and the reduction f̄′ is nondegenerate. The composition of ρ_{1} with the reduction map gives a faithful representation ρ̄_{1} : N ↪ Aut(W_{0}, f̄′), where the corresponding k[N]-module W_{0} is absolutely simple and symplectic. By Schur’s Lemma, every N-invariant bilinear form on W_{0} is alternating. If tr((ρ̄_{1} ⊕ ρ̄_{1})(h)) = tr(ρ̄(h)) for every h ∈ N, then the multiplicities of the simple components of the L[N]-modules V_{0} and W_{0} ⊕ W_{0} are congruent modulo ℓ (see Theorem 17.3 of [4]). Then V_{0} and W_{0}^{2} are isomorphic L[N]-modules, since W_{0} is simple and ℓ > 2. Since End_{N}(W_{0}) = F, we have End_{N}(V_{0}) = M_{2}(F). Fix a generator c of μ_{ℓ}. Then c is an element of End_{N}(V_{0}) of multiplicative order ℓ. We can therefore identify V_{0} with W_{0} ⊕ W_{0} in such a way that c(x, y) = (x + y, y) for every (x, y) ∈ W_{0} ⊕ W_{0} = V_{0}. For x, y ∈ W_{0} we have

f_{0}((x, 0), (0, y)) = f_{0}(c(x, 0), c(0, y)) = f_{0}((x, 0), (y, y)) =
f_{0}((x, 0), (0, y)) + f_{0}((x, 0), (y, 0)).

Therefore,

f_{0}((x, 0), (y, 0)) = 0 for all x, y ∈ W_{0}. (1)

Further,

f_{0}((0, x), (0, y)) = f_{0}(c(0, x), c(0, y)) = f_{0}((x, x), (y, y)) =
f_{0}((0, x), (0, y)) + f_{0}((x, 0), (y, 0)) + f_{0}((x, 0), (0, y)) + f_{0}((0, x), (y, 0)).

Therefore, f_{0}((x, 0), (0, y)) + f_{0}((0, x), (y, 0)) = 0. Since f_{0} is alternating,

f_{0}((x, 0), (0, y)) = −f_{0}((0, x), (y, 0)) = f_{0}((y, 0), (0, x)). (2)

Define h : W_{0} × W_{0} → L by h(x, y) = f_{0}((x, 0), (0, y)). By (2), h is symmetric. Since f_{0} is N-invariant, so is h. The nondegeneracy of h follows from (1) and the nondegeneracy of f_{0}. Since h is a N-invariant pairing on W_{0}, it is alternating. Since h is both alternating and symmetric we have h = 0, giving a contradiction.

The next result shows that the bound ℓ > de + 1 in Theorem 4.3 is sharp with respect to e (subject to the restrictions imposed by Remark 4.4). Here we have e = (ℓ − 1)/2, d = 2, and ℓ − 1 = de. A special case of Propositions 6.5 and 6.6 is (in the notation of Proposition 6.3) when ℓ is odd, F = Q_{ℓ}(ζ_{q}) (resp., Q_{ℓ}), W = Q_{ℓ}(ζ_{q})^{2} (resp., Q_{ℓ}(ζ_{q})), and N is the generalized quaternion group Q_{p}. In this case the group G is of inertia type.

Proposition 6.6. Suppose ℓ is an odd prime number, F is a complete discrete valuation field of characteristic zero and residue characteristic ℓ which is an unramified extension of Q_{ℓ}, and N is a finite group of order not divisible by ℓ. Let K = F (ζ_{ℓ} + ζ_{ℓ}^{−1}), and suppose W is a non-zero K-vector space which is also an absolutely simple faithful symplectic K[N]-module. Let V be the K-vector space W ⊗_{K} K(ζ_{ℓ}), and let G = N × μ_{ℓ}. Then V carries a natural structure of a faithful symplectic K[G]-module, and there is no G-stable lattice T in V with a G-invariant perfect alternating pairing f′ : T × T → O_{K}.

Proof. Let 𝔪 denote the maximal ideal of O_{K}, let k = O_{K}/𝔪, let e = e(K), and write dim_{K}(W) = 2t. By Proposition 6.5, V carries a natural structure of a faithful symplectic K[G]-module. Suppose there existed a G-stable lattice T in V and a G-invariant perfect alternating pairing f′ : T ×T → O_{K}, and consider the resulting representation ρ : G ↪ Aut_{K}(T, f′). The composition of ρ with reduction modulo 𝔪 gives a representation ρ̄: G → Aut_{k}(k^{4t}, f̄′), where f̄′ is the reduction of f′. If A ∈ ker(ρ̄), then A − 1 ∈ 𝔪End(T). By Proposition 3.12b we have A = 1, since e = (ℓ − 1)/2 < ℓ − 1. Therefore ρ̄ is injective, contradicting Proposition 6.5.

7. Polarizations and isogeny classes of abelian varieties

In Theorem 7.3 we use results of §6 to construct certain interesting isogeny classes of abelian varieties in characteristics 2 and 3. From now on, let p = 2 or 3, let F̄_{p} denote an algebraic closure of F_{p}, and let F = F̄_{p}((t)). Suppose ℓ is an odd prime different from p. In Theorem 7.3 we will show that for every integer d ≥ ℓ − 1, there exists an F -isogeny class of d-dimensional supersingular abelian varieties B such that the degree of every polarization on B is divisible by ℓ. In particular, B does not have a principal polarization.

Remark 7.1. E. Howe [7] constructed isogeny classes of ordinary abelian varieties over finite fields which do not have principal polarizations (see also [8]).

Let F^{s} denote a separable closure of F, and let G_{F} denote the Galois group Gal(F^{s}/F).

Lemma 7.2. If p and ℓ are distinct primes, then there exists a Galois extension L of F such that Gal(L/F) ≅ Q_{p} × μ_{ℓ}.

Proof. This follows immediately from the explicit description of G_{F} given in [11], §2, Theorem 1; see also [9].

Suppose A is an abelian variety over F and n is a positive integer. Let A_{n} denote the kernel of multiplication by n in A(F^{s}). Let T_{ℓ}(A) denote the ℓ-adic Tate module lim_{←}A_{ℓ^{m}}, and let V_{ℓ}(A) = T_{ℓ}(A) ⊗_{Z_{ℓ}} Q_{ℓ}. Let Z_{ℓ}(1) denote the projective limit of

the groups of ℓ^{m}-th roots of unity in F̄_{p}. An F -polarization λ on A induces a nondegenerate alternating bilinear form

e_{λ} : T_{ℓ}(A) × T_{ℓ}(A) → Z_{ℓ}(1) ≅ Z_{ℓ},

which is perfect if and only if deg(λ) is not divisible by ℓ. One may extend e_{λ} by Q_{ℓ}-linearity to a nondegenerate alternating bilinear form

e_{λ} : V_{ℓ}(A) × V_{ℓ}(A) → Q_{ℓ}.

Since λ is defined over F, and F contains all ℓ-power roots of unity, therefore e_{λ} is G_{F} -invariant. If A is an elliptic curve, then Aut(A) ⊂ Aut(V_{ℓ}(A), e_{λ}) = SL(V_{ℓ}(A)). Recall (Definition 2.7) that Q_{p} denotes the generalized quaternion group of order 4p.

Theorem 7.3. Let p = 2 or 3, let E be a supersingular elliptic curve over F̄_{p}, and let F = F̄_{p}((t)). Suppose ℓ is an odd prime and ℓ ≠ p. Let G = Q_{p} × μ_{ℓ}, and let L be a Galois extension of F such that Gal(L/F) ≅ G. Suppose b is a non-negative integer. Then there is an injective homomorphism

χ : Gal(L/F) ≅ G ↪ Aut(E^{ℓ−1})

such that if A is the twist of E^{ℓ−1} by χ, and B is an abelian variety over F which is F -isogenous to A × E^{b}, then the degree of every F -polarization of B is divisible by ℓ.

Proof. Since E is a supersingular elliptic curve over F̄_{p}, and p is 2 or 3, therefore Q_{p} ⊆ Aut(E) ⊆ SL(V_{ℓ}(E)) (see [20], §2). Since Q_{p} is a non-abelian finite group and V_{ℓ}(E) is 2-dimensional, therefore V_{ℓ}(E) is an absolutely simple faithful symplectic Q_{ℓ}[Q_{p}]-module. Fixing an isomorphism of additive groups Z[ζ_{ℓ}] ≅ Z^{ℓ−1} gives rise to an embedding μ_{ℓ} ↪ GL_{ℓ−1}(Z). We thus have

G = Q_{p} × μ_{ℓ} ⊂ Aut(E) × GL_{ℓ−1}(Z) ⊂ Aut(E^{ℓ−1}),

defining the homomorphism χ. Let A be the twist of E^{ℓ−1} by χ. Suppose B is an abelian variety over F, and β : B → A × E^{b} is an F -isogeny. By functoriality, β induces an isomorphism

V_{ℓ}(B) ≅ V_{ℓ}(A × E^{b}) = V_{ℓ}(A) ⊕ V_{ℓ}(E)^{b}

of Q_{ℓ}[G_{F}]-modules. Since E is defined over the algebraically closed field F̄_{p}, therefore the ℓ-power torsion on E is defined over F̄_{p}, and thus over F. Therefore, V_{ℓ}(E) is a trivial Q_{ℓ}[G_{F}]-module. It follows that the ℓ-adic representation G_{F} → Aut(T_{ℓ}(A)) factors through Gal(L/F) = G. From the definitions of χ and A, it follows that

V_{ℓ}(A) ≅ V_{ℓ}(E) ⊗_{Q_{ℓ}} Q_{ℓ}(ζ_{ℓ})

as Q_{ℓ}[G]-modules (here we view V_{ℓ}(E) as a Q_{p}-module and Q_{ℓ}(ζ_{ℓ}) as a μ_{ℓ}-module). Suppose λ is an F -polarization on B of degree prime to ℓ. Then e_{λ} is a perfect alternating G-invariant pairing on the lattice T_{ℓ}(B) in

V_{ℓ}(B) ≅ (V_{ℓ}(E) ⊗_{Q_{ℓ}} Q_{ℓ}(ζ_{ℓ})) ⊕ V_{ℓ}(E)^{b}.

This contradicts Proposition 6.2 (when b = 0) and Corollary 6.4, with N = Q_{p}, K = Q_{ℓ}, W = V_{ℓ}(E), and U = V_{ℓ}(B).

Remark 7.4. One may easily check that L is the smallest extension of F over which B acquires good reduction [17].

Remark 7.5. When p = 2 and ℓ ≠ 3, we may replace Q_{2} by SL_{2}(F_{3}) in Lemma 7.2 and Theorem 7.3 (since Aut(E) ≅ SL_{2}(F_{3})).

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Department of Mathematics, Ohio State University, Columbus, Ohio 43210, 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