Hasse Principle for G-Quadratic Forms

Eva Bayer–Fluckiger, Nivedita Bhaskhar, and Raman Parimala

Received: August 19, 2012 Revised: February 14, 2013

Communicated by Ulf Rehmann

Abstract.

Introduction.

Let k be a global field of characteristic ≠ 2. The classical Hasse–Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether this is true for G–quadratic forms, where G is a finite group. In the case of number fields the Hasse principle for G–quadratic forms does not hold in general, as shown by J. Morales [M 86]. The aim of the present paper is to study this question when k is a global field of positive characteristic. We give a sufficient criterion for the Hasse principle to hold (see th. 2.1.), and also give counter–examples. These counter–examples are of a different nature than those for number fields : indeed, if k is a global field of positive characteristic, then the Hasse principle does hold for G–quadratic forms on projective k[G]– modules (see cor. 2.3), and in particular if k[G] is semi–simple, then the Hasse principle is true for G–quadratic forms, contrarily to what happens in the case of number fields. On the other hand, there are counter–examples in the non semi–simple case, as shown in §3. Note that the Hasse principle holds in all generality for G–trace forms (cf. [BPS 13]).

The third named author is partially supported by National Science Foundation grant DMS-1001872.

§1. Definitions, notation and basic facts

Let k be a field of characteristic ≠ 2. All modules are supposed to be left modules.

G–quadratic spaces

Let G be a finite group, and let k[G] be the associated group ring. A G– quadratic space is a pair (V, q), where V is a k[G]–module that is a finite dimensional k–vector space, and q : V × V → k is a non–degenerate symmetric bilinear form such that

q(gx, gy) = q(x, y)

for all x, y ∈ V and all g ∈ G.

Two G–quadratic spaces (V, q) and (V′, q′) are isomorphic if there exists an isomorphism of k[G]–modules f : V → V′ such that q′(f(x), f(y)) = q(x, y) for all x, y ∈ V. If this is the case, we write (V, q) ≃_{G} (V′, q′), or simply q ≃_{G} q′.

Hermitian forms

Let R be a ring endowed with an involution r ↦ \overline{r}. For any R–module M, we denote by M^{∗} its dual Hom_{R}(M, R). Then M^{∗} has an R–module structure given by (rf)(x) = f(x)\overline{r} for all r ∈ R, x ∈ M and f ∈ M^{∗}. If M and N are two R–modules and if f : M → N is a homomorphism of R–modules, then f induces a homomorphism f^{∗} : N^{∗} → M^{∗} defined by f^{∗}(g) = gf for all g ∈ N^{∗}, called the adjoint of f.

A hermitian form is a pair (M, h) where M is an R–module and h : M ×M → R is biadditive, satisfying the following two conditions: (1.1) h(rx, sy) = rh(x, y)\overline{s} and \overline{h(x, y)} = h(y, x) for all x, y ∈ M and all r, s ∈ R. (1.2) The homomorphism h : M → M^{∗} given by y ↦ h( , y) is an isomorphism. Note that the existence of h implies that M is self-dual, i.e. isomorphic to its dual.

If G is a finite group, then the group algebra R = k[G] has a natural k–linear involution, characterized by the formula \overline{g} = g^{−1} for every g ∈ G. We have the following dictionary (see for instance [BPS 13, 2.1, Example]

a) R–module M ⟺ k-module M with a k–linear action of G;

b) R–dual M^{∗} ⟺ k–dual of M, with the contragredient (i.e. dual) action of G.

c) hermitian space (M, h) ⟺ symmetric bilinear form on M, which is G– invariant and defines an isomorphism of M onto its k–dual.

Therefore a hermitian space over k[G] corresponds to a G–quadratic space, as defined above.

Hermitian elements

Let E be a ring with an involution σ : E → E and put

E^{0} = {z ∈ E^{×} | σ(z) = z}.

Hasse Principle for G-Quadratic Forms 385

If z ∈ E^{0}, the map h_{z} : E×E → E defined by h_{z}(x, y) = x.z.σ(y) is a hermitian space over E; conversely, every hermitian space over E with underlying module E is isomorphic to h_{z} for some z ∈ E^{0}.

Define an equivalence relation on E^{0} by setting z ≡ z′ if there exists e ∈ E^{×} with z′ = σ(e)ze; this is equivalent to (E, h_{z}) ≃ (E, h_{z′}). Let H(E, σ) be the quotient of E^{0} by this equivalence relation. If z ∈ E^{0}, we denote by [z] its class in H(E, σ).

Classifying hermitian spaces via hermitian elements

Let (M, h_{0}) be a hermitian space over R. Set E_{M} = End(M). Let τ : E_{M} → E_{M} be the involution of E_{M} induced by h_{0}, i.e.

τ(e) = h_{0}^{−1}e^{∗}h_{0}, for e ∈ E_{M},

where e^{∗} is the adjoint of e. If (M, h) is a hermitian space (with the same underlying module M), we have τ(h_{0}^{−1}h) = h_{0}^{−1}(h_{0}^{−1}h)^{∗}h_{0} = h_{0}^{−1}h^{∗}(h_{0}^{−1})^{∗}h_{0} = h_{0}^{−1}h. Hence h_{0}^{−1}h is a hermitian element of (E_{M}, τ); let [h_{0}^{−1}h] be its class in H(E_{M}, τ).

Lemma 1.1. (see for instance [BPS 13, lemma 3.8.1]) Sending a hermitian space (M, h) to the element [h_{0}^{−1}h] of H(E_{M}, τ) induces a bijection between the set of isomorphism classes of hermitian spaces (M, h) and the set H(E_{M}, τ).

Components of algebras with involution

Let A be a finite dimensional k–algebra, and let ι : A → A be a k–linear involution. Let R_{A} be the radical of A. Then A/R_{A} is a semi–simple k– algebra, hence we have a decomposition A/R_{A} = ∏_{i=1,...,r} M_{n_{i}}(D_{i}), where D_{1}, …, D_{r} are division algebras. Let us denote by K_{i} the center of D_{i}, and let D_{i}^{op} be the opposite algebra of D_{i}.

Note that ι(R_{A}) = R_{A}, hence ι induces an involution ι : A/R_{A} → A/R_{A}. Therefore A/R_{A} decomposes into a product of involution invariant factors. These can be of two types : either an involution invariant matrix algebra M_{n_{i}}(D_{i}), or a product M_{n_{i}}(D_{i}) × M_{n_{i}}(D_{i}^{op}), with M_{n_{i}}(D_{i}) and M_{n_{i}}(D_{i}^{op}) exchanged by the involution. We say that a factor is unitary if the restriction of the involution to its center is not the identity : in other words, either an involution invariant M_{n_{i}}(D_{i}) with ι|K_{i} not the identity, or a product M_{n_{i}}(D_{i})× M_{n_{i}}(D_{i}^{op}) . Otherwise, the factor is said to be of the first kind. In this case, the component is of the form M_{n_{i}}(D_{i}) and the restriction of ι to K_{i} is the identity. We say that the component is orthogonal if after base change to a separable closure ι is given by the transposition, and symplectic otherwise. A component M_{n_{i}}(D_{i}) is said to be split if D_{i} is a commutative field.

Completions If k is a global field and if v is a place of k, we denote by k_{v} the completion of k at v. For any k–algebra E, set E_{v} = E ⊗_{k} k_{v}. If K/k is a field extension of finite degree and if w is a place of K above v, then we use the notation w|v.

§2. Hasse principle

In this section, k will be a global field of characteristic ≠ 2. Let us denote by Σ_{k} the set of all places of k. The aim of this section is to give a sufficient criterion for the Hasse principle for G–quadratic forms to hold. All modules are left modules, and finite dimensional k–vector spaces.

Theorem 2.1. Let V be a k[G]–module , and let E = End(V). Let R_{E} be the radical of E, and set \overline{E} = E/R_{E}. Suppose that all the orthogonal components of \overline{E} are split, and let (V, q), (V, q′) be two G–forms. Then q ≃_{G} q′ over k if and only if q ≃_{G} q′ over all the completions of k.

This is announced in [BP 13], and replaces th. 3.5 of [BP 11]. The proof of th. 2.1 relies on the following proposition

Proposition 2.2. Let E be a finite dimensional k–algebra endowed with a k– linear involution σ : E → E. Let R_{E} be the radical of E, and set \overline{E} = E/R_{E}. Suppose that all the orthogonal components of \overline{E} are split. Then the canonical map H(E, σ) → ∏_{v∈Σ_{k}} H(E_{v}, σ_{v}) is injective.

Proof. The case of a simple algebra. Suppose first that E is a simple k– algebra. Let K be the center of E, and let F be the fixed field of σ in K. Let Σ_{F} denote the set of all places of F. For all v ∈ Σ_{k}, set E_{v} = E ⊗_{k} k_{v}, and note that E_{v} = ∏_{w|v} E_{w}, therefore ∏_{v∈Σ_{k}} H(E_{v}, σ_{v}) = ∏_{w∈Σ_{F}} H(E_{w}, σ_{w}). By definition, H(E, σ) is the set of isomorphism classes of one dimensional hermitian forms over E. Moreover, if σ is orthogonal, then the hypothesis implies that E is split, in other words we have E ≃ M_{n}(F). Therefore the conditions of [R 11, th. 3.3.1] are fulfilled, hence the Hasse principle holds for hermitian forms over E with respect to σ. This implies that the canonical map H(E, σ) → ∏_{v∈Σ_{k}} H(E_{v}, σ_{v}) is injective.

The case of a semi–simple algebra. Suppose now that E is semi–simple. Then

E ≃ E_{1} × … × E_{r} × A × A^{op},

where E_{1}, …, E_{r} are simple algebras which are stable under the involution σ, and where the restriction of σ to A × A^{op} exchanges the two factors. Applying [BPS 13, lemmas 3.7.1 and 3.7.2] we are reduced to the case where E is a simple algebra, and we already know that the result is true in this case.

General case. We have \overline{E} = E/R_{E}. Then \overline{E} is semi–simple, and σ induces a k–linear involution \overline{σ} : \overline{E} → \overline{E}. We have the following commutative diagram

H(E, σ) ⟶^{f} ∏_{v∈Σ_{k}} H(E_{v}, σ)
↓  ↓
H(\overline{E}, \overline{σ}) ⟶^{\overline{f}} ∏_{v∈Σ_{k}} H(\overline{E}_{v}, \overline{σ}),

where the vertical maps are induced by the projection E → \overline{E}. By [BPS 13, lemma 3.7.3], these maps are bijective. As \overline{E} is semi–simple, the map \overline{f} is injective, hence f is also injective. This concludes the proof.

Proof of th. 2.1. It is clear that if q ≃_{G} q′ over k, then q ≃_{G} q′ over all the completions of k. Let us prove the converse. Let (V, h) be the k[G]-hermitian space corresponding to (V, q), and let σ : E → E be the involution induced by (V, h) as in §1. Let (V, h′) be the k[G]–hermitian space corresponding to (V, q′), and set u = h^{−1}h′. Then u ∈ E^{0}, and by lemma 1.1. the element [u] ∈ H(E, σ) determines the isomorphism class of (V, q′); in other words, we have q ≃_{G} q′ if and only if [u] = [1] in H(E, σ). Hence the theorem is a consequence of proposition 2.2.

Corollary 2.3 Suppose that char(k) = p > 0, and let V be a projective k[G]– module.. Let (V, q), (V, q′) be two G–forms. Then q ≃_{G} q′ over k if and only if q ≃_{G} q′ over all the completions of k.

Proof. Since V is projective, there exists a k[G]–module W and n ∈ N such that V ⊕ W ≃ k[G]^{n}. The endomorphism ring of k[G]^{n} is M_{n}(k[G]), and as char(k) = p > 0, we have k[G] = F_{p}[G] ⊗_{F_{p}} k. Hence M_{n}(k[G]) is isomorphic to M_{n}(F_{p}[G]) ⊗_{F_{p}} k. Let E = End(V), let R_{E} be the radical of E, and let \overline{E} = E/R_{E}. Let us show that all the components of \overline{E} are split. Let e be the idempotent endomorphism of V ⊕ W which is the identity of V. Set Λ = End(V ⊕ W) and let R_{Λ} be the radical of Λ. Then eΛe = E and eR_{Λ}e = R_{E}. Set \overline{Λ} = Λ/R_{Λ}, and and let \overline{e} be the image of e in \overline{Λ}. Set \overline{k[G]} = k[G]/rad(k[G]). Then we have \overline{E} ≃ \overline{e}\overline{Λ}\overline{e} ≃ \overline{e}M_{n}(\overline{k[G]})\overline{e}. This implies that \overline{E} is a component of the semi–simple algebra M_{n}(\overline{k[G]}). Let us show that all the components of M_{n}(\overline{k[G]}) are split. As F_{p} is a finite field, F_{p}[G]/(rad(F_{p}[G]) is a product of matrix algebras over finite fields. Moreover, for any finite field F of characteristic p, the tensor product F ⊗_{F_{p}} k is a product of fields. This shows that (F_{p}[G]/(rad(F_{p}[G]))⊗_{F_{p}} k is a product of matrix algebras over finite extensions of k; in particular, it is semi–simple. The natural isomorphism F_{p}[G] ⊗_{F_{p}} k → k[G] induces an isomorphism [F_{p}[G]/(rad(F_{p}[G]))] ⊗_{F_{p}} k → k[G]/(rad(F_{p}[G]).k[G]). Therefore rad(F_{p}[G].k[G]) is the radical of k[G], and we have an isomorphism [F_{p}[G]/(rad(F_{p}[G]))]⊗_{F_{p}} k → k[G]/(rad(k[G])). Hence all the components of k[G]/(rad(k[G])) are split. This implies that all the components of \overline{E} are split as well. Therefore the corollary follows from th. 2.1.

The following corollary is well–known (see for instance [R 11, 3.3.1 (b)]).

Corollary 2.4 Suppose that char(k) = p > 0, and that the order of G is prime to p. Then two G–quadratic forms are isomorphic over k if and only if they become isomorphic over all the completions of k.

Proof. This follows immediately from cor. 2.3.

§3. Counter–examples to the Hasse principle

Let k be a field of characteristic p > 0, let C_{p} be the cyclic group of order p, and let G = C_{p} × C_{p} × C_{p}. In this section we give counter–examples to the Hasse principle for G×G–quadratic forms over k in the case where k is a global field. We start with some constructions that are valid for any field of positive characteristic.

3.1 A construction

Let D be a division algebra over k. It is well–known that there exist indecomposable k[G]–modules such that their endomorphism ring modulo the radical is isomorphic to D. We recall here such a constuction, brought to our attention by R. Guralnick, in order to use it in 3.2 in the case of quaternion algebras.

The algebra D can be generated by two elements (see for instance [J 64, Chapter VII, §12, th. 3, p. 182]). Let us choose i, j ∈ D be two such elements. Let us denote by D^{op} the opposite algebra of D, and let d be the degree of D. Then we have D ⊗_{k} D^{op} ≃ M_{d^{2}}(k). Let us choose an isomorphism f : D ⊗_{k} D^{op} ≃ M_{d^{2}}(k), and set a_{1} = f(1 ⊗ 1) = 1, a_{2} = f(i ⊗ 1) and a_{3} = f(j ⊗ 1).

Let g_{1}, g_{2}, g_{3} ∈ G be three elements of order p such that the set {g_{1}, g_{2}, g_{3}} generates G.and let us define a representation G → GL_{2d^{2}}(k) by sending g_{m} to the matrix

( \begin{matrix}I & a_{m} \\ 0 & I \end{matrix} )

for all m = 1, 2, 3. Note that this is well–defined because char(k) = p. This endowes k^{2d^{2}} with a structure of k[G]–module. Let us denote by N this k[G]– module, and let E_{N} be its endomorphism ring. Then

E_{N} = \{( \begin{matrix}x & y \\ 0 & x \end{matrix} ) | x ∈ D^{op} ⊂ M_{d^{2}}(k), y ∈ M_{d^{2}}(k)\} ,

and its radical is

R_{N} = \{( \begin{matrix}0 & y \\ 0 & 0 \end{matrix} ) | y ∈ M_{d^{2}}(k)\} ,

hence E_{N}/R_{N} ≃ D^{op}.

3.2. The case of a quaternion algebra

Let H be a quaternion algebra over k. Then by 3.1, we get a k[G]–module N = N_{H} with endomorphism ring E_{N} such that E_{N}/R_{N} ≃ H^{op}, where R_{N} is the radical of E_{N}. We now construct a G–quadratic form q over N in such a way that the involution it induces on E_{N}/R_{N} ≃ H^{op} is the canonical involution.

Let i, j ∈ H such that i^{2}, j^{2} ∈ k^{×} and that ij = −ji. Let τ : H → H be the orthogonal involution of H obtained by composing the canonical involution of H with Int(ij). Let σ : H^{op} → H^{op} be the canonical involution of H^{op}. Let us consider the tensor product of algebras with involution

(H, τ) ⊗ (H^{op}, σ) = (M_{4}(k), ρ).

Then ρ is a symplectic involution of M_{4}(k) satisfying ρ(a_{m}) = a_{m} for all m = 1, 2, 3, since τ(i) = (ij)(−i)(ij)^{−1} = i, τ(j) = (ij)(−j)(ij)^{−1} = j. Let α ∈ M_{4}(k) be a skew–symmetric matrix such that for all x ∈ M_{4}(k), we have

ρ(x) = α^{−1}x^{T} α, where x^{T} denotes the transpose of x. Set A = ( \begin{matrix}0 & α \\ −α & 0 \end{matrix} ).

Then A^{T} = A. Let q : N × N → k be the symmetric bilinear form defined by A :

q(v, w) = v^{T} Aw

for all v, w ∈ N. Let γ : M_{8}(k) → M_{8}(k) be the involution adjoint to q, that is

γ(X) = A^{−1}X^{T} A

for all X ∈ M_{8}(k), i.e. q(fv, w) = q(v, γ(f)w) for all f ∈ M_{8}(k) and all v, w ∈ N. The involution γ restricts to an involution of E_{N}, as for all x, y ∈ M_{4}(k), we have

γ ( \begin{matrix}x & y \\ 0 & x \end{matrix} ) = ( \begin{matrix}α^{−1}x^{T} α & −α^{−1}y^{T} α \\ 0 & α^{−1}x^{T} α \end{matrix} ) .

It also sends R_{N} to itself, and induces an involution \overline{γ} on H^{op} ≃ E_{N}/R_{N} that coincides with the canonical involution of H^{op}.

We claim that q : N × N → k is a G–quadratic form. To check this, it suffices to show that q(g_{m}v, g_{m}w) = q(v, w) for all v, w ∈ N and for all m = 1, 2, 3. Since ρ(a_{m}) = a_{m} for all m = 1, 2, 3, we have

γ ( \begin{matrix}I & a_{m} \\ 0 & I \end{matrix} ) = ( \begin{matrix}I & a_{m} \\ 0 & I \end{matrix} )^{−1}

and hence

q(g_{m}v, g_{m}w) = q(v, γ(g_{m})g_{m}w) = q(v, w)

for all m = 1, 2, 3 and all v, w ∈ N. Thus q is a G–quadratic form, and by construction, the involution of E_{N} induced by q is the restriction of γ to E_{N}.

3.3. Two quaternion algebras

Let H_{1} and H_{2} be two quaternion algebras over k. By the construction of 3.2, we obtain two indecomposable k[G]–modules N_{1} and N_{2}. Set E_{1} = E_{N_{1}} and E_{2} = E_{N_{2}}. Let R_{i} be the radical of E_{i} for i = 1, 2, and set \overline{E}_{i} = E_{i}/R_{i}. We also obtain G–quadratic spaces q_{i} : N_{i} × N_{i} → k inducing involutions γ_{i} : E_{i} → E_{i} such that the involutions \overline{γ}_{i} : \overline{E}_{i} → \overline{E}_{i} coincide with the canonical involution of H_{i}^{op}, for all i = 1, 2.

Let us consider the tensor product (N, q) = (N_{1}, q_{1}) ⊗_{k} (N_{2}, q_{2}). Then (N, q) is a G × G–quadratic space. Set E = End_{k[G×G]}(N_{1} ⊗ N_{2}). Then E ≃ E_{1} ⊗ E_{2}. Let I be the ideal of E generated by R_{1} and R_{2}. Then there is a natural isomorphism f : E_{1} ⊗ E_{2} → E with f(I) = R_{E}, where R_{E} is the radical of E. Set \overline{E} = E/R_{E}. Then \overline{E} ≃ \overline{E}_{1} ⊗ \overline{E}_{2} ≃ H_{1}^{op} ⊗ H_{2}^{op}.

Set γ = γ_{1} ⊗ γ_{2}. Then γ : E → E is the involution induced by the G × G– quadratic space (N, q). We obtain an involution \overline{γ} : \overline{E} → \overline{E}, and \overline{γ} = \overline{γ}_{1} ⊗ \overline{γ}_{2}. Let us recall that \overline{E}_{i} = H_{i}^{op} for i = 1, 2, and that \overline{γ}_{i} is the canonical involution of H_{i}^{op}. Hence \overline{γ} : \overline{E} → \overline{E} is an orthogonal involution.

3.4. A counter–example to the Hasse principle

Suppose now that k is a global field of characteristic p, with p > 2, and suppose that H_{i} is ramified at exactly two places v_{i}, v′_{i} of k, such that v_{1}, v′_{1}, v_{2}, v′_{2} are all distinct. We have H_{1}^{op} ⊗ H_{2}^{op} ≃ M_{2}(Q) where Q is a quaternion division algebra over k, and Q is ramified exactly at the places v_{1}, v′_{1}, v_{2}, v′_{2} of k. Recall that the involution \overline{γ} : M_{2}(Q) → M_{2}(Q) is the tensor product of the canonical involutions of H_{i}^{op}. In particular, \overline{γ} is of orthogonal type. Note that at all v ∈ Σ_{k}, one of the algebras H_{1}^{op} or H_{2}^{op} is split. This implies that at all v ∈ Σ_{k}, the involution \overline{γ} is hyperbolic.

Let δ : Q → Q be an orthogonal involution of the division algebra Q. Then \overline{γ} is induced by some hermitian space h : Q^{2} × Q^{2} → Q with respect to the involution δ. As for all v ∈ Σ_{k}, the involution \overline{γ} is hyperbolic at v, the hermitian form h is also hyperbolic at v. By lemma 1.1 the set of isomorphism classes of hermitian spaces on Q^{2} is in bijection with the set H(\overline{E}, \overline{γ}), the hermitian space (Q^{2}, h) corresponding to the element [1] ∈ H(\overline{E}, \overline{γ}).

Let (Q^{2}, h′) be a hermitian space which becomes isomorphic to (Q^{2}, h) over Q_{v} for all v ∈ Σ_{k}, but is not isomorphic to (Q^{2}, h) over Q (this is possible by [Sch 85, 10.4.6]). Let u ∈ \overline{E}^{0} such that [u] ∈ H(\overline{E}, \overline{γ}) corresponds to (Q^{2}, h′) by the bijection of lemma 1.1. Then [u] ≠ [1] ∈ H(\overline{E}, \overline{γ}), and the images of [u] and [1] coincide in ∏_{v∈Σ_{k}} H(\overline{E}_{v}, \overline{γ}).

Recall that H(E, γ) is in bijection with the isomorphism classes of (G × G)– quadratic forms over N, the element [1] ∈ H(E, γ) corresponding to the isomorphism class of (N, q). Let π : E → \overline{E} be the projection, and let ũ ∈ E^{0} be such that π(ũ) = u (cf. lemma 1.1). Let (N, q′) be a (G × G)–quadratic form corresponding to ũ. The diagram

H(E, γ) ⟶^{f} ∏_{v∈Σ_{k}} H(E_{v}, γ)
↓  ↓
H(\overline{E}, \overline{γ}) ⟶^{\overline{f}} ∏_{v∈Σ_{k}} H(\overline{E}_{v}, \overline{γ}),

is commutative, and the vertical maps are bijective by [BPS 13, lemma 3.7.3]. Hence (N, q) and (N, q′) are become isomorphic over all the completions of k, but are not isomorphic over k.

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Eva Bayer–Fluckiger École Polytechnique Fédérale de Lausanne EPFL/FSB/MATHGEOM/CSAG Station 8 1015 Lausanne, Switzerland eva.bayer@epfl.ch Nivedita Bhaskhar Department of Mathematics & Computer Science Emory University Atlanta, GA 30322, USA. nbhaskh@emory.edu

Raman Parimala

Department of Mathematics & Computer Science Emory University Atlanta, GA 30322, USA. parimala@mathcs.emory.edu