Arithmetic invariant theory II

Transcription

1 Arithmetic invariant theory II Manjul Bhargava, Benedict H. Gross, and Xiaoheng Wang November 3, 2013 Contents 1 Introduction 1 2 Liting results Pure inner orms Twisting the representation Rational orbits in the twisted representation A cohomological obstruction to liting invariants Examples with trivial stabilizer SO(n + 1) acting on n copies o the standard representation SL(5) acting on 3 copies o the representation 2 (5) U(n 1) acting on the adjoint representation u(n) o U(n) Examples with nontrivial stabilizer and nontrivial obstruction Spin(n + 1) acting on n copies o the standard representation o SO(n + 1) SL n acting on Sym 2 (n) Sym 2 (n) Some inite group schemes and their cohomology SL n /µ 2 acting on Sym 2 (n) Sym 2 (n) The two obstructions Integral orbits 23 1 Introduction Geometric invariant theory involves the study o invariant polynomials or the action o a reductive algebraic group G on a linear representation V over a ield k, and the relation between these invariants and the G-orbits on V, usually under the hypothesis that the base ield k is algebraically closed. In avorable cases, one can determine the geometric quotient V /G = Spec(Sym (V )) G and identiy certain ibers o the morphism V V /G with certain G-orbits on V. For general ields k the situation is more complicated. The additional complexity in the orbit picture, when k is not separably closed, is what we reer to as arithmetic invariant theory. 1

2 In a previous paper [4], we studied the arithmetic invariant theory o a reductive group G acting on a linear representation V over a general ield k. Let k s denote a separable closure o k. When the stabilizer G v o a vector v is smooth, the k-orbits inside o the k s -orbit o v are parametrized by classes in the kernel o the map o pointed sets in Galois cohomology γ : H 1 (k, G v ) H 1 (k, G) (c. [22]). We produced elements in the kernel o γ or three representations o the split odd orthogonal group G = SO(W ) = SO(2n+1): the standard representation V = W, the adjoint representation V = 2 (W ), and the symmetric square representation V = Sym 2 W. For all three representations the ring o G-invariant polynomials on V is a polynomial ring and the categorical quotient V /G is isomorphic to aine space. Furthermore, in each case there is a natural section o the morphism π : V V /G, so the k-rational points o V /G lit to k-rational orbits o G on V. Such a section may not exist or the action o the odd orthogonal groups G = SO(W ) that are not split over k. The corresponding representations V = W, 2 (W ), and Sym 2 W have the same ring o polynomial invariants, so V /G = V /G, but there may be rational points in this aine space that do not lit to rational orbits o G on V. The groups G = SO(W ) are the pure inner orms o G, i.e., twists o G using classes in H 1 (k, G) as opposed to inner orms o G which are obtained using classes in H 1 (k, G ad ). The representations V o G are then the corresponding twists o V. These pure inner orms and their representations are determined by a class c in the pointed set H 1 (k, G). Suppose that the image o v in V /G is equal to, and that G(k s ) acts transitively on the k s -rational points o the iber above. Then we show that the k-orbits or G on V with invariant are parametrized by the elements in the iber o the map γ : H 1 (k, G v ) H 1 (k, G) above the class c. We also consider representations where there is an obstruction to liting k-rational invariants in V /G to k-rational orbits on V, or all pure inner orms o G. Let be a rational invariant in V /G, and assume that there is a single orbit over k s with invariant, whose stabilizers G v are abelian. We show that these stabilizers are canonically isomorphic to a ixed commutative group scheme G, which is determined by and is deined over k. We then construct a class d in the cohomology group H 2 (k, G ), whose non-vanishing obstructs the descent o the orbit to k, or all pure inner orms o G. On the other hand, i d = 0, we show that there is least one pure inner orm o G that has k-rational orbits with invariant. When the stabilizer G v is trivial, so the action o G(k s ) on elements with invariant over k s is simply transitive, the obstruction d clearly vanishes. In this case, we show that there is a unique pure inner orm G or which there exists a unique k-rational orbit on V with invariant. We give a number o examples o such representations, such as the action o SO(W ) = SO(n + 1) on n copies o the standard representation W, and the action o SL(W ) = SL(5) on three copies o the exterior square representation 2 (W ). It is also possible that the stabilizer G v is abelian and nontrivial, and yet the obstruction d still vanishes. This scenario occurs requently; or example, it occurs or all representations arising in Vinberg s theory o θ-groups (see [19] and [17]). These representations are remarkable in that the morphism π : V V /G has an (algebraic) section (called the Kostant section). This implies that the obstruction d vanishes. The representations 2 (W ) and Sym 2 W o the odd split orthogonal group SO(W ) studied in [4] indeed shared this property. (For a treatment o many such representations o arithmetic interest, involving rational points and Selmer groups o Jacobians o algebraic curves, see [7], [5], [10], [24], and [25].) Finally, it is possible that the stabilizer G v o a stable vector v is abelian and nontrivial, and the obstruction class d is also nontrivial in H 2 (k, G v ). Fewer such representations occur in the lit- 2

3 erature, but they too appear to be extremely rich arithmetically. In this paper, we give a detailed study o such a representation, namely the action o G = SL(W ) = SL(n) on the vector space V = Sym 2 W Sym 2 W o pairs o symmetric bilinear orms on W. Like the representation Sym 2 W o SO(W ), the ring o polynomial invariants is a polynomial ring, and there are stable orbits in the sense o geometric invariant theory. In act, the stabilizer G v o any vector v in one o the stable orbits is a inite commutative group scheme isomorphic to (Z/2Z) n 1 over k s, and G(k s ) acts transitively on the vectors in V (k s ) with the same invariant as v. However, when the dimension n = 2g +2 o W is even, it may not be possible to lit k-rational points o the quotient V /G to k-rational orbits o G on V. We relate this obstruction to the arithmetic o 2-coverings o Jacobians o hyperellipic curves o genus g over k. In [3], this connection with hyperelliptic curves was used to show that most hyperelliptic curves over Q o genus g 2 have no rational points. In a orthcoming paper [6], we will use the ull connection with 2-coverings o Jacobians o hyperelliptic curves to study the arithmetic o hyperelliptic curves; in particular, we will prove that a positive proportion o hyperelliptic curves over Q have points locally over Q ν or all places ν o Q, but have no points globally over any odd degree extension o Q. This paper is organized as ollows. In Section 2, we describe the notion o a pure inner orm G o a reductivegroup G over a ield k, and the corresponding twisted orm V o a given representation V o G. We also discuss in detail the problem o liting k-rational points o V /G to k-rational orbits o G (and its pure inner orms) in the case where the generic stabilizer G v is abelian, and we describe the cohomological obstruction to liting invariants lying in H 2 (k, G ). The obstruction element in H 2 (k, G ) can also be deduced rom the theory o residual gerbes on algebraic stacks (see [9] and [14, Chapter 11]). Since we have not seen any concise reerence to the speciic results needed in this context, we elt it would be useul to give a sel-contained account here. In Section 3, we then consider three examples o representations where the stabilizer G v is trivial. These representations are: 1. the split orthogonal group SO(W ) acting on n copies o W, where dim(w ) = n + 1; 2. SL(W ) acting on three copies o 2 W, where dim(w ) = 5; 3. the unitary group U(n) acting on the adjoint representation o U(n + 1). In each o these three cases, the cohomological obstruction clearly vanishes and we see explicitly how the orbits, over all pure inner orms o the group G, are classiied by the elements o the space V /G o invariants. The third representation and its orbits have played an important role in the work o Jacquet Rallis [13] and Wei Zhang [29] in connection with the relative trace ormula approach to the conjectures o Gan, Gross, and Prasad [11]. In Section 4, we study three examples o representations where the stabilizer G v is nontrivial and abelian, and where there are cohomological obstructions to liting invariants. These representations are: 1. Spin(W ) acting on n copies o W, where dim(w ) = n + 1; 2. SL(W ) acting on Sym 2 W Sym 2 W ; 3. (SL /µ 2 )(W ) acting on Sym 2 W Sym 2 W (this group acts only when dim(w ) is even). 3

4 In the irst case, we show that the obstruction is the Brauer class o a Cliord algebra determined by the invariants. In the second and third cases, we show that when n is odd, there is no cohomological obstruction to liting invariants, but when n is even, the obstruction can be nontrivial. We parametrize the orbits or both groups in terms o arithmetic data over k, and describe the resulting criterion or the existence o orbits over k. Finally, we describe the connection between the cohomological obstruction and the arithmetic o two-covers o Jacobians and hyperelliptic curves over k, which will play an important role in [6]. As in [4], the heart o this paper lies in the examples that illustrate the various scenarios that can occur, and how one can treat each scenario in order to classiy the orbits, over a ield that is not necessarily algebraically closed, in terms o suitable arithmetic data. We thank Jean-Louis Colliot-Thélène, Jean-Pierre Serre, Bas Edixhoven, and Wei Ho or useul conversations and or their help with the literature. 2 Liting results In this section, we assume that G is a reductive group with a linear representation V over the ield k. We will study the general problem o liting k-rational points o V /G to k-rational orbits o pure inner orms G o G on the corresponding twists V o V. For stable orbits over the separable closure k s with smooth abelian stabilizers G v, we will show how these stabilizers descend to a group scheme G over k and describe a cohomological obstruction to the liting problem lying in H 2 (k, G ). 2.1 Pure inner orms We begin by recalling the notion o a pure inner orm G c o G and the action o G c on a twisted representation V c ([22, Ch 1 5]). Suppose (σ c σ ) is a 1-cocycle on Gal(k s /k) with values in the group G(k s ). That is, c στ = c σ σc τ or any σ, τ Gal(k s /k). We deine the pure inner orm G c o G over k by giving its k s -points and describing a Galois action. Let G c (k s ) = G(k s ) with action σ(h) = c σ σ hc 1 σ. (1) Since c is a cocycle, we have στ(h) = σ(τ(h)). Let g be an element o G(k s ). I b σ = g 1 c σ σ g is a cocycle in the same cohomology class as c, then the map on k s -points G b G c deined by h ghg 1 commutes with the respective Galois actions, so deines an isomorphism over k. Hence the isomorphism class o the pure inner orm G c over k depends only on the image o c in the pointed set H 1 (k, G). 2.2 Twisting the representation I we compose the cocycle c with values in G(k s ) with the homomorphism ρ : G GL(V ), we obtain a cocycle ρ(c) with values in GL(V )(k s ). By the generalization o Hilbert s Theorem 90, we have H 1 (k, GL(V )) = 1 ([23, Ch X]). Hence there is an element g in GL(V )(k s ), well-deined up to let multiplication by GL(V )(k), such that ρ(c σ ) = g 1σ g (2) 4

5 or all σ in Gal(k s /k). We use the element g to deine a twisted representation o the group G c on the vector space V over k. The homomorphism ρ g : G c (k s ) GL(V )(k s ) deined by ρ g (h) = gρ(h)g 1 commutes with the respective Galois actions, so deines a representation over k. We emphasize that the Galois action on G c (k s ) is as deined in (1), whereas the Galois action on GL(V )(k s ) is the usual action. The isomorphism class o the representation ρ g : G c GL(V ) over k is independent o the choice o g in (2) which trivializes the cocycle. I g = ag is another choice, with a in GL(V )(k), then conjugation by a gives an isomorphism rom ρ g to ρ g. Since the isomorphism class o this representation depends only on the cocycle c, we will write V c or the representation ρ g o G c. The act that the cocycle c σ takes values in G, and not in the adjoint group, is crucial to deining the twist V c o the representation V. For 1-cocycles c with values in G ad Aut(G), one can deine the inner orm G c, but one does not always obtain a twisted representation V c. For example, consider the case o G = SL 2 with V the standard two-dimensional representation. The nontrivial inner orms o G are obtained rom nontrivial cohomology classes in H 1 (k, PGL 2 ). These are the groups G c o invertible elements o norm 1 in quaternion division algebras D over k. The group G c does not have a aithul two-dimensional representation over k this representation is obstructed by the quaternion algebra D. Since H 1 (k, SL 2 ) is trivial, there are no nontrivial pure inner orms o G. 2.3 Rational orbits in the twisted representation We now ix a rational point in the canonical quotient V /G, and let V be the iber in V. For the rest o this subsection, we assume that the set V (k) o rational points in the iber is nonempty, and that G(k s ) acts transitively on the points in V (k s ). In particular, this orbit is closed (as it is deined by the values o the invariant polynomials). Let v be a point in V (k) and let G v denote its stabilizer in G. The group G(k) acts on the rational points o the iber over. In Proposition 1 o [4] we showed that the orbits o G(k) on the set V (k) correspond bijectively to elements in the kernel o the map γ : H 1 (k, G v ) H 1 (k, G) o pointed sets in Galois cohomology. In this section, we will generalize this to a parametrization o certain orbits o G c (k), where c H 1 (k, G). Note that by our hypothesis and the deinition o G c, the group G c (k s ) = G(k s ) acts transitively on the set gv (k s ) in V (k s ), where g is as in (2). We deine the set V c (k) := V (k) gv (k s ), which admits an action o the rational points o the pure inner orm G c. Here is a simple example, which illustrates many elements o the theory o orbits or pure inner twists with a ixed rational invariant. Assume that the characteristic o k is not equal to 2, and let G be the étale group scheme µ 2 o order 2 over k. Let V be the nontrivial one-dimensional representation o G on the ield k. (This is the standard representation o the orthogonal group O(1) over k.) The polynomial invariants o this representation are generated by q(x) = x 2, so the canonical quotient V /G is the aine line. Let be a rational invariant in k with 0. Then the iber V is the subscheme o V deined by {x : x 2 = }, so V (k) is nonempty i and only i is a square in k. This is certainly true over the separable closure k s o k, and the group G(k s ) acts simply transitively on V (k s ). 5

6 An element c in k deines a cocycle c σ = σ c/ c with values in G(k s ), whose class in the cohomology group H 1 (k, G) = k /k 2 depends only on the image o c modulo squares. The element g = c in GL(V )(k s ) trivializes this class in the group H 1 (k, GL(V )). Although the inner twist G c and the representation V c remain exactly the same, we ind that V c (k) = V (k) gv (k s ) = {x k : x 2 = c}. Hence the set V c(k) is nonempty i and only i the element c is a square in k. Note that there is a unique inner twisting G c where the iber V c has k-rational points, and in that case the group Gc (k) acts simply transitively on V c(k). Returning to the general case, we have the ollowing generalization o Proposition 1 in [4] (which is the case c = 1 below). Proposition 1 Let G be a reductive group with representation V. Suppose there exists v V (k) with invariant (V /G)(k) and stabilizer G v such that G(k s ) acts transitively on V (k s ). Then there is a bijection between the set o G c (k)-orbits on V c (k) and the iber γ 1 (c) o the map γ : H 1 (k, G v ) H 1 (k, G) above the class c H 1 (k, G). In particular, the image o H 1 (k, G v ) in H 1 (k, G) determines the set o pure inner orms o G or which the k-rational invariant lits to a k-rational orbit o G c on V c. Beore giving the proo, we illustrate this with an example rom [4]. Let W be a split orthogonal space o dimension 2n + 1 and signature (n + 1, n) over k = R, let G = SO(W ) = SO(n + 1, n). The pure inner orms o G are the groups G c = SO(p, q) with p + q = 2n + 1 and q n (mod 2), and the representation W c o G c is the standard representation on the corresponding orthogonal space W (p, q) o signature (p, q). The group G = SO(W ) acts aithully on the space V = Sym 2 (W ) o sel-adjoint operators T on W. For this representation, the inner twists G c o G are exactly the same, and the twisted representation V c o G c is isomorphic to Sym 2 W c. The polynomial invariants in (V /G)(R) are given by the coeicients o the characteristic polynomial o T. Assume that this characteristic polynomial is separable, with 2m + 1 real roots. Then the stabilizer o a point v 0 V (R) is the inite commutative group scheme (µ 2m+1 2 (Res C/R µ 2 ) n m ) N=1. Hence H 1 (R, G v0 ) is an elementary abelian 2-group o order 2 2m. This group maps under γ to the pointed set H 1 (R, SO(W )), which is inite o cardinality n + 1. The iber over the class o SO(p, q) is nonempty i and only i both p and q are greater than or equal to n m. In this case, write q = n m + a, with a m (mod 2). Then the iber has cardinality ( ) ( 2m+1 a. For example, the kernel has cardinality 2m+1 ) m. When pq = 0, so the space W c = W (p, q) is deinite, there are orbits in V c (R) only in the case when m = n, so the characteristic polynomial splits completely over R. In that case there is a single orbit. This is the content o the classical spectral theorem. Proo o Proposition 1: Suppose c is a 1-cocycle with values in G(k s ) and ix g GL(V )(k s ) such that c σ = g 1σ g or all σ Gal(k s /k). When V c (k) is nonempty we must show that c is in the image o H 1 (k, G v ). Indeed, suppose gw V c(k) or some w V (k s ). By our assumption on the transitivity o the action on k s points, there exists h G(k s ) such that w = hv. The rationality condition on gw translates into saying that, or any σ Gal(k s /k), we have c σ σ hv = hv. That is, h 1 c σ σ h G v or any σ Gal(k s /k). In other words, c is in the image o γ. 6

7 Now suppose c H 1 (k, G) is in the image o γ. Without loss o generality, assume that c σ G v (k s ) or any σ Gal(k s /k). Pick any g GL(V )(k s ) as in (2) above and set w = gv V c(ks ). Then or any σ Gal(k s /k), we have σ w = gc σ v = gv = w. This shows that w V c (k). Hence there is a bijection between Gc (k)\v c (k) and ker γ c where γ c is the natural map o sets H 1 (k, G c w) H 1 (k, G c ). To prove Proposition 1, it suices to establish a bijection between γ 1 (c) and ker γ c. Consider the ollowing two maps: γ 1 (c) ker γ c ker γ c γ 1 (c) (σ d σ ) (σ d σ c 1 σ ) (σ a σ ) (σ a σ c σ ) We need to check that these maps are well-deined. First, suppose (σ d σ ) γ 1 (c). Then we need to show that (σ d σ c 1 σ ) is a 1-cocycle in the kernel o γ c. Note that, or any σ, τ Gal(k s /k), we have (d σ c 1 σ ) σ(d τ c 1 τ ) (d στ c 1 στ ) 1 = d σ c 1 σ (c σ σ d σ τ c 1 τ c 1 σ )(d στ c 1 στ ) 1 = 1. Moreover, there exists h G(k s ) such that d σ = h 1 c σ σ h or any σ Gal(k s /k), and thus h 1 σ(h) = h 1 c σ σ hc 1 σ = d σ c 1 σ. This shows that (σ d σ c 1 σ ) is in the kernel o γ c. Likewise, one can show that the second map is also well-deined. The composition o these two maps in either order yields the identity map, and this completes the proo. 2.4 A cohomological obstruction to liting invariants Suppose (V /G)(k) is a rational invariant. We continue to assume that the group G(k s ) acts transitively on the set V (k s ). In this section, we consider the problem o determining when the set V c(k) is nonempty or some c H1 (k, G). That is, when does a rational invariant lit to a rational orbit or some pure inner orm o G? We resolve this problem under the additional assumption that the stabilizer G v o any point in the orbit V (k s ) is abelian. For σ Gal(k s /k), the vector σ v also lies in V (k s ), so there is an element g σ with g σ σ v = v. The element g σ is well-deined up to let multiplication by an element in the subgroup G v. Since we are assuming that the stabilizers are abelian, the homomorphism θ σ : Gσ v G v deined by mapping α to g σ αgσ 1 is independent o the choice o g σ. This gives a collection o isomorphisms θ σ : σ (G v ) G v that satisy the 1-cocycle condition θ στ = θ σ σ θ τ, and hence provide descent data or the group scheme G v. We let G be the corresponding commutative group scheme over k which depends only on the rational invariant. Let ι v : G (k s ) G v denote the canonical isomorphisms. More precisely, i h G(k s ) and v V (k s ) then ι hv (b) = hι v (b)h 1 b G (k s ). (3) The descent data translates into saying or any σ Gal(k s /k) and v V (k s ), we have σ (ι v (b)) = ισ v( σ b) b G (k s ). (4) 7

8 Beore constructing a class in H 2 (k, G ) whose vanishing is intimately related to the existence o rational orbits, we give an alternate method (shown to us by Brian Conrad) to obtain the inite group scheme G over k using pp descent. Suppose G is a group scheme o inite type over k such that the orbit map G V V is pp. Suppose also that the stabilizer G v G(k a ) or any v V (k a ) is abelian where k a denotes an algebraic closure o k. Let H denote the stabilizer subscheme o G V. In other words, H is the pullback o the action map G V V V over the diagonal o V. Note that H is a V -scheme and its descent to k will be G. The descent datum amounts to a canonical isomorphism p 1H p 2H where p 1, p 2 denote the two projection maps V V V. The commutativity o G v or any v V (k a ) implies the commutativity o (G R ) x or any k-algebra R and any element x V (R). Thereore, there are canonical isomorphisms (G R ) x (G R ) y or any x, y V (R). This gives canonical isomorphisms p 1H p 2H locally over V V. Being canonical, these local isomorphisms patch together to a global isomorphism and hence yield the desired descent datum. We now construct a class d in H 2 (k, G ) that will be trivial whenever a rational orbit exists. Choose v and g σ as above, with g σ σ v = v. Deine d σ,τ = ι 1 v (g σ σ g τ gστ 1 ). Standard arguments show that d σ,τ is a 2-cocycle whose image d in H 2 (k, G ) does not depend on the choice o g σ. We also check that the 2-cochain d σ,τ does not depend on the choice o v V (k s ). Suppose v = hv V (k s ) or some h G(k s ). For any σ Gal(k s /k), we have Moreover, or any σ, τ Gal(k s /k), we compute hence, by (3), we have ι 1 v hg σ σ h 1σ v = hg σ σ v = hv = v. hg σ σ h 1 σ (hg τ τ h 1 ) (hg στ στ h 1 ) 1 = hg σ σ g τ g 1 στ h 1 ; (hg σ σ h 1 σ (hg τ τ h 1 ) (hg στ στ h 1 ) 1 ) = ι 1 (g σ σ g τ g 1 I V (k) is nonempty, then one can take v in V (k). Then one can take g σ = 1 and hence d = 0. We have thereore obtained the ollowing necessary condition or liting invariants to orbits. Proposition 2 Suppose that is a rational invariant, and that G(k s ) acts transitively on V (k s ) with abelian stabilizers. I V (k) is nonempty, then d = 0 in H 2 (k, G ). This necessary condition is not always suicient. As shown by the ollowing cocycle computation, the class d in H 2 (k, G ) does not depend on the pure inner orm o G. Indeed, suppose c H 1 (k, G) and g GL(V )(k s ) such that c σ = g 1σ g or all σ Gal(k s /k). Note that gv V c(ks ) and A direct computation then gives (g σ c 1 σ (gg σ c 1 σ g 1 ) σ(gv) = gv. ) σ(g τ c 1 τ ) (c στ gστ 1 ) = g σ σ g τ gστ 1. The act that d is independent o the pure inner orm suggests that d = 0 might be suicient or the existence o a rational orbit or some pure inner twist. Indeed, this is the case. 8 v στ ).

9 Theorem 3 Suppose that is a rational invariant, and that G(k s ) acts transitively on V (k s ) with abelian stabilizers. Then d = 0 in H 2 (k, G ) i and only i there exists a pure inner orm G c o G such that V c(k) is nonempty. That is, the condition d = 0 is necessary and suicient or the existence o rational orbits or some pure inner twist o G. In particular, when H 1 (k, G) = 1, the condition d = 0 in H 2 (k, G ) is necessary and suicient or the existence o rational orbits o G(k) on V (k). Proo: Necessity has been shown in Proposition 2 and the above computation. It remains to prove suiciency. Fix v V (k s ) and g σ such that g σ σ v = v or any σ Gal(k s /k). The idea o the proo is that i d = 0, then one can pick g σ so that (σ g σ ) is a 1-cocycle and that rational orbits exist or the pure inner twist associated to this 1-cocycle. Suppose d = 0 in H 2 (k, G ). Then there exists a 1-cochain (σ b σ ) with values in G (k s ) such that g σ σ g τ g 1 στ = ι v (b σ σ b τ b 1 στ ) σ, τ Gal(k s /k). Lemma 4 There exists a 1-cochain e σ with values in G v (k s ) such that (σ e σ g σ ) is a 1-cocycle. To see how Lemma 4 implies Theorem 3, we consider the twist o G and V using the 1- cocycle c = (σ e σ g σ ) H 1 (k, G). Choose any g GL(V )(k s ) such that g 1σ g = e σ g σ or any σ Gal(k s /k). Then gv V c (k). Indeed, σ (gv) = ge σ g σ σ v = ge σ v = gv σ Gal(k s /k). We now prove Lemma 4. Consider e σ = ι v (b 1 σ ) or any σ Gal(k s /k). Since g σ σ v = v, we have by (3) and (4) that Hence or any σ, τ Gal(k s /k), we have g σ σ (ι v (b))g 1 σ = ι v ( σ b) σ Gal(k s /k), b G (k s ). (e σ g σ ) σ (e τ g τ )(e στ g στ ) 1 = ι v (b 1 σ = ι v (b 1 σ = ι v (b 1 σ = 1 )g σ σ (ι v (b 1 τ )) σ g τ gστ 1 ι v (b στ ) )ι v ( σ b 1 τ )g σ σ g τ gστ 1 ι v (b στ ) )ι v ( σ b 1 )ι v (b σ σ b τ b 1 στ )ι v (b στ ) τ where the last equality ollows because G (k s ) is abelian. Corollary 5 Suppose that is a rational orbit and that G(k s ) acts simply transitively on V (k s ). Then there is a unique pure inner orm G c o G such that V c(k) is nonempty. Moreover, the group Gc (k) acts simply transitively on V c(k). Proo: Since G = 1, we have H 2 (k, G ) = 0 and so the cohomological obstruction d vanishes. We conclude that rational orbits exist or some pure inner twist G c. Let v 0 V c (k) denote any k-rational lit. Since H 1 (k, G ) = 0, the image o γ : H 1 (k, G c v 0 ) H 1 (k, G c ) is a single point, and hence no other pure inner twist has a rational orbit with invariant. Since the kernel o γ has cardinality 1, there is a single orbit o G c (k) on V c(k). 9

10 3 Examples with trivial stabilizer In this section, we give several examples o representations G GL(V ) over k where there are stable orbits which are determined by their invariants in V /G and which have trivial stabilizer over k s. Thus G(k s ) acts simply transitively on the set V (k s ). When is rational, Corollary 5 implies that there is a unique pure inner orm G o G over k or which V (k) is nonempty, and that G (k) acts simply transitively on V (k). We will describe this pure inner orm, using the ollowing results on classical groups [12]. Since H 1 (k, GL(W )) and H 1 (k, SL(W )) are both pointed sets with a single element, there are no nontrivial pure inner orms o GL(W ) and SL(W ). On the other hand, when the characteristic o k is not equal to 2 and W is a nondegenerate quadratic space over k, the pointed set H 1 (k, SO(W )) classiies the quadratic spaces W with dim(w ) = dim(w ) and disc(w ) = disc(w ). The corresponding pure inner orm is the group G = SO(W ). Similarly, i W is a nondegenerate Hermitian space over the separable quadratic extension E o k, then the pointed set H 1 (k, U(W )) classiies Hermitian spaces W over E with dim(w ) = dim(w ), and the corresponding pure inner orm o G is the group G = U(W ). 3.1 SO(n + 1) acting on n copies o the standard representation In this subsection, we assume that k is a ield o characteristic not equal to 2. We irst consider the action o the split group G = SO(W ) = SO(4) on three copies o the standard representation V = W W W. Let q(w) = w, w /2 be the quadratic orm on W and let v = (w 1, w 2, w 3 ) be a vector in V. The coeicients o the ternary quadratic orm (x, y, z) = q(xw 1 + yw 2 + zw 3 ) give six invariant polynomials o degree 2 on V, which reely generate the ring o polynomial invariants, and an orbit is stable i the discriminant () o this quadratic orm is nonzero in k s. In this case, the group G(k s ) acts simply transitively on V (k s ). Indeed, the quadratic space U 0 o dimension 3 with orm embeds isometrically into W over k s, and the subgroup o SO(W ) that ixes U 0 acts aithully on its orthogonal complement, which has dimension 1. The condition that the determinant o an element in SO(W ) is equal to 1 orces it to act trivially on the orthogonal complement. The set V (k) is nonempty i and only i the quadratic orm represents zero over k. Indeed, i v = (w 1, w 2, w 3 ) is a vector in this orbit over k, then the vectors w 1, w 2, w 3 are linearly independent and span a 3-dimensional subspace o W. This subspace must have a nontrivial intersection with a maximal isotropic subspace o W, which has dimension 2. Conversely, i the quadratic orm represents zero, let U 0 be the 3-dimensional quadratic space with this bilinear orm, and U the orthogonal direct sum o U 0 with a line spanned by a vector u with u, u = det(u 0 ). Then U is a quadratic space o dimension 4 and discriminant 1 containing an isotropic line (rom U 0 ). It is thereore split, and isomorphic over k to the quadratic space W. Choosing an isometry θ : U W, we obtain three vectors (w 1, w 2, w 3 ) as the images o the basis elements o U 0, and this gives the desired element in V (k). Note that θ is only well-deined up to composition by an automorphism o W, so we really obtain an orbit or the orthogonal group o W. Since the stabilizer o this orbit is a simple relection, we obtain a single orbit or the subgroup SO(W ). I the orm does not represent zero, let W be the quadratic space o dimension 4 that is the orthogonal direct sum o the subspace U 0 o dimension 3 with quadratic orm and a nondegenerate space o dimension 1, chosen so that the discriminant o W is equal to 1. Then G = SO(W ) is 10

11 the unique pure inner orm o G (guaranteed to exist by Corollary 5) where V (k) is nonempty. The construction o an orbit or G is the same as above. The same argument works or the action o the group G = SO(W ) = SO(n + 1) on n copies o the standard representation: V = W W W. The coeicients o the quadratic orm (x 1, x 2,..., x n ) = q(x 1 w 1 +x 2 w 2 + +x n w n ) give polynomial invariants o degree 2, which reely generate the ring o invariants. The orbit o v = (w 1, w 2,..., w n ) is stable, with trivial stabilizer, i and only i the discriminant () is nonzero in k s. I W is the quadratic space o dimension n + 1 with disc(w ) = disc(w ), that is the orthogonal direct sum o the space U 0 o dimension n with quadratic orm and a nondegenerate space o dimension 1, then G = SO(W ) is the unique pure inner orm with V (k) nonempty. 3.2 SL(5) acting on 3 copies o the representation 2 (5) Let k be a ield o characteristic not equal to 2, U a k-vector space o dimension 3, and W a k-vector space o dimension 5. In this subsection, we consider the action o G = SL(W ) on V = U 2 W. Choosing bases or U and W, we may identiy U(k) and W (k) with k 3 and k 5, respectively, and thus V (k) with 2 k 5 2 k 5 2 k 5. We may then represent elements o V (k) as a triple (A, B, C) o 5 5 skew-symmetric matrices with entries in k. For indeterminates x, y, and z, we see that the determinant o Ax + By + Cz vanishes, being a skew-symmetric matrix o odd dimension. To construct the G-invariants on V, we consider instead the 4 4 principal sub-paians o Ax+By +Cz; this yields ive ternary quadratic orms Q 1,..., Q 5 in x, y, and z, which are generically linearly independent over k. In basis-ree terms, we obtain a G-equivariant map U 2 W Sym 2 U W. (5) Now an SL(W )-orbit on Sym 2 U W may be viewed as a ive-dimensional subspace o Sym 2 U; hence we obtain a natural G-equivariant map Sym 2 U W Sym 2 U. (6) The composite map π : U 2 W Sym 2 U is thus also G-equivariant, but since G acts trivially on the image o π, we see that the image o π gives (a 6-dimensional space o) G-invariants, and indeed we may identiy V /G with Sym 2 U. A vector v V is stable precisely when det(π(v)) 0. Now since SL(W ) acts with trivial stabilizer on W, it ollows that SL(W ) acts with trivial stabilizer on Sym 2 U W too. Since the map (5) is G-equivariant, it ollows that the generic stabilizer in G(k) o an element in V (k) is also trivial! Since SL(W ) has no other pure inner orms, by Corollary 5 we conclude that every Sym 2 U o nonzero determinant arises as the set o G-invariants or a unique G(k)-orbit on V (k). 3.3 U(n 1) acting on the adjoint representation u(n) o U(n) In this subsection, we assume that the ield k does not have characteristic 2 and that E is an étale k-algebra o rank 2. Hence E is either a separable quadratic extension ield, or the split algebra k k. Let τ be the nontrivial involution o E that ixes k. Let Y be a ree E-module o rank n 2, and let, : Y Y E 11

12 be a nondegenerate Hermitian symmetric orm on Y. In particular y, z = τ z, y. Let e be a vector in Y with e, e 0, and let W be the orthogonal complement o e in Y. Hence Y = W Ee. The unitary group G = U(W ) = U(n 1) embeds as the subgroup o U(Y ) that ixes the vector e. In particular, it acts on the Lie algebra u(y ) = u(n) via the restriction o the adjoint representation. Deine the adjoint T o an E-linear map T : Y Y by the usual ormula T y, z = y, T z. The elements o the group U(Y ) are the maps g that satisy g = g 1. Dierentiating this identity, we see that the elements o the Lie algebra are those endomorphisms o Y that satisy T + T = 0. The group acts on the space o skew sel-adjoint operators by conjugation: T gt g 1 = gt g. I T is skew sel-adjoint and δ is an invertible element in E satisying δ τ = δ, then the scaled operator δt is sel-adjoint. Hence the adjoint representation o U(Y ) on its Lie algebra is isomorphic to its action by conjugation on the vector space V, o dimension n 2 over k, consisting o the sel-adjoint endomorphisms T : Y Y. In this subsection, we consider the restriction o this representation to the subgroup G = U(W ). The ring o polynomial invariants or G = U(W ) on V is a polynomial ring, reely generated by the n coeicients c i (T ) o the characteristic polynomial o T (which are invariants or the larger group U(Y )) as well as the n 1 inner products e, T j e or j = 1, 2,..., n 1 ([29, Lemma 3.1]). Note that all o these coeicients and inner products take values in k, as T is sel-adjoint. In particular, the space V /G is isomorphic to the aine space o dimension 2n 1. Note that the inner products T i e, T j e are all polynomial invariants or the action o G. Let D be the invariant polynomial that is the determinant o the n n symmetric matrix with entries T i e, T j e or 0 i, j n 1. Clearly D is nonzero i and only i the vectors {e, T e, T 2 e,..., T n 1 e} orm a basis or the space Y over E. Rallis and Shiman [20, Theorem 6.1] show that the condition D() 0 is equivalent to the condition that G(k s ) acts simply transitively on the points o V (k s ). We can thereore conclude that when D() is nonzero, there is a unique pure inner orm G o G = U(W ) that acts simply transitively on the corresponding points in V (k), and that these spaces are empty or all other pure inner orms. To determine the pure inner orm G = U(W ) or which V (k) is nonempty, it suices to determine the Hermitian space W over E o rank n 1. The rational invariant determines the inner products T i e, T j e, and hence a Hermitian structure on Y = Ee + E(T e) + + E(T n 1 e). Since the nonzero value e, e is ixed, this gives the Hermitian structure on its orthogonal complement W in Y, and hence the pure inner orm G such that V (k) is nonempty. When the algebra E is split, the Hermitian space Y = X + X decomposes as the direct sum o an n-dimensional vector space X over k and its dual. The group U(Y ) is isomorphic to GL(X) = GL(n). The vector e gives a nontrivial vector x in X as well as a nontrivial unctional in X with (x) 0. Let X 0 be the kernel o, so X = X 0 + kx. The subgroup U(W ) is isomorphic to GL(X 0 ) = GL(n 1). In this case, the representation o U(W ) on the space o sel-adjoint endomorphisms o Y is isomorphic to the representation o G = GL(n 1) by conjugation on the space V = End(X) o all k-linear endomorphisms o X. Since GL(n 1) has no pure inner orms, Corollary 5 implies that GL(n 1) acts simply transitively on the points o V (k) whenever D() 0. Once we have chosen an invertible element δ in E o trace zero, the rational invariants or the action o U(W ) = U(n 1) on the Lie algebra o U(n) match the rational invariants or the action o GL(X) = GL(n 1) on the Lie algebra o GL(n). Since the stable orbits or the pure inner orms U(W ) and GL(X) are determined by these rational invariants, we obtain a matching o orbits. This gives a natural explanation or the matching o orbits that plays an important role in the work o Jacquet and Rallis [13] on the relative trace ormula, where they establish a comparison o the corresponding orbital integrals, and in the more recent work o Wei Zhang [29] on the global conjecture o Gan, Gross, 12

13 and Prasad [11]. 4 Examples with nontrivial stabilizer and nontrivial obstruction In this section, we will provide some examples o representations with a nontrivial abelian stabilizer G, and calculate the obstruction class d in H 2 (k, G ). The irst example is a simple modiication o a case we have already considered, namely, the non-aithul representation V o Spin(W ) = Spin(n + 1) on n copies o the standard representation W o the special orthogonal group SO(W ). In this case, the stabilizer G o the stable orbits is the center µ 2. We will also describe the stable orbits or the groups G = SL(W ) and H = SL(W )/µ 2 acting on the representation V = Sym 2 W Sym 2 W. (The group H exists and acts when the dimension o W is even.) In these cases, the stabilizer G is a inite elementary abelian 2-group, related to the 2-torsion in the Jacobian o a hyperelliptic curve. 4.1 Spin(n + 1) acting on n copies o the standard representation o SO(n + 1) In this subsection, we reconsider the representation V = W n o SO(W ) studied in 3.1. There we saw that the orbits o vectors v = (w 1, w 2,..., w n ), where the quadratic orm = q(x 1 w 1 + x 2 w x n w n ) has nonzero discriminant, have trivial stabilizer. I we consider V as a representation o the two-old covering group G = Spin(W ), then these orbits have stabilizer G = µ 2. In the ormer case, we ound that the unique pure inner orm SO(W ) or which V (k) is nonempty corresponded to the quadratic space W o dimension n + 1 and disc(w ) = disc(w ) that is the orthogonal direct sum o the subspace U 0 with quadratic orm and a nondegenerate space o dimension 1. The group Spin(W ) will have orbits with invariant, but this group may not be a pure inner orm o the group G = Spin(W ). I it is not a pure inner orm, the invariant d must be non-trivial in H 2 (k, G ). Assume, or example, that the orthogonal space W is split and has odd dimension 2m + 1, so that the spin representation U o G = Spin(W ) o dimension 2 m is deined over k. Then a necessary and suicient condition or the group G = Spin(W ) to be a pure inner orm o G is that the even Cliord algebra C + (W ) o W is a matrix algebra over k. In this case, the spin representation U o G can also be deined over k. Hence the obstruction d is given by the Brauer class o the even Cliord algebra o the space W determined by. Note that the even Cliord algebra C + (W ) has an anti-involution, so its Brauer class has order 2 and lies in the group H 2 (k, G ) = H 2 (k, µ 2 ). 4.2 SL n acting on Sym 2 (n) Sym 2 (n) Let k be a ield o characteristic not equal to 2 and let W be a vector space o dimension n over k. Let e be a basis vector o the one-dimensional vector space n W. The group G = SL n acts linearly on W and trivially on n W. The action o G on the space Sym 2 W o symmetric bilinear orms v, w on W is given by the ormula g v, v = gv, gv This action preserves the discriminant o the bilinear orm A =,, which is deined by the ormula: disc(a) = ( 1) n(n 1)/2 e, e n. 13

14 Here, n is the induced symmetric bilinear orm on n (W ). I {w 1, w 2,..., w n } is any basis o W with w 1 w 2... w n = e, then e, e n = det( w i, w j ). The discriminant is a polynomial o degree n = dim(w ) on Sym 2 W which reely generates the ring o G-invariant polynomials. Now consider the action o G on the representation V = Sym 2 W Sym 2 W. I A =, A and B =, B are two symmetric bilinear orms on W, we deine the binary orm o degree n over k by the ormula (x, y) = disc(xa yb) = 0 x n + 1 x n 1 y + + n y n. The coeicients o this orm are each polynomial invariants o degree n on V, and the n+1 coeicients j reely generate the ring o polynomial invariants or G on V. (This will ollow rom our determination o the orbits o G over k s in Theorem 6.) We call (x, y) the invariant binary orm associated to (the orbit o) the vector v = (A, B). The discriminant () o the binary orm is deined by writing (x, y) = (α i x β i y) over the algebraic closure o k and setting () = i<j(α i β j α j β i ) 2. Then () is a homogeneous polynomial o degree 2n 2 in the coeicients j, so is a polynomial invariant o degree 2n(n 1) on V. For example, the binary quadratic orm ax 2 + bxy + cy 2 has discriminant = b 2 4ac and the binary cubic orm ax 3 + bx 2 y + cxy 2 + dy 3 has discriminant = b 2 c abcd 4ac 3 4b 3 d 27a 2 d 2. The irst result shows how the invariant orm and its discriminant determine the stable orbits or G on V over k s. Theorem 6 Let k s be a separable closure o k, and let (x, y) be a binary orm o degree n over k s with 0 0 and () 0. Then there are vectors (A, B) in V (k s ) with invariant orm (x, y), and these vectors all lie in a single orbit or G(k s ). This orbit is closed, and the stabilizer o any vector in the orbit is an elementary abelian 2-group o order 2 n 1. To begin the proo, we make a simple observation. Let A and B denote two symmetric bilinear orms on W over k s with disc(xa yb) = (x, y). Then both A and B give k s -linear maps W W. Our assumption that 0 is nonzero implies that the linear map A : W W is an isomorphism, so we obtain an endomorphism T = A 1 B : W W. The act that both A and B are symmetric with respect to transpose implies that T is sel-adjoint with respect to the bilinear orm, A on W. Write (x, 1) = 0 g(x) with g(x) monic o degree n. The characteristic polynomial det(xi T ) is equal to the monic polynomial g(x), and our assumption that the discriminant o (x, y) is nonzero in k implies that the polynomial g(x) is separable. Hence the endomorphism T o V is regular and semisimple. The group G(k s ) acts transitively on the bilinear orms with discriminant 0, and the stabilizer o A is the orthogonal group SO(W, A). Since the group SO(W, A)(k s ) acts transitively on the sel-adjoint operators T with a ixed separable characteristic polynomial g(x), there is a single G(k s )-orbit on the vectors (A, B) with invariant orm (x, y). The stabilizer is the centralizer o T in SO(W, A), which is an elementary abelian 2-group o order 2 n 1. For proos o these assertions, see [4, Prop. 4]. Having classiied the stable orbits o G on V over the separable closure, we now turn to the problem o classiying the orbits with a ixed invariant polynomial (x, y) over k. 14

15 Theorem 7 Let (x, y) = 0 x n + 1 x n 1 y + + n y n be a binary orm o degree n over k whose discriminant and leading coeicient 0 are both nonzero in k. Write (x, 1) = 0 g(x) and let L be the étale algebra k[x]/(g) o degree n over k. Then there is a canonical bijection (constructed below) between the set o orbits (A, B) o G(k) on V (k) having invariant binary orm (x, y) and the equivalence classes o pairs (α, t) with α L and t k, satisying 0 N(α) = t 2. The pair (α, t) is equivalent to the pair (α, t ) i there is an element c L with c 2 α = α and N(c)t = t. The group scheme G obtained by descending the stabilizers G A,B or (A, B) V (k s ) to k is the inite abelian group scheme (Res L/k µ 2 ) N=1 o order 2 n 1 over k. As a corollary, we see that the set o orbits with invariant orm (x, y) is nonempty i and only i the element 0 k lies in the subgroup N(L )k 2. In this case, we obtain a surjective map (by orgetting t) rom the set o orbits to the set (L /L 2 ) N 0, where the subscript indicates that the norm is congruent to 0 in the group k /k 2. This map is a bijection when there is an element c L that satisies c 2 = 1 and N(c) = 1. Such an element c will exist i and only i the polynomial g(x) has a monic actor o odd degree over k. I no such element c exists, then the two orbits (α, t) and (α, t) are distinct and map to the same class α in (L /L 2 ) N 0. In that case, the map is two-to-one. When n is odd, the set o orbits is always nonempty and has a natural base point (α, t) = ( 0, (n+1)/2 0 ). Using this base point, and the existence o an element c with c 2 = 1 and N(c) = 1, we can identiy the set o orbits with the group (L /L 2 ) N 1. When n is even, 0 may not lie in the subgroup N(L )k 2 o k. In this case, there may be no orbits over k with invariant polynomial (x, y). For example, when n = 2 there are no orbits over R with invariant orm (x, y) = x 2 y 2. Even when orbits exist, there is no natural base point and we can not identiy the orbits with the elements o a group. There is a close relation between the existence o an orbit with invariant (x, y) in the even case n = 2g +2 and the arithmetic o the smooth hyperelliptic curve C o genus g over k with equation z 2 = (x, y). For example, every k-rational point P = (u, 1, v) on C with v 0 (so P is not a Weierstrass point) gives rise to an orbit [3, 2]. Indeed, write (x, 1) = 0 g(x) and let θ be the image o x in the algebra L = k[x]/(g(x)). The orbit associated to P has α = u θ L and t = v k. Then N(α) = g(u), so t 2 = 0 N(α). This is the association used in [3] to show that most hyperelliptic curves over Q have no rational points. Proo o Theorem 7: Assume that we have a vector (A, B) in V (k) with disc(xa yb) = (x, y). Using the k-linear maps W W given by the bilinear orms A and B and the assumption that 0 is nonzero, we obtain an endomorphism T = A 1 B : W W which is sel-adjoint or the pairing, A and has characteristic polynomial g(x). Since () is nonzero, the polynomial g(x) is separable and W has the structure o a ree L = k[t ] = k[x]/(g) module o rank one. Let β denote the image o x in L, and let {1, β, β 2,, β n 1 } be the corresponding power basis o L over k. The k-bilinear orms A and B both arise as the traces o L-bilinear orms on the rank one L module W. Choose a basis vector m o W over L and consider the k-linear map L k deined by λ m, λm A. Since g(x) is separable, the element g (β) is a unit in L and the trace map rom L to k is nonzero. Hence there is a unique element κ in L such that m, λm A = Trace(κλ/g (β)) or all λ in L. Since all elements o L are sel-adjoint with respect to the orm, A, we ind that the ormula µm, λm A = Trace(κµλ/g (β)) 15

16 holds or all µ and λ in L. Since the discriminant 0 o the bilinear orm, A is nonzero in k, we conclude that κ is a unit in the algebra L, so is an element o the group L. We deine α = κ 1 L, so that µm, λm A = Trace(µλ/αg (β)). A amous ormula due to Euler [23, Ch III, 6] then shows that or all µ and λ in L, the value µm, λm A is the coeicient o β n 1 in the basis expansion o the product µλ/α. It ollows that the value µm, λm B is the coeicient o β n 1 in the basis expansion o the product βµλ/α. We deine the element t k by the ormula t(m βm β 2 m... β n 1 m) = e in the one-dimensional vector space n (W ). Then e, e n = t 2 det( β i m, β j m A ). Since e, e n = ( 1) n(n 1)/2 0 and det( β i m, β j m A ) = ( 1) n(n 1)/2 N(α) 1, we have that t 2 = 0 N(α). We have thereore associated to the binary orm (x, y) an étale algebra L, and to the vector (A, B) with discriminant (x, y) an element α L and an element t k satisying t 2 = 0 N(α). The deinition o α and t required the choice o a basis vector m or W over L. I we choose instead m = cm with c in L, then α = c 2 α and t = N(c)t. Hence the vector (A, B) only determines the equivalence class o the pair (α, t) as deined above. It is easy to see that every equivalence class (α, t) determines an orbit. Since the dimension n o L over k is equal to the dimension n o W, we can choose a linear isomorphism θ : L W that maps the element 1 β β 2... β n 1 in n (L) to the element t 1 e in n (V ). Every other isomorphism with this property has the orm hθ, where h is an element in the subgroup G = SL(W ). Using θ we deine two bilinear orms on W : θ(µ), θ(λ) A = Trace(µλ/(αg (β))) θ(µ), θ(λ) B = Trace(βµλ/(αg (β))). The G(k)-orbit o the vector (A, B) in V (k) is well-deined and has invariant polynomial (x, y). To complete the proo, we need to determine the stabilizer o a point (A, B) V (k s ) in an orbit with binary orm (x, y). Let L s = k s [x]/(g(x)) denote the k s -algebra o degree n. Since the bilinear orm, A is nondegenerate, the stabilizer o A in G is the special orthogonal group SO(W, A) o this orm. The stabilizer o B in the special orthogonal group SO(W, A) is the subgroup o those g that commute with the sel-adjoint transormation T. Since T is regular and semisimple, the centralizer o T in GL(W ) is the subgroup k s [T ] = L s, and the operators in L s are all sel-adjoint. Hence the intersection o L s with the special orthogonal group SO(W, A)(k s ) consists o those elements g that are simultaneously sel-adjoint and orthogonal, so consists o those elements g in L s with g 2 = 1 and N(g) = 1. The same argument works over any k s -algebra E. The elements in G(E) stabilizing (A, B) are the elements h in (E L s ) with h 2 = 1 and N(h) = 1. Hence the stabilizer G A,B is isomorphic to to the inite étale group scheme (Res L s /k sµ 2) N=1 over k s. To show that these group schemes descend to (Res L/k µ 2 ) N=1, it remains to construct isomorphisms ι v : (Res L/k µ 2 ) N=1 (k s ) G v compatible with the descent data or every v V (k s ), i.e., satisying (3) and (4). Let α 1,..., α n k s denote the roots o g(x). For any i = 1,..., n, deine h i (x) = g(x) x α i, g i (x) = 1 2 h i(x) h i (α i ). 16