Compositio Math. 140 (2004) 287–300 DOI: 10.1112/S0010437X03000484 On representations of spinor genera Wai Kiu Chan and Fei Xu Abstract We determine exactly when a quadratic form is represented by a spinor genus of another quadratic form of three or four variables. We apply this to extend the embedding theorem for quaternion and also answer a question by Borovoi. 1. Introduction It is a basic question to decide when a quadratic form is represented by the spinor genus of another quadratic form. Some effort has been made in [EH82]and[HSX98]. In this paper, we make some progress on this subject. As an application, we show that the result in [EH82] is a special case of our investigation. There are two other motivations for this paper. First of all, we explain that the embedding theorem for quaternion algebras proved in [CF99] is a consequence of the representation theory of ternary quadratic forms and hence, in principle, the main results in [CF99] can be extended to arbitrary orders. To illustrate our point, we generalize Theorem 3.3 in [CF99] to Eichler orders. The second motivation is to explain how the results in [Xu01] are related to those in [BR95], and answer a question raised in [Bor01] concerning the representation mass of an integer by indefinite ternary quadratic forms over Z. The notation and terminology are standard if not explained, or adopted from [Xu01], [Ome73] and [HSX98]. Let V be a quadratic space over a number field F with a non-degenerate symmetric bilinear form B(x, y). The quadratic map on V is denoted by Q and its special orthogonal group by SO(V ). Let oF be the ring of integers of F . For any prime p of F , Vp (respectively Fp,etc.) denotes the local completion of V (respectively F ,etc.).Ifp is a finite prime, the group of units of o u ∈ × Fp is denoted by p,andπp is a uniformizer of Fp. For any two elements a, b Fp ,(a, b)p is the Hilbert symbol. Let SOA(V )bethead`elic group of SO(V ), θA = p∈Ω θp be the ad`elic spinor norm map of SOA(V ), and IF (respectively AF ) be the group of id`eles (respectively ad`eles) of F . For an oF -lattice L on V ,gen(L) (respectively spn(L)andcls(L)) is defined as the orbit of L under the natural action of SOA(V ) (respectively SO(V )kerθA and SO(V )), and SOA(L) is the stabilizer of L under the action of SOA(V ). We also use n(L), s(L)andv(L) to denote scale, norm and volume of L, respectively. Throughout this paper, all scales of lattices are integral. For two oF -modules L1 ⊆ L2 of the same rank, [L2 : L1] is the module index ideal. For any two lattices K and L in V, define XA(L/K)={σA ∈ SOA(V ):K ⊆ σAL} and X(Lp/Kp)={σ ∈ SO(Vp):Kp ⊆ σLp}. It is clear that XA(L/K) (respectively X(Lp/Kp)) is non-empty if and only if K (respectively Kp) is represented by gen(L) (respectively Lp). Received 23 November 2001, accepted in final form 26 June 2003. 2000 Mathematics Subject Classification 11Exx (primary); 11Fxx, 11Gxx (secondary). Keywords: spinor genera, representation mass, Hardy–Littlewood varieties. This journal is c Foundation Compositio Mathematica 2004. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 01 Oct 2021 at 14:07:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X03000484 W. K. Chan and F. Xu 2. Representation by spinor genera I, codimension less than or equal to one Suppose K is represented by gen(L). Then there exists σA ∈ SOA(V ) such that K ⊆ σAL. By the reduction formulae for computing the relative spinor norms in [HSX98]and[Xu99], θA(X(σAL/K)) depends only on the Jordan splittings of Kp and Lp for all p < ∞ and is inde- pendent of the choice of σA. Definition 2.1. θA(gen(L):K) is defined as θA(X(σAL/K)). By [HSX98]or[Xu99], K is represented by spn(L) if and only if × θA(σA) ∈ F θA(gen(L):K). (2.2) We extended some results in [HSX98] to the representation setting for low-dimensional cases. r Proposition 2.3. Suppose Kp and Lp are unimodular and [Kp : Kp ∩Lp]=p for some non-dyadic prime p.Furthermore: 1) rank(L)=rank(K);or 2) rank(K)+1=rank(L) 4. Then πp ∈ θp(X(Lp/Kp)) if and only if r is odd. Proof. 1) The argument is already in [HSX98]. 2) We separate the discussion according to the rank of L. We present the proofs for the quaternary and ternary cases in below, and leave that for the binary case to the readers. −s 2i) rank(Lp)=3and rank(Kp)=2.WriteB(K, L)oFp = p .Thens 0. When s =0,Kp splits Lp and the proposition is obvious. Therefore, we assume that s>0. −s Let K = oFp x ⊥ oFp y and B(x, L)oFp = p .Write Lp =(oFp e + oFp f) ⊥ oFp g, Q(e)=Q(f)=0,B(e, f)=1 and Q(g) ∈ up. Let a, b,andc be in Fp so that x = ae+bf +cg.Thenmin{ord(a), ord(b), ord(c)} = −s.Iford(c) < 0, then ord(a)+ord(b)=2ord(c). This implies that min{ord(a), ord(b)} =min{ord(a), ord(b), ord(c)} = −s. −s Without loss of generality, we can assume that ord(a)=−s.Thenx = πp (αe + βf + γg), where s α ∈ up and β and γ are in oFp . Therefore, oFp πpx + oFp f splits Lp and s Lp =(oFp πpx + oFp f) ⊥ oFp h for some h ∈ Lp. If y = ξx + ηf + δh,thenξQ(x)+ηαπ−s =0andy = ξ(x − α−1πsQ(x)f)+δh.Since −s −s 2 2 απp ξ = ξB(x, f)=B(y,f) ∈ p and δ Q(h) − ξ Q(x)=Q(y) ∈ up, both ξ and δ are in oFp , and at least one of ξ and δ is a unit. As a result, s Lp ∩ Kp = p x + oFp (y − ξx), and hence r = s. Define σ ∈ SO(Vp)by r 1 −1 r 1 −1 r r σ : πp(x − 2 α Q(x)πpf) → x − 2 α Q(x)πpf, f → πpf, h → h. It is clear that o r o − 1 −1 r ⊥ o σ(Lp)=( Fp πpf + Fp (x 2 α Q(x)πpf)) Fp h r =(oFp x + p f) ⊥ oFp h ⊃ Kp. 288 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 01 Oct 2021 at 14:07:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X03000484 Representations of spinor genera r × 2 and X(Lp/Kp)=σSO(Lp). Since θp(σ) ∈ πpup(Fp ) , we conclude that πp ∈ θp(X(Lp/Kp)) if and only if r is odd. 2ii) rank(Lp)=4and rank(Kp) = 3. Without loss of generality, we can assume that Kp = −s (oFp x + oFp y) ⊥ oFp z,whereQ(x)=Q(y)=0,B(x, y)=1andB(Kp,Lp)oFp = B(x, Lp)oFp = p s with s>0. It is clear that Q(z) ∈ up. The isotropic vector p x is primitive in Lp. Therefore, there are f,u,v ∈ Lp such that s −s Lp =(p x + oFp f) ⊥ (oFp u + oFp v)withQ(f)=0andB(x, f)=πp . If y = ax + bf + cu + dv,then −s −s −s aπp = aB(x, f)=B(y,f) ∈ p and 1 = B(x, y)=bB(x, f)=bπp . s These imply that a ∈ oFp and b = πp. Similarly, we have z = αx + βu + γv with α ∈ oFp . When Fpu + Fpv is anisotropic, then oFp u + oFp v = {x ∈ Fpu + Fpv : Q(x) ∈ oFp }. Since s Q(cu + dv)=−Q(ax + πpf) ∈ oFp and Q(βu + γv)=Q(z) ∈ up, therefore c, d, β,andγ are all in oFp and s s Kp ∩ Lp =(p x + oFp (πpf + cu + dv)) ⊥ oFp z. s −s s It is clear that [Kp : Kp ∩ Lp]=p . Define σ ∈ SO(Vp)byσ : x → πp x, f → πpf,u → u, v → v. s × 2 Then σ ∈ X(Lp/Kp)=σSO(Lp) and we are done since θp(σ) ∈ πpup(Fp ) . When oFp u + oFp v is a hyperbolic plane, we can assume that Q(u)=Q(v)=0andB(u, v)=1. Then −1 cd = −a ∈ oFp ,βγ=2 Q(z) ∈ up,βd+ γc = −α ∈ oFp . Without loss of generality, we assume that ord(β) 0, and hence ord(γ) 0 and ord(d) 0. If ord(c) ord(β), there is ξ ∈ oFp such that ξc + β =0.Then s s Kp = oFp x + oFp (πpf + cu + dv)+oFp (πpξf +(dξ + γ)v) and s −ord(c) s s Kp ∩ Lp = p x + p (πpf + cu + dv)+oFp (ξπpf +(ξd + γ)v). s−ord(c) Therefore, [Kp : Kp ∩ Lp]=p .Letσ ∈ SO(Vp) be defined by −s s ord(c) −ord(c) σ : x → πp x, f → πpf,u → πp u, v → πp v. Since dξ + γ =2dξ +(γ − dξ)=−2aξc−1 − αc−1 ∈ p−ord(c), s−ord(c) we see that Kp ⊂ Lp and X(Lp/Kp)=σSO(Lp). The lemma now follows because θ(σ) ∈ πp up × 2 (Fp ) . If ord(β) < ord(c), there is η ∈ oFp such that c + ηβ =0.Then s Kp = oFp x + oFp (πpf +(d + ηγ)v)+oFp (βu + γv) and s s −ord(β) Kp ∩ Lp = p x + oFp (πpf +(d + γη)v)+p (βu + γv).