aa r X i v : . [ m a t h . N T ] A p r Spinor class fields for sheaves of lattices
Luis Arenas-Carmona ∗ Universidad de Chile,Facultad de Ciencias,Casilla 653, Santiago,ChileE-mail: [email protected] 13, 2018
Abstract
We extend the theory of spinor class field and representation fieldsto any linear algebraic group over a global function field satisfyingsome technical conditions that ensure the existence of a suitable spinornorm. This is the analog of a result given by the author in the numberfield case. Spinor class fields can still be defined for lattices definedover a projective curve in a sheaf-theoretical context. Spinor genus isa rather weak invariant in this context, by it can be used to study thebehavior of the genus at affine subsets. Examples are provided.
Let n be a positive integer and let k be a number field with ring of integers O k . Every conjugacy class of maximal orders in the matrix algebra M n ( k )has a representative of the form D n ( I ) = ( I δ ,i − δ ,j ) i,j , where ( I i,j ) i,j is thelattice of matrices ( a i,j ) i,j satisfying a i,j ∈ I i,j for ideals I i,j ([9], p. 18), i.e., D ( I ) = (cid:18) O k II − O k (cid:19) , D ( I ) = O k I II − O k O k I − O k O k , . . . ∗ Supported by Fondecyt, proyecto No. 1085017. D n ( I ) and D n ( I ′ ) are conjugate if and only if J n I ′ I − isprincipal for some fractional ideal J of k ([9], p. 23). It follows that the setof conjugacy classes of maximal orders in M n ( k ) is in correspondence withthe quotient g / g n of the class group g of k . In [4] we extended this theory tothe case where k is a global function field and O k is an arbitrary Dedekinddomain whose field of quotients is k . We can even consider the projective caseif we replace the ring O k by the structure sheaf O X of a smooth projectivecurve X whose field of rational functions is k , and we interpret maximalorders and ideals in the preceding statements in a sheaf-theoretical setting.In that context, it can only be proved that every spinor genera of maximalorders can have a representative of the form D n ( I ). In fact, the existence ofnon-split maximal orders in n -th of all spinor genera is a consequence of theTheory of representation by spinor genera ( § X be a smooth projective curve over a finite field F . Let K ( X ) bethe field of rational functions on X , and let G be an algebraic subgroupof Aut K ( X ) V , for some K ( X )-vector space V . Let C be an arbitrary non-empty Zariski-open subset of X . We allow the case C = X . Let Λ and M be C -lattices on V . We say that Λ and M are:1. in the same G -class if g (Λ) = M for some g ∈ G ,2. in the same G -genus if for every point ℘ ∈ C there exists an element g ℘ ∈ G ℘ such that g ℘ (Λ ℘ ) = M ℘ , where Λ ℘ and M ℘ denote the com-pletions at ℘ ,3. in the same G -spinor genus if there exists a lattice P satisfying thefollowing conditions:(a) There exists an element h ∈ G such that h (Λ) = P ,(b) For every point ℘ ∈ C there exists an element g ℘ ∈ G ℘ with trivialspinor norm ( §
2) such that g ℘ ( P ℘ ) = M ℘ .Spinor genera and classes coincide whenever G is non-compact at some infi-nite point of C if any (Proposition 7). In § Theorem 1.
Let G be any semi-simple linear algebraic group G , defined overthe field of functions K = K ( X ) of a smooth projective curve X . Assume G atisfies the technical conditions SN and RU in § G -spinor genera in the G -genus of any C -lattice Λ is a principal homogeneousspace over the Galois group G of an Abelian extension Σ C Λ /K ( X ) called thespinor class field of the lattice Λ . This extension splits completely at everyinfinite point of C if any. If D ⊆ C is open and M is the restriction of λ to D as a sheaf (or equivalenlty the O X ( D ) -lattice generated by Λ ), then Σ C Λ /K ( X ) is the maximal subextension of Σ C Λ /K ( X ) splitting at every placein C \ D . In other words, we have a natural action of G on the set Ω of spinorgenera in the genus, and for every pair of lattices ( M, N ) there exists aunique element ρ ( M, N ) ∈ G taking the spinor genus of M to the spinorgenus of N . The map ρ satisfies the relation ρ ( M, P ) = ρ ( M, N ) ρ ( N, P ) forany three lattices M , N , and P in the genus. The class field Σ C Λ dependsonly on the genus of the lattice Λ. Example A . Assume q is odd. Let { C i } mi =1 be an affine cover of anirreducible smooth projective curve X over F q . For i = 1 , . . . , m , let Λ i bea C i -lattice in a fixed regular quadratic or skew-hermitian K ( X )-space W .Assume that the completions Λ i,℘ and Λ j,℘ coincide whenever ℘ ∈ C i ∩ C j .Then there exists a class field Σ such that, for every affine subset D of X , if M is the D -lattice satisfying M ℘ = Λ i,℘ for ℘ ∈ C i ∩ D , then the spinor class fieldΣ DM is the maximal subfield of Σ splitting completely at the infinite placesof D . We simply define Σ as the spinor class field of the lattice obtained bypasting together the lattices Λ i as sheaves.It is apparent that Σ ⊇ Σ C Λ · · · Σ C m Λ m , but equality does not need to hold.For example, assume X = P ( F q ) = C ∪ C with either C i affine. Let Λbe a free X -lattice with a basis v , . . . , v n for some n ≥
3, and let Λ i be therestriction of Λ to C i , for i = 1 ,
2. Let Q be the quadratic form defined by Q n X i =1 g i v i ! = n − X i =1 g i + g n − g n . Then every spinor genus has a representative of the form L ( I ) = n − ⊥ i =1 O C v i ⊥ ( Iv n − ⊕ I − v n ) , I .This shows that Σ C i Λ i = K ( X ) for either affine set C i ,but we claim that Σ must contain a quadratic extension. The claim followsby considering an open subset D isomorphic to the rational affine curve withequation y = P ( x, y ) where P is an irreducible quadratic form on F q [ x, y ]. Itsuffices to prove that the class group of the ring O X ( D ) has even order. Infact, the ideal I = ( x, y ) is not principal, but I = ( y ), as an easy computationshows.In fact, the spinor genus of the lattice L ( I ) can be computed from thedivisor class defining the ideal I . We show in § L ( O X ) has a representative of this type. We do thisby extending the theory of representation fields to the sheaf setting. Whenstrong approximation fails, as is always the case for X -lattices, representationfields give only information on the number of spinor genera representing agiven lattice. Just as in the number field case, representation fields mightfail to exists, but they do exist in many important families of examples. To fix ideas, let X be the irreducible smooth projective algebraic curve de-fined over a finite field F = F q , and let K ( X ) be its field of rational functions.We say that a semi-simple linear algebraic group G ⊆ GL (cid:16) n, K ( X ) (cid:17) , definedover K ( X ), satisfies condition SN if:1. The extension of the universal cover φ : ˜ G → G of G to any separablyclosed field E containing K ( X ) is surjective. For almost all points ℘ ∈ X , any integral element g of the completion G ℘ has a pre-image in e G Z for some unramified extension Z/K ℘ . Let F = ker φ be the fundamental group of a semi-simple group G satisfy-ing condition SN . The cohomology function θ : G → H (cid:16) K ( X ) , F E (cid:17) arisingfrom the short exact sequence F E ֒ → ˜ G E ։ G E , where E is the separableclosure of K ( X ), is called the spinor norm. It is also defined at any fieldcontaining K ( X ). The spinor norm on a completion K ℘ of K ( X ) is denoted θ ℘ . There exists also an adelic version of the spinor norm. It is the mapΘ : G A → Y ℘ ∈ X H ( K ℘ , F ) , Θ( g ) = (cid:16) θ ℘ ( g ℘ ) (cid:17) ℘ , G A ⊆ Q ℘ G ℘ is the adelization of the group G [12]. Lemma 1.
For any semi-simple group G satisfying condition SN the spinornorm is surjective over K ( X ) and over any localization K ℘ .Proof. Recall that H ( K, e G ) = { } for both, the global field K = K ( X ) andthe local field K = K ℘ [17]. Now the result follows by applying cohomologyto the short exact sequence F E ֒ → e G E ։ G E , where E is the separableclosure of K ( X ) or K ℘ .Let E be the separable closure of K ( X ). In what follows, we say that G satisfies condition RU if it satisfies the following conditions:1. The fundamental group F E of G E is cyclic, and its order n is notdivisible by the characteristic of F . In particular, it is isomorphic tothe group µ n of n -roots of unity in E . There exists an isomorphism between F E and µ n commuting with thenatural action of the Galois group Gal (cid:16)
E/K ( X ) (cid:17) on either group. Condition RU implies that H (cid:16) K ( X ) , F E (cid:17) ∼ = K ( X ) ∗ n /K ( X ) ∗ . Note that if E ( ℘ ) is the separable closure of K ℘ , then F E ( ℘ ) = F E , since K ℘ contains noinseparable extensions of K ( X ) ([18], § VIII.6.). It follows that H ( K ℘ , F E ) ∼ = K ∗ n℘ /K ∗ ℘ for any ℘ ∈ X . Lemma 2.
For any semi-simple group G satisfying conditions SN and RU the image of the adelic spinor norm Θ is contained in J X /J nX , where J X isthe idele group of K ( X ) .Proof. Observe that for any extension
E/K ℘ such that a given element g ∈ G K ℘ = G ℘ is in the image of the cover φ : e G E → G E , the spinor norm θ ℘ ( g )can be computed from the short exact sequence F E ֒ → e G E ։ φ ( e G E ), whenceit lies in H ( E/K ℘ , F E ) ∼ = ( E ∗ n ∩ K ∗ ℘ ) /K ∗ n℘ . The latter isomorphism is justthe coboundary map of the short exact sequence F E ֒ → E ∗ ։ E ∗ n .By condition SN every g ∈ G O ℘ is the image of some h ∈ e G E for someunramified extension E/K ℘ , for almost all points ℘ ∈ X . This implies θ ℘ ( g ) ∈ ( E ∗ n ∩ K ∗ ℘ ) /K ∗ n℘ ⊆ O ∗ ℘ K ∗ n℘ /K ∗ n℘ , whence θ ℘ ( G O ℘ ) ⊆ O ∗ ℘ K ∗ n℘ /K ∗ n℘ for almost all ℘ .5he automorphism groups of the structures mentioned in the introductionindeed satisfy these conditions. Lemma 3.
Orthogonal groups of regular quadratic forms and unitary groupsof regular quaternionic skew-hermitian forms satisfy conditions SN and RU if the characteristic of the base field F is not .Proof. The universal cover of the Orthogonal group of a quadratic space(
V, q ) over a field K whose characteristic is not 2 is the spin group Spin( q ).It is defined as the set of elements u in the Even Clifford Algebra C + ( q ) [11]satisfying uu = 1 and uV u − = V . Condition RU follows since any element u in the spin group satisfying uvu − = v for any v ∈ V is in the base field K ( X ) ([11], § uu = 1 implies u = ± E ⊇ K is separably closed, then every product of two symme-tries τ v τ w is the image of vw √ q ( v ) q ( w ) ∈ Spin( q ) E . This elements generate thespecial orthogonal group [11]. Furthermore, if K = K ℘ is a local field and q is a unimodular integral quadratic form at ℘ , the integral orthogonal groupis generated by products of 2 reflections τ v τ w where q ( v ) and q ( w ) are units([11], § vw √ q ( v ) q ( w ) is defined over an unramified extension.For unitary groups of quaternionic skew-hermitian forms the proof followsfrom the previous case, since any quaternion algebra splits on some separablequadratic extension of K ( X ) ([14], Thm. 7.15), which ramifies at only finitelymany places (Theorem 1 in § VIII.4 of [18]), and the unitary group of a skew-hermitian form on a split quaternion algebra is isomorphic to an orthogonalgroup ([7], Lemma 3).A similar result can be proved for the automorphism group of a centralsimple algebra of dimension n when n is not divisible by the characteristicof F . However, we have a stronger result: Lemma 4. If G is the automorphism group of a central simple algebra A ,the reduced norm map Θ = N : G A → J X /J nX satisfies the conclusions ofLemma 2 and Lemma 1 regardless of the characteristic.Proof. See ([18], § X.2, Prop 6) and ([18], § XI.3, Prop 3) for Lemma 1.Passing to a separable extension if needed we assume that A is isomorphicto a matrix algebra M t (cid:16) K ( X ) (cid:17) . Restricting to a smaller set of points ℘ ifneeded, we assume that the isomorphism maps the standard basis of the6atrix algebra to a basis of the lattice of integral elements in A . Then theintegral elements of G are just the automorphisms of M t ( K ℘ ) fixing M t ( O ℘ ),i.e. P GL ( n, O ℘ ). Any element g ∈ P GL ( n, O ℘ ) has reduced norm in O ∗ ℘ K ∗ n℘ and the conclusion of Lemma 2 follows.We define the spinor norm for the automorphism group of an algebra A asthe reduced norm as above. When char( F ) divides n , the map SL ( A E ) → Aut E A E fails to be surjective, so we cannot interpret the spinor norm asa co-boundary. However, the explicit construction of Θ is not used in theremaining of this work. X -lattices In this section we recall the properties of X -lattices that are used in thesequel. Let X be a smooth irreducible projective curve over a finite field F .Let K = K ( X ) be the field of rational functions on X , and for every place ℘ ∈ X we let K ℘ be the completion at ℘ of K . We let O ℘ be the ring ofintegers at ℘ , i.e., the completion of the ring of rational functions defined at ℘ . Let V be a vector space over K . A coherent system of lattices in X is afamily { Λ ℘ } ℘ ∈ X satisfying the following conditions ([18], Ch. VI, p.97):1. Every Λ ℘ is a O ℘ -lattice in V ℘ .2. There exists an affine set C ⊂ X and a lattice L over the ring O X ( C ),of rational functions defined everywhere in C , such that L ℘ = Λ ℘ forevery ℘ ∈ C .Let V A be the adelization of the space V , and let us identify V with a discretesubgroup of V A as in [18]. Then for any coherent system ˜Λ = { Λ ℘ } ℘ ∈ X , theproduct Λ A = Q ℘ ∈ X Λ ℘ is an open and compact subgroup of V A , and everyopen and compact O A -submodule of V A arises in this way ([18], § VI, Prop1). For every affine subset C of X we define Λ A ,C = Q ℘ ∈ X \ C V ℘ × Q ℘ ∈ C Λ ℘ .Then Λ( C ) = Λ A ,C ∩ V defines a sheaf Λ on X . We call a sheaf of thistype an X -lattice. Equivalently, an X -lattice is a locally free sub-sheaf of V ,where V is identified with the corresponding constant sheaf. Thus defined, X -lattices share some of the properties of usual lattices, namely: • An X -lattice Λ is completely determined by the coherent system { Λ ℘ } ℘ .7 A coherent system can be modified at a finite number of places todefine a new X -lattice. In particular, X -lattices can be defined bygluing together lattices defined over an affine cover. • The adelization GL A ( V ) of the general linear group GL( V ) of V actson the set of lattices by acting on the family of compact and open O A -submodules of V A . • If V = A is an algebra, an X -lattice D is an order (i.e., a sheaf oforders) if and only if every completion is an order. The same holds formaximal orders.For the proofs see [11] or [12] . For any linear algebraic group acting on thespace V , we have an induced action of the adelic group G A on the set of X -lattices in V . Two X -lattices are in the same G -genus if they are in the sameorbit under this action. Similarly, classes are characterized as G K ( V ) -orbitsand spinor genera as G K ( V ) ker(Θ)-orbits. It follows from our main theoremthat there exist only a finite number of spinor genera in a genus for any group G satisfying SN and RU . On the contrary, the number of classes in a genusis frequently infinite, as in the examples we show in § X -lattices are locally free sheaves over the structure sheaf of X , and therefore they are associated to vector bundles [5]. The assumptionmade here that an X -lattice is contained in the constant sheaf V is notrestrictive since for any locally free sheaf Λ on X the sheaf V = Λ ⊗ O X K is constant and a finite vector space over K . Furthermore, any isomorphismbetween two X -lattice in a space V can be extended to a linear map on V ,whence next result follows: Proposition 5.
The set of isomorphism classes of vector bundles of rank n over a curve X defined over a finite field is in correspondence with the set ofdouble cosets GL(
K, n ) \ GL( A K , n ) / GL( O A , n ) . It is not true in general that this set of double cosets is finite or thattheir elements are parameterized by their images under the reduced normas it is the case for lattices over affine subsets. This follows easily from theclassification results for vector bundles over arbitrary fields. See for example[5]. Although [12] assumes characteristic 0 throughout, this hypotheses is not used for theresults quoted here. X -lattice Λ in a space V is completely decomposable if Λ = L i J i v i ,where { v , . . . , v n } is a basis of the space V and J , . . . , J n are X -lattices in K ( X ). Note that every such lattice has the form J i = L B i , where L B ( C ) = n f ∈ K ( X ) (cid:12)(cid:12)(cid:12) div( f ) | C ≥ − B | C o , for some divisor B on X . Not every X -lattice is completely decomposable, asfollows from the corresponding result for vector bundles [5]. In next sectionwe need the following result: Lemma 6.
There is a correspondence between conjugacy classes of maximal X -orders in M n ( K ) and isomorphism classes of n -dimensional vector bundlesover X up to multiplication by invertible bundles.Proof. Since all maximal orders are locally conjugate at all places, any max-imal X -order on A has the form b D b − where b ∈ A A is a matrix with adeliccoefficients and D ∼ = M n ( O X ) is the sheave of matrices with regular coeffi-cients. We know that the adelization D A is the ring of all adelic matrices c satisfying c ( O nX ) = O nX . It follows that b D b − is the ring of all adelicmatrices c satisfying c Λ = Λ where Λ = b Λ = b ( O nX ). Since the stabilizerof a local order D ℘ is D ∗ ℘ K ∗ ℘ , it follows that two X -lattices Λ and Λ corre-sponds to the same maximal order, if and only if Λ = d Λ for some d in thegroup J X of ideles on X . The result follows since the idele d generates theinvertible bundle L − div( d ) .Note that split orders correspond to completely decomposable bundles.In particular, not all maximal orders are split. Proof of Theorem 1
The set of spinor genera in a genus is in one toone correspondence with the Abelian group G A /G Λ A G K ( V ) ker(Θ) . (1)If the group G satisfies condition RU , then H ( K ℘ , F ) = K ∗ ℘ /K ∗ n℘ . By Lemma2, the image of Θ is contained in J X /J nX , where J X is the idele group of K ( X ).Since the spinor norm is surjective in both K ( X ) and K ℘ (Lemma 1), thegroup (1) is isomorphic to J X /K ( X ) ∗ H (Λ), where H (Λ) is the pre-image in J X of the group Θ( G Λ A ) ⊆ J X /J nX . We let Σ C Λ be the class field associated tothe open subgroup K ( X ) ∗ H (Λ) ([18], § XIII.9). The set of G -spinor genera9n the G -genus of Λ is a principal homogeneous space, via Artin Map, for thegroup G = Gal (cid:16) Σ C Λ /K ( X ) (cid:17) . The element of G sending the spinor genus ofan X -lattice M to the spinor genus of a second X -lattice M ′ is defined by ρ ( M, M ′ ) = [ a, Σ Λ /K ( X )], where x [ x, Σ Λ /K ( X )] denotes the Artin map,and a is any element of J X satisfying Θ( g ) = aJ nX for some g ∈ G A such that M ′ = g ( M ). The las statement follows from the identity H ( L ) = H (Λ) Y ℘ ∈ C \ D θ ℘ ( G ℘ ) = H (Λ) Y ℘ ∈ C \ D K ∗ ℘ , which follows from condition SN and the surjectivity of θ ℘ . Example A (continued).
Let Λ be the free X -lattice with basis { v i } ni =1 inthe quadratic space described in the introduction. Then Σ Λ = L ( t ), where L is the only quadratic extension of F , by a straightforward local computation.To find a representative in every spinor genus we observe that the adelicorthogonal element g = a ( λ ) defined by g ℘ ( v i ) = v i for i = 1 , . . . , n − g ℘ ( v n − ) = λ ℘ v n − , and g ℘ ( v n ) = λ − ℘ v n has spinor norm λ = ( λ ℘ ) ℘ . Bowtake u ∈ J X , and let B = div( u ) be the corresponding divisor. To finda representative L u of the spinor genus corresponding to the class of u in J X /J X H (Λ) we set λ = u above, whence L u = a ( u )Λ = n − ⊥ i =1 O X v i ⊥ ( L B v n − ⊕ L − B v n ) . Note that L u = L ( B ) depends only on the divisor B of u . Proposition 7.
Assume that C is a (necessarily proper) open set such that G X \ C is non-compact. Then two C -lattices N and M in the same G -genusare in the same G -class if and only if ρ ( N, M ) is the trivial element in Gal (cid:16) Σ CN /K ( X ) (cid:17) .Proof. By the strong approximation theorem over function fields [13], theuniversal cover e G of G has the strong approximation property with respectto the set S = X \ C . Assume that ρ ( N, M ) is trivial. Then N and M arein the same spinor genus, i.e., there exist g ∈ G and h ∈ ker Θ such that gh ( N ) = M . Then any pre-image ˜ h of h can be arbitrarily approximatedby an element ˜ f in e G whose image f ∈ G approximates h . Since latticestabilizers in G A are open, the result follows.10ote that in example A , the group G has strong aproximation with re-spect two every non-empty finite subset of X . It follows that two lattices arein the same spinor genus if and only if they are conjugate over every affinesubset of X . Corollary 7.1.
Let D and D ′ be two maximal C -orders in the central simplealgebra A that is not totally ramified at one or more infinite places of C .Then D and D ′ are conjugate if and only if ρ ( D , D ′ ) = id . Corollary 7.2.
Assume char( F ) = 2 . Let N and M be two C -lattices in thequadratic space W of dimension at least (or a skew-hermitian space of rankat least 2) that belong to the same genus. Assume that W is isotropic at oneor more infinite places of C . Then N and M are in the same G -class if andonly if ρ ( N, M ) = id . Next corollary follows now from ([11], p.170) and ([16], p.363).
Corollary 7.3.
Let N and M be two C -lattices in the quadratic space W ofdimension at least (or the skew-hermitian space W of rank at least 4) thatbelong to the same genus. If C is any proper open subset of X , then N and M are in the same G -class if and only if ρ ( N, M ) = id . Recall that the space λ ( X ) of global sections of a lattice λ is a finitedimensional vector space over finite field F = O X ( X ) ([18], Chapter VI).Furthermore, for any n -linear map τ : V n → W satisfying τ ( λ n ) ⊆ M ,where λ is a lattice in V and M is a lattice in the K ( X )-vector space W ,there exists an induced map ˜( τ ) : λ ( X ) n → M ( X ) that is n -linear over F . Inparticular:1. If λ is an order, then λ ( X ) is a F -algebra.2. If λ is an integral quadratic lattice, then λ ( X ) is, naturally, a quadraticspace over F .This observation is used throughout next section. Example: Maximal orders in M ( K ) . If n = 2 every split maximalorder has the form D B = (cid:18) O X L B L − B O X (cid:19) , B is a divisor of X defined over F . If B is a principal divisor, the ringof global sections D B ( X ) is isomorphic to the matrix algebra M ( F ). If B isnot principal, then L B and L − B cannot have a global section simultaneously.In fact, if div( f ) + B ≥ g ) − B ≥ B = div( g ) = div( f − ).We conclude that either D B ( X ) ∼ = F × F or D B ( X ) ∼ = ( F × F ) ⊕ V , where V is an ideal of nilpotency degree 2. Note that the dimension of V tends to ∞ with the degree of B by Riemann Roch’s Theorem, whence we concludethat there exist infinite many conjugacy classes of maximal X -orders in A .In fact, we can give a more precise result: Proposition 8.
The maximal orders D B and D D defined in the previousexample are conjugate if and only if B is linearly equivalent to either D or − D .Proof. If B is principal, and if D B is conjugate to D D , we must have D D ( X ) ∼ = M ( F ), and therefore D is principal. We can assume therefore that neither B nor D is principal. Replacing B or D by − B or − D if needed, we mayassume B, D ≥
0. Let U be a global matrix such that D B = U D D U − .From the explicit description of D B ( X ) given earlier, we conclude that the F -vector spaces L B ( X ) and L D ( X ) have the same dimension. Furthermore,if W B and W D denote the K -vector spaces spanned by D B ( X ) and D D ( X )respectively, then W B = U W D U − . There are two cases two be considered:1. If L B ( X ) = { } , then W B = W D = KE , ⊕ KE , ⊕ KE , .2. If L B ( X ) = { } , then W B = W D = KE , ⊕ KE , .In the first case, we conclude that U has the form (cid:18) a b c (cid:19) . In particular wemust have L − B E , = E , D B E , = E , ( U D D U − ) E , = a − c L − D E , . We conclude that B = D + div( ac − ), and therefore B and D are linearlyequivalent. In the second case U has either the form (cid:18) a c (cid:19) , which is similarto the previous case, or the form (cid:18) ac (cid:19) , so that B = − D + div( ac − ), and B is linearly equivalent to − D . 12 Representation fields
Let Λ be an X -lattice in a K ( X )-vector space V as before, and let M bean X -lattice in a subspace W ⊆ V . We allow the case V = W . Assume M ⊆ Λ in all that follows. Assume that G is a semi-simple linear algebraicsub-group of GL( V ) satisfying the conditions SN and RU . Following thenotations in [1] we call an element u ∈ G A a generator for Λ | M if M ⊆ u Λ.Local generators are defined analogously. Note that u ∈ G A is a generatorif and only if u ℘ is a local generator for every place ℘ . As usual we saythat M is G -represented by an X -lattice N in V , or that N G -represents M , if N contains a lattice in the the G -orbit of M . We say that a set Ψof lattices G -represents M if some element of Ψ does. In this setting, wehave the following proposition, whose proof is transliteration of the one inthe number field case [1], and therefore is omited. Proposition 9.
In the above notations, let Λ ′ be an X -lattice in the G -genusof Λ . The lattice M is G -represented by the spinor genus of Λ ′ if and onlyif there exists a generator u for Λ | M such that ρ (Λ , Λ ′ ) = [Θ( u ) , Σ Λ /K ( X )] . We denote by H (Λ | M ) the pre-image in J X of the set of spinor norms Θ( u )of all generators u for Λ | M . If K ( X ) ∗ H (Λ | M ) is a group, the correspondingclass field is called the representation field F (Λ | M ) for Λ | M . The spinorgenus of a lattice Λ ′ represents M if and only if ρ (Λ , Λ ′ ) is trivial on F (Λ | M ).The proof of the following fact is also completely analogous to the numberfield case ([10] and [3]): Proposition 10.
The representation field always exist for lattices in quadraticor quaternionic skew-hermitian spaces.
The corresponding result for orders in quaternion algebras follows fromthe case of quadratic forms just as in the number field case, but it cannot beextended to algebras of higher dimension [4].It must be kept in mind, however, that the classification of the latticesin a genus into spinor genera is a much coarser invariant than in the numberfield case, as the number of classes in a genus is usually infinite. The followingexample illustrate this: 13 xample B (continued).
Let X and Λ be as in §
4, but assume n = 4and the field of definition of X is F = F q with q = 4 t + 3. In particular, thespace of global sections of L ( B ) isΛ( X ) = O X ( X ) v + O X ( X ) v + L B ( X ) v + L − B ( X ) v . By Riemann-Rochs Theorem, the F -dimension of Λ( X ) tends to infinity withthe degree of B . In particular there exists infinitely many classes of suchlattices, while they belong to the same spinor genus as long as deg( B ) iseven. We claim that none of these lattices represents M = O X v + O X v when B is not principal. In fact, if B fails to be principal, then either L B ( X ) or L − B ( X ) has dimension 0, whence the space of global sectionsΛ( X ) = [ O X ( X ) v + O X ( X ) v ] ⊥ U where U = L B ( X ) v + L − B ( X ) v isthe radical, and Z = O X ( X ) v + O X ( X ) v is anisotropic by the choice of q , while M ( X ) is a hyperbolic plane. We note however that the theory ofrepresentation by spinor genera tell us that half of the spinor genera in thegenus of Λ must represent M . It is not hard to see that the image of thespinor norm in this case is H (Λ) = J nX O ∗ A . It follows that there are morethan one spinor genus representing M whenever the torsion subgroup of thePicard group of X has even order. In this case there must exist classes inthe genus of Λ that are not in the class of any of the lattices L ( B ).Recall that an order of maximal rank in a central simple algebra A K ( V ) issaid to be split if it represents the n-fold cartesian product O X × · · · × O X .A maximal X -order D in M n ( K ) is split if and only if the correspondingvector bundle is a direct product of one dimensional vector bundles, i.e., itcorresponds to an X -lattice of the typeΛ = L B × · · · × L B n . Note that if the order of diagonal matrices L i O X E i,i is contained in D ,every diagonal matric unit E i,i is a global section of D , an therefore D = P i,j J i,j E i,j for some invertible bundle J i,j ⊆ K , and the same holds for therings of global sections. A simple computation shows J i,j ∼ = L B i − B j . Anargument similar to that in the previous example can be used to prove nextresult: Proposition 11. If N is the total number of spinor genera of maximal X -orders in the matrix algebra M n ( K ) , where K is the field of functions ona smooth projective curve X over a finite field, then at least Nn − spinorgenera contain non-split X -orders. roof. One particular example of split order is the maximal order D B corre-sponding to the X -lattice L B × O X × · · · × O X . We claim that1. Every spinor genera of maximal X -orders contains the order D B forsome divisor B .2. The maximal orders D B and D D are in the same spinor genus if andonly if B and D + nC are linearly equivalent for some divisor C .The stabilizer of the local maximal X -order D ℘ is K ∗ ℘ D ∗ ℘ and its set ofnorms is K ∗ n℘ O ∗ ℘ . We conclude that H ( D ) = J nK O ∗ A . It follows that theclass group J K /K ∗ H ( D ) is isomorphic to the divisor group of X modulo n -powers. Observe that D D = u D B u − where u = diag( b, , . . . , b = n ( u ) ∈ J K satisfies div( b ) = D − B . It follows that D D and D B are in the same spinor genus if and only if D − B is 0 modulo n -powers inthe divisor group of X . Furthermore, any spinor genera can be obtained inthis way for an apropiate choice of b in J X /K ( X ) ∗ H ( D ). In particular, everyspinor genus contains a split order.It follows from the previous argumennt that the class modulo n of thedivisor B depends only on the spinor genera of the maximal order D B . Inparticular, the degree of B is well defined for a particular spinor genus as anelement of Z /n Z . We use this in all that follows.Let L be the only field extension of the finite field F of degree n . Weclaim that, if L embeds into D ( X ) for a split order D , then D ∼ = M n ( O X ).In fact, let Λ = L B × · · · × L B n be the lattice corresponding to D . Then D = ( L B i − B j ) i,j . We define an order in the group of divisor classes by D (cid:22) C if D ≤ C + div( f ) for some f ∈ K . Note that L C − D has a non-trivial globalsection if and only if D (cid:22) C . We assume that the B i ’s have been re-arrangedin a way that B n is minimal with respect to this order, and B r +1 , . . . , B n areall the divisors that are linearly equivalent to B n . Then any global sectionof D has the form (cid:18) A B C (cid:19) where A is an r -times- r block. It follows that K L has a representation ofdimension r < n over K , and therefore r = 0.Now, the proposition follows if we prove that for any divisor B such thatdeg B ≡ n ), there exists a maximal X -order D in the same spinorgenus as D B for which there is an embedding L ֒ → D ( X ), since we knowthat D cannot be a split order unless B is principal.15o prove this we let L = K L = L ⊗ F K , and let H = L ⊗ F O X be the onlymaximal order in the K -algebra L . Note that if Y is the projective curveover L defined by the same equations defining X over F , and φ : Y → X isthe natural morphism of schemes, then H is the push-forward to X of thestructure sheaf on Y . In particular H ( X ) = L .Consider the natural embedding φ : H = L ⊗ F O X ֒ → M ( F ) ⊗ F O X = D induced by an arbitrary embedding L ֒ → M ( F ). Then the order H ′ = φ ( H )is contained in some maximal order in the spinor genera of D B if and onlyif we can write D B = u D u − where the image of the reduced norm n ( u )in the quotient J K /K ∗ H ( D ) coincide with the image of a generator. Notethat if we identify L with the sub-algebra of A spanned by H ′ , the group ofinvertible elements L ∗ A (all of which are generators for D | H ′ ) is isomorphicto the group of ideles J L of L , and the reduced norm n : J L → J K is justthe field norm n L/K . It follows from Theorem 7 in chapter XIII of [18], that H L = K ∗ n L/K ( J L ) is the kernel of the Artin map t [ t, L/K ] on ideles.In particular H L has index at most n in J K , and we can check that thedivisor of every idele in H L has degree in n Z by computing the degrees ofthe generators. We conclude that H L is the group of all ideles whose divisorshave degrees in n Z , whence the result follows. Remark 1.
Since for every curve Pic( X ) ∼ = Z × T where T ∼ = Pic ( X ) isa finite group ([18], § IV.4, Theorem 7), we conclude that J K /K ∗ H ( D ) ∼ =( Z /n Z ) × ( T /nT ). In particular, the bound in the proposition is | T /nT | − Remark 2.
Assume for simplicity that K has odd characteristic and n = 2.Let B = div( b ) be a divisor of even degree, with b ∈ J K . An order D inthe same spinor genera as D B , representing the maximal order of L , is givenas follows: Tchebotarev Density Theorem ([15], Thm. 9.13A) implies theexistence of a place ℘ ∈ X , such that any idele j , where j ℘ is a uniformizerof K ℘ , and j q = 1 if q = ℘ , satisfies bj − ∈ K ∗ H ( D ). Note that ℘ haseven degree, whence the field L embeds into K ℘ . We may assume L = F ( u ),where u is a root of x = δ for some δ ∈ F . Then L K embeds into M ( K ) bysending u to the matrix (cid:18) δ (cid:19) . Let P be the divisor corresponding to ℘ .Then we may choose D = b D b − where b q is the identity matrix for q = ℘ and b ℘ = A (cid:18) j ℘
00 1 (cid:19) A − , where A (cid:18) u − u (cid:19) A − = (cid:18) δ (cid:19) . b = j . Example 1.
When X = P is the projective plane, the lower bound givenby this result is 0, so in principle there could be no non-split X -orders in M n ( K ). In fact, the Birkhoff-Grothendieck Theorem implies that everysuch X -lattice is a sum of X -lattices of rank 1. It follows that non-splitorders fail to exist and the bound is sharp in this case. Example 2.
Consider the plane curve X of genus 1 with projective equation y z − x ( x − z ) = 0 over a finite field of odd characteristic. Then div( x ) =2( P − P ∞ ), while there is no element in K = K ( X ) whose divisor is P − P ∞ ,or such element would generate the field K . We conclude that Pic ( X ) has anelement of order 2, and therefore its order is even. We conclude the existenceof at least one class of non-split maximal orders in M ( K ), or equivalently,a non-split vector bundle defined over F . References [1]
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