The local-global principle for symmetric determinantal representations of smooth plane curves
aa r X i v : . [ m a t h . N T ] J a n THE LOCAL-GLOBAL PRINCIPLE FOR SYMMETRICDETERMINANTAL REPRESENTATIONS OF SMOOTH PLANECURVES
YASUHIRO ISHITSUKA AND TETSUSHI ITO
Abstract.
A smooth plane curve is said to admit a symmetric determinantal repre-sentation if it can be defined by the determinant of a symmetric matrix with entriesin linear forms in three variables. We study the local-global principle for the ex-istence of symmetric determinantal representations of smooth plane curves over aglobal field of characteristic different from two. When the degree of the plane curveis less than or equal to three, we relate the problem of finding symmetric determi-nantal representations to more familiar Diophantine problems on the Severi-Brauervarieties and mod 2 Galois representations, and prove that the local-global principleholds for conics and cubics. We also construct counterexamples to the local-globalprinciple for quartics using the results of Mumford, Harris, and Shioda on thetacharacteristics. Introduction
Let C ⊂ P K be a smooth plane curve of degree n ≥ K . If there is atriple of symmetric matrices ( M , M , M ) of size n with entries in K such that C isdefined by the equation det (cid:0) X M + X M + X M (cid:1) = 0 , we say C admits a symmetric determinantal representation over K . In this paper, westudy the local-global principle for the existence of symmetric determinantal represen-tations of smooth plane curves over a global field of characteristic different from two.We prove that the local-global principle holds for smooth plane conics and smoothplane cubics. We also construct counterexamples to the local-global principle forsmooth plane quartics using the results of Mumford, Harris, and Shioda on thetacharacteristics.The following theorems are the main results of this paper. Theorem 1.1 (see Theorem 5.1) . Let K be a global field of characteristic differentfrom two, and C ⊂ P K be a smooth plane conic or cubic (i.e., smooth plane curveof degree n = 2 or 3) over K . If C admits a symmetric determinantal representationover the completion K v for each place v of K , the smooth plane curve C admits asymmetric determinantal representation over K . Theorem 1.2 (see Subsection 6.5) . Let K be a global field of characteristic differentfrom two. Let C ⊂ P K be a smooth plane quartic (i.e., smooth plane curve of degree Date : June 7, 2018.2010
Mathematics Subject Classification.
Primary 14H50; Secondary 11D41, 14F22, 14G17,14K15, 14K30.
Key words and phrases. plane curve, determinantal representation, local-global principle, thetacharacteristic.
4) such that the associated mod 2 Galois representation on the 2-torsion points on theJacobian variety Jac( C ) ρ C, : Gal( K sep /K ) −→ Sp (cid:0) Jac( C )[2]( K sep ) (cid:1) ∼ = Sp ( F )is surjective. Then there is a finite extension L/K such that C admits a symmetricdeterminantal representation over L w for each place w of L , and C does not admit asymmetric determinantal representation over L .By the results of Harris, and Shioda, there are smooth plane quartics with surjectiveassociated mod 2 Galois representations over Q and over F p ( T ) for p / ∈ { , , , , , , } ([17, p. 721], [34, Theorem 7]). Hence we have the following corollary. Corollary 1.3 (see Subsection 6.5) . Let p be an integer which is either zero or aprime number different from 2 , , , , , , K ofcharacteristic p and a smooth plane quartic C ⊂ P K such that C admits a symmetricdeterminantal representation over K v for each place v of K , and C does not admit asymmetric determinantal representation over K . Remark 1.4.
In Theorem 1.1, we can replace “for each place” by “for all but oneplaces” when C is a conic. Also, we can replace “for each place” by “for all butfinitely many places” when C is a cubic. In Theorem 1.2, the assumption on thesurjectivity of ρ C, can be weakened when C has a K -rational point. (See Theorem5.1 and Subsection 6.5.) Remark 1.5.
In this paper, we only consider global fields of characteristic differentfrom two. The story is completely different in characteristic two. The local-globalprinciple holds for any smooth plane curves of any degree over a global field of char-acteristic two ([23]).Concerning the local-global principle for the existence of symmetric determinantalrepresentations of smooth plane curves, it seems interesting to study the followingproblems.
Problem 1.6. (1) Are there counterexamples to the local-global principle for smoothplane quartics over Q or F p ( T ) for p = 2? In other words, can we take K = Q or F p ( T ) for p = 2 in Corollary 1.3?(2) Are there counterexamples to the local-global principle in degree n ≥ K is algebraically closed of char-acteristic zero, in 1902, Dixon proved the existence of symmetric determinantal rep-resentations for generic plane curves of any degree ([9]). Since then, this problemhas been re-examined by many people ([31], [36], [8], [7], [37], [38], [1], [39], [28]). In2000, Beauville systematically studied minimal resolutions of coherent sheaves on theprojective spaces, and proved that all plane curves (including singular ones) admitsymmetric determinantal representations when K is algebraically closed of character-istic zero. The situation is quite different when K is not algebraically closed. In 1938, OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 3
Edge studied the field of definition of symmetric determinantal representations of theFermat quartic and related curves called
Edge’s quartics ([13]). In 2009, Wei Ho stud-ied, among other things, certain linear orbits of triples of matrices related to symmetricdeterminantal representations of smooth plane curves over a field of characteristic notdividing 3 n ( n −
1) ([21]). The results of Beauville and Ho were generalized by the firstauthor to include the case of higher dimensional hypersurfaces over arbitrary fields([22]). For symmetric determinantal representations of Fermat curves of prime degreeand the Klein quartic over Q , see [24].Let us give a sketch of the proof of our results. The existence of a symmetricdeterminantal representation of a smooth plane curve C of degree n ≤ n = 2, it is related to the existence ofa K -rational point on the conic (Proposition 4.1). When n = 3, it is related to theexistence of a non-trivial K -rational 2-torsion point on the Jacobian variety Jac( C )(Proposition 4.2). We prove Theorem 1.1 using these relations. The proof of Theorem1.2 depends on a group theoretic lemma on the action of subgroups of Sp m ( F ) onquadratic forms over F (Lemma 6.6). We give a sufficient condition for a smooth planecurve of any degree to violate the local-global principle in terms of the associated mod 2Galois representation (Proposition 6.10). For a smooth plane quartic whose associatedmod 2 Galois representation is surjective, it is not difficult to see that it satisfies thesufficient condition after taking a finite extension of the base field.The outline of this paper is as follows: In Section 2, we recall a relation betweensymmetric determinantal representations and certain line bundles called non-effectivetheta characteristics . In Section 3, we recall the basic facts on the relative Picardfunctors and Picard schemes. In Section 4, we examine the case of n ≤ F , we prove Theorem 1.2. Notation.
For a field K , an algebraic closure of it is denoted by K , and a separableclosure of it is denoted by K sep . A global field is a field isomorphic to a finite extensionof Q or F p ( T ), where p is a prime number, F p is the finite field of order p , and T is anindeterminate. For a place v of a global field K , the completion of K at v is denotedby K v . For a morphism of schemes X −→ S and S ′ −→ S , the base change X × S S ′ is denoted by X S ′ . When S = Spec K and S ′ = Spec L for a field extension L/K , thebase change X S ′ is also denoted by X ⊗ K L or X L .2. Theta characteristics and symmetric determinantalrepresentations
We recall the definition of theta characteristics on proper smooth curves and itsrelation to symmetric determinantal representations. In this section, K is a field ofarbitrary characteristic. Definition 2.1 ([29]) . Let C be a proper smooth geometrically connected curve over K . (1) A theta characteristic on C is a line bundle L on C satisfying L ⊗ L ∼ = Ω C ,where Ω C is the canonical sheaf on C . YASUHIRO ISHITSUKA AND TETSUSHI ITO (2) A theta characteristic L on C is effective (resp. non-effective ) if H ( C, L ) = 0(resp. H ( C, L ) = 0).(3) A theta characteristic L on C is even (resp. odd ) if dim K H ( C, L ) is even (resp. odd ). Theorem 2.2.
Let ι : C ֒ → P K be a smooth plane curve over K . Then C admits asymmetric determinantal representation over K if and only if there is a non-effectivetheta characteristic on C . Proof . This result is well known when the characteristic of K is different from two([2, Proposition 4.2], [12, Ch 4], [21]). Although the proofs in [2], [21] are writtenunder the additional assumptions on the base field, it is not difficult to modify thearguments to cover the case of characteristic two ([2, Remark 2.2]). For a proof of thisproposition which works over arbitrary fields, see [22]. (cid:3) Remark 2.3.
Let C be a smooth plane quartic over K . It is well known that all eventheta characteristics on C are non-effective. This can be seen as follows. Assume thatthere is an effective even theta characteristic L on C . Then L is isomorphic to O C ( D )for an effective divisor D of degree g ( C ) − K H ( C, O C ( D )) ≥ f with div( f ) + D ≥
0. Then f defines amorphism f : C −→ P K of degree 2. It contradicts to the well-known fact that smoothplane quartics are non-hyperelliptic ([18, IV, Exercise 3.2]).3. Relative Picard functors and Picard schemes
We recall the basic definitions and properties of relative Picard functors and Picardschemes ([4], [26]).For a morphism of schemes f : X −→ S , the relative Picard functor Pic
X/S is thefppf sheaf associated with the functor (cid:0)
Schemes /S (cid:1) op −→ (cid:0) Sets (cid:1) , S ′ Pic( X S ′ ) , where the Picard group Pic( X S ′ ) is the group of isomorphism classes of line bundleson X S ′ := X × S S ′ ([4, § f : X −→ S is quasi-compact, quasi-separated, and f ∗ ( O X ) = O S holds, we have the following exact sequence for each flat S -scheme S ′ :(3.1) 0 / / Pic( S ′ ) / / Pic( X S ′ ) / / Pic
X/S ( S ′ ) / / Br( S ′ ) / / Br( X S ′ ) , where Br( S ′ ) := H ( S ′ , G m ) , Br( X S ′ ) := H ( X S ′ , G m )are the cohomological Brauer groups calculated in the fppf topology ([4, § G m is representable by a smooth scheme, the cohomologicalBrauer groups Br( S ′ ) , Br( X S ′ ) can be calculated using ´etale topology ([16, Th´eor`eme11.7]). In particular, when S ′ := Spec K is the spectrum of a field K , the cohomo-logical Brauer group Br(Spec K ) is isomorphic to the Brauer group of K defined byGalois cohomology ([33, Ch. X, § K ) ∼ = Br( K ) := H (cid:0) Gal( K sep /K ) , ( K sep ) × (cid:1) . When S = S ′ = Spec K for a field K , the Picard group Pic(Spec K ) is trivial, and theexact sequence (3.1) becomes(3.2) 0 / / Pic( X ) / / Pic
X/K ( K ) / / Br( K ) / / Br( X ) . OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 5
We have Pic( X ) = Pic X/K ( K ) if Br( K ) is trivial or X has a K -rational point ([4, § X is a proper scheme over a field K , the relative Picard functor Pic X/K isrepresentable by a scheme which is locally of finite type over K ([4, § X/K is called the
Picard scheme .When X is a proper smooth geometrically connected curve over K , the identitycomponent Jac( X ) := Pic X/K ⊂ Pic
X/K is called the
Jacobian variety . It is an abelianvariety whose dimension is equal to the genus g ( X ) of X ([4, § Remark 3.1.
For a proper scheme X over a field K whose geometric fiber is connectedand reduced, the following are well known.(1) We have Pic X L /L ∼ = (Pic X/K ) L because the formation of Picard schemes com-mutes with base change ([26, Exercise 9.4.4]).(2) The map Pic( X ) −→ Pic( X L ) is injective for any field extension L/K .(3) For a Galois extension
L/K , we have Pic
X/K ( L ) Gal(
L/K ) = Pic X/K ( K ). ButPic( X L ) Gal(
L/K ) = Pic( X ) does not hold in general. Proposition 3.2.
Let X be a proper scheme over a field K whose geometric fiber isconnected and reduced. Let L/K be a Galois extension which is not necessarily finite.Let L be a line bundle on X L such that its class [ L ] ∈ Pic( X L ) is fixed by the actionof Gal( L/K ). Assume that at least one of the following conditions is satisfied:(a) Br( K ) is trivial,(b) X has a K -rational point,(c) there is a finite extension M/K and an integer r ≥ M : K ] suchthat X has an M -rational point and [ L ⊗ r ] comes from a line bundle on X , or(d) K is a global field and there is a place v of K such that, for any place v = v of K and a place w of L above v , [ L L w ] comes from a line bundle on X K v .Then [ L ] comes from a line bundle on X . Proof . Since the image of [ L ] in Pic X/K ( L ) is fixed by Gal( L/K ), it comes from anelement α K ∈ Pic
X/K ( K ). We shall show that the image of α K in Br( K ) is trivial. (a)is obvious; (b), (c) are standard; (d) follows from the injectivity of the mapBr( K ) −→ M v = v Br( K v ) . in global class field theory ([35, § § (cid:3) We give an application of Picard schemes to theta characteristics.
Proposition 3.3.
Let C be a proper smooth geometrically connected curve over afield K , and L a line bundle on C . Let L/K be an extension of fields, and L L be thepullback of L to C L .(1) L is a theta characteristic on C if and only if L L is a theta characteristic on C L .(2) L is effective (resp. non-effective) if and only if L L is effective (resp. non-effective). Proof . (1) The element [ L L ] (resp. [Ω C L ]) is the image of [ L ] (resp. [Ω C ]) by thecanonical map Pic( C ) −→ Pic( C L ). Since this map is injective by Remark 3.1 (2), wesee that 2[ L ] = [Ω C ] holds if and only if 2[ L L ] = [Ω C L ] holds. YASUHIRO ISHITSUKA AND TETSUSHI ITO (2) This follows from the equality dim K H ( C, L ) = dim L H ( C L , L L ). (cid:3) Symmetric determinantal representations in degree n ≤ n ≤ K ofarbitrary characteristic. Let C ⊂ P K be a smooth plane curve of degree n ≤ Lines ( n = 1 ). Obviously, every line over K admits a symmetric determinantalrepresentation over K . In order to illustrate the methods of this paper, let us confirmit using Theorem 2.2. The plane curve C ⊂ P K of degree 1 is isomorphic to P K over K . The Picard group of P K is isomorphic to Z generated by [ O P K (1)]. Sincedeg Ω P K = −
2, the line bundle O P K ( −
1) is a unique theta characteristic on P K , upto isomorphism. It is non-effective because its degree is negative. Hence C admits asymmetric determinantal representation over K by Theorem 2.2.4.2. Conics ( n = 2 ). There is a natural map from the set of smooth plane conics over K to the set of elements of Br( K ) killed by 2. There are several ways to constructit. The following method seems most suitable for our purposes: since C is smoothover K , the curve C has a K sep -rational point ([4, § C K sep isisomorphic to P K sep over K sep . The Picard group of P K sep is isomorphic to Z generatedby a line bundle of degree 1. Since the action of Gal( K sep /K ) on Pic C/K ( K sep ) doesnot change the degree of line bundles, we have the following isomorphisms:Pic C/K ( K ) = Pic C/K ( K sep ) Gal( K sep /K ) = Pic C/K ( K sep ) deg ∼ = Z . (See Remark 3.1 (3) for the first equality.) There is a unique element s ∈ Pic
C/K ( K )of degree 1. We define α C ∈ Br( K ) to be the image of s in Br( K ) by the followingexact sequence (cf. (3.2)):0 / / Pic( C ) / / Pic
C/K ( K ) / / Br( K ) . Since deg Ω C = −
2, we see that α C is trivial if and only if C has a line bundle of odddegree. The element α C ∈ Br( K ) is killed by 2. Proposition 4.1.
The following are equivalent:(a) C admits a symmetric determinantal representation over K .(b) C is isomorphic to P K over K .(c) C has a K -rational point.(d) C has a line bundle of odd degree.(e) α C ∈ Br( K ) is trivial. Proof . The equivalence (d) ⇔ (e) follows from the construction of α C ∈ Br( K )recalled as above. The implications (b) ⇒ (c) ⇒ (d) are obvious. If there is a linebundle L on C of odd degree, there is a line bundle L ′ on C of degree 1 becausedeg Ω C = −
2. The complete linear system of L ′ gives an isomorphism C ∼ = P K by Riemann-Roch ([27, Proposition 7.4.1]). Hence (d) ⇒ (b) follows. Finally, byProposition 3.3, a line bundle L on C is a theta characteristic if and only if deg L = − L is negative. By Theorem 2.2, the existence a linebundle L with deg L = − C over K . Hence (d) ⇔ (a) follows. (cid:3) OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 7
Cubics ( n = 3 ). Since C ⊂ P K is a smooth plane cubic, the Jacobian varietyJac( C ) is an elliptic curve over K . It is well known that C is isomorphic to Jac( C ) ifand only if C has a K -rational point. Proposition 4.2.
The following are equivalent:(a) C admits a symmetric determinantal representation over K .(b) Jac( C ) has a non-trivial K -rational 2-torsion point. Proof . Since Ω C is trivial, by Theorem 2.2 and Proposition 3.3, C admits a symmetricdeterminantal representation over K if and only if there is a non-trivial line bundle L on C satisfying L ⊗ L ∼ = O C .(a) ⇒ (b) If L is a non-trivial line bundle satisfying L ⊗ L ∼ = O C , its class [ L ] ∈ Jac( C )( K ) is a non-trivial K -rational 2-torsion point.(b) ⇒ (a) Let α ∈ Jac( C )( K ) be a non-trivial K -rational 2-torsion point. There is afinite extension M/K of odd degree with C ( M ) = ∅ because C has odd degree. Hence α comes from a line bundle L α on C by Proposition 3.2 (c) for r = 2. (cid:3) The local-global principle for conics and cubics
The following theorem is slightly more general than Theorem 1.1.
Theorem 5.1.
Let K be a global field of characteristic different from two, and C ⊂ P K be a smooth plane curve of degree 2 or 3 over K .(1) Assume that C has degree 2. If there is a place v of K such that C admitssymmetric determinantal representations over K v for all places v = v of K ,the smooth plane curve C admits a symmetric determinantal representationover K .(2) Assume that C has degree 3. If C admits symmetric determinantal represen-tations over K v for all but finitely many places v of K , the smooth plane curve C admits a symmetric determinantal representation over K . Proof . (1) Let α C ∈ Br( K ) be the element associated with C in Subsection 4.2.Since the image of α C in Br( K v ) is trivial for each v = v , we see that α C is trivial bythe structure of the Brauer group of K ([35, § § C admits a symmetric determinantal representation over K by Proposition 4.1.(2) The non-trivial 2-torsion points on Jac( C ) are defined by a cubic polynomial f ( X ) ∈ K [ X ]. The assertion follows from the well known fact that f ( X ) = 0 has asolution in K if it has a solution in K v for all but finitely many v by Chebotarev’sdensity theorem ([32, I.2.2]). (cid:3) Remark 5.2.
Theorem 5.1 (1) is optimal in the following sense: for any global field K and any finite set S of places of K of cardinality ≥
2, there is a smooth planeconic C ⊂ P K such that C admits a symmetric determinantal representation over K v for each v / ∈ S , and C does not admit a symmetric determinantal representationover K . To see this, let α ∈ Br( K ) be a non-trivial element killed by 2 satisfyinginv v ( α ) = 0 for all v / ∈ S ([35, § § § Remark 5.3.
The following argument seems instructive to understand how to con-struct counterexamples to the local-global principle for quartics in Section 6 (see also
YASUHIRO ISHITSUKA AND TETSUSHI ITO
Remark 6.8). The proof of Theorem 5.1 (2) can be rephrased in terms of mod 2 Ga-lois representations. Choose an F -basis on Jac( C )[2]( K sep ), and consider the mod 2Galois representation on it: ρ C, : Gal( K sep /K ) −→ GL ( F ) . Let v be a finite place of K such that C admits a symmetric determinantal represen-tation over K v and Jac( C ) has good reduction at v . By Proposition 4.2, the image ofthe geometric Frobenius element ρ C, (Frob v ) has a non-zero fixed vector. Let G := ρ C, (cid:0) Gal( K sep /K ) (cid:1) ⊂ GL ( F )be the image of ρ C, , which is generated by ρ C, (Frob v ) for all but finitely many v byChebotarev’s density theorem ([32, I.2.2]). By an easy group theoretic lemma (seeLemma 5.4 below), there is a non-zero vector fixed by all elements of G . Hence Jac( C )has a non-trivial K -rational 2-torsion point, and C admits a symmetric determinantalrepresentation over K by Proposition 4.2.The proof of the following lemma is easy and omitted. Lemma 5.4.
Let G ⊂ GL ( F ) be a subgroup such that, for each element g ∈ G , theaction of g on F ⊕ has a non-zero fixed vector. Then there is a non-zero vector v ∈ F ⊕ fixed by all elements of G .6. Counterexamples to the local-global principle for quartics
We recall the basic results on quadratic forms over F in Subsection 6.1. Then werecall the Mumford’s results on a relation between theta characteristics and quadraticforms over F in Subsection 6.2. In Subsection 6.3, we prove a group theoretic lemmaon the action of a subgroup of Sp m ( F ) on quadratic forms over F (Lemma 6.6). InSubsection 6.4, we give a sufficient condition for a smooth plane curve over a globalfield to violate the local-global principle in terms of the associated mod 2 Galois repre-sentation (Proposition 6.10). In Subsection 6.5, we prove Theorem 1.2 and constructcounterexamples to the local-global principle for quartics.6.1. Quadratic forms over F . We recall the basic results on quadratic forms overthe finite field F of order two. (For more details on the group theoretic properties ofthe action of Sp m ( F ) on quadratic forms, see [11], [10, § F ⊕ m be the 2 m -dimensional vector space over F . Let { e , . . . , e m , f , . . . , f m } be the standard basis of F ⊕ m , and define the alternating bilinear form h , i on F ⊕ m by h e i , e j i = 0 , h f i , f j i = 0 , h e i , f j i = h f j , e i i = ( i = j, i = j. . The symplectic group Sp m ( F ) is defined to be the group of F -linear automor-phisms of F ⊕ m preserving the alternating form h , i . A quadratic form on F ⊕ m withpolar form h , i is a map Q : F ⊕ m −→ F satisfying Q ( x + y ) − Q ( x ) − Q ( y ) = h x, y i for all x, y ∈ F ⊕ m . OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 9
There are 2 m quadratic forms on F ⊕ m with polar form h , i . The symplectic groupSp m ( F ) acts on the set of quadratic forms with polar form h , i by( g · Q )( x ) := Q ( g − x )for g ∈ Sp m ( F ) , x ∈ F ⊕ m . This action has two orbits Ω + , Ω − of size 2 m − (2 m +1) , m − (2 m − Arf invariant .There are several equivalent definitions of the Arf invariant. One definition of the Arfinvariant of a quadratic form Q is m X i =1 Q ( e i ) Q ( f i ) ∈ F , which is shown to be independent of the choice of the symplectic basis. Anotherimpressive definition is this: the Arf invariant of Q is a ( a ∈ { , } ) if and only if thenumber of elements x ∈ F ⊕ m with Q ( x ) = a is equal to 2 m − (2 m + 1) ([15, Corollary1.12]).6.2. Theta characteristics and quadratic forms over F . We recall a relationbetween theta characteristics and quadratic forms over F due to Mumford ([29]).Let C be a proper smooth geometrically connected curve of genus g over a field K . Assume that the characteristic of K is different from two . The Jacobian varietyJac( C ), which is the identity component of the Picard scheme Pic C/K , is an abelianvariety of dimension g (cf. Section 3). The multiplication-by-2 isogeny[2] : Pic C/K −→ Pic
C/K is ´etale because 2 is invertible in K . The group scheme Jac( C )[2] is finite and ´etaleover K of order 2 g . All K -rational points on Jac( C )[2] are defined over K sep . We seethat Jac( C )[2]( K sep ) = Jac( C )[2]( K )is an F -vector space of dimension 2 g . We have the Weil pairing e : Jac( C )[2]( K sep ) × Jac( C )[2]( K sep ) −→ {± } ∼ = F , which is an alternating bilinear form over F . Here the multiplicative group {± } isisomorphic to the additive group of F . Mumford proved that, for a theta characteristic L on C K sep , the map Q L : Jac( C )[2]( K sep ) −→ F defined by[ M ] (cid:0) dim K sep H ( C K sep , L ⊗ M ) + dim K sep H ( C K sep , L ) (cid:1) (mod 2)is a quadratic form with polar form e . Any quadratic form on Jac( C )[2]( K sep ) withpolar form e can be written as Q L for a theta characteristic L on C K sep , and theisomorphism class of L is uniquely determined by its associated quadratic form. TheArf invariant of Q L is 0 (resp. 1) if and only if L is even (resp. odd) (see Definition2.1). Remark 6.1.
A cautious reader might note that Mumford worked over an algebraicclosure K of K rather than a separable closure K sep of K ([29]). It is easy to see thathis results are valid over K sep as well. To see this, it is enough to note the following:(1) Jac( C )[2]( K sep ) = Jac( C )[2]( K ) because 2 is invertible in K .(2) For a theta characteristic L on C K sep , we have Q L = Q L K . (3) Every K -rational (resp. K sep -rational) point on the Picard scheme Pic C/K comes from a line bundle on C K (resp. C K sep ) because the Brauer group Br( K )(resp. Br( K sep )) is trivial ([4, § C K (resp. C K sep )is identified with (cid:0) [2] − ([Ω C ]) (cid:1) ( K ) (resp. (cid:0) [2] − ([Ω C ]) (cid:1) ( K sep )). Since [2] is an´etale isogeny, these two sets are equal. Hence the set of isomorphism classes oftheta characteristics on C K and on C K sep are canonically identified.From the above remarks, we have the following proposition. Proposition 6.2.
For each a ∈ { , } , the association L 7→ Q L gives a bijectionbetween the following sets: • The set of quadratic forms on Jac( C )[2]( K sep ) of Arf invariant a whose polarform is the Weil pairing e . • The set of isomorphism classes of theta characteristics L on C K sep satisfyingdim K sep H ( C K sep , L ) ≡ a (mod 2).The bijection is equivariant with respect to the action of Gal( K sep /K ).From Proposition 3.2, we have the following corollary. Corollary 6.3.
Let Q be a quadratic form on Jac( C )[2]( K sep ) whose polar form isthe Weil pairing e . If Q is fixed by the action of Gal( K sep /K ), there is a thetacharacteristic L on C K sep with Q = Q L by Proposition 6.2. Then the line bundle L on C K sep comes from a theta characteristic on C if at least one of the following conditionsis satisfied:(a) Br( K ) is trivial,(b) C has a K -rational point,(c) there is a finite extension M/K of odd degree such that C has an M -rationalpoint, or(d) K is a global field, and there is a place v of K such that L K sep v comes from atheta characteristic on C K v for any place v = v of K .6.3. Group theoretic lemmas.
We prove a group theoretic lemma on the action ofsubgroups of Sp m ( F ) on quadratic forms over F (Lemma 6.6). The case of m = 3of Lemma 6.6 will be used to construct counterexamples to the local-global principlefor quartics (cf. Subsection 6.5).In the following, we use the same notation as in Subsection 6.1. Lemma 6.4.
Fix an integer m ≥
1, and a quadratic form Q on F ⊕ m of Arf invariant 1whose polar form is the standard alternating bilinear form h , i . Let O( Q ) ⊂ Sp m ( F )be the orthogonal group associated with Q , which is the group of F -linear automor-phisms of F ⊕ m preserving Q . We denote the identity element by e ∈ O( Q ). Thenthere are elements σ, τ ∈ O( Q ) satisfying all of the following conditions:(a) σ = e ,(b) τ = e ,(c) τ στ = σ − ,(d) σ i has no non-zero fixed vector in F ⊕ m for any i with σ i = e , and(e) τ σ i has a non-zero fixed vector x ∈ F ⊕ m with Q ( x ) = 1 for any i .Note that, in the condition (e), the fixed vector x ∈ F ⊕ m of τ σ i may depend on i . OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 11
Remark 6.5.
The subgroup of O( Q ) generated by σ, τ is isomorphic to the dihedralgroup of order 2 n ( σ ), where n ( σ ) is the order of σ . Proof . Since Sp m ( F ) acts transitively on the set of quadratic forms with Arf in-variant 1, it is enough to prove the assertion for a particular quadratic form with Arfinvariant 1.Let F m be the finite field of order 2 m . Let us consider F m as an F -vector spaceof dimension 2 m . We shall construct an alternating bilinear form h , i and a quadraticform Q with Arf invariant 1 as follows. In order to shorten the notation, we put F ( k ) := F k . For x ∈ F (2 m ), the conjugate of x over F ( m ) is denoted by x . For x, y ∈ F (2 m ), we define h x, y i and Q ( x ) by h x, y i := Tr F (2 m ) /F (1) ( xy ) ,Q ( x ) := Tr F ( m ) /F (1) (cid:0) N F (2 m ) /F ( m ) ( x ) (cid:1) . It is a routine exercise to check that h , i is an alternating bilinear form on F (2 m ), and Q is a quadratic form on F (2 m ) with polar form h , i .We shall show that the Arf invariant of Q is 1. We count the number of elements x ∈ F (2 m ) with Q ( x ) = 1 as follows. Since F ( m ) /F (1) is a separable extension, thetrace map Tr F ( m ) /F (1) : F ( m ) −→ F (1)is surjective. The number of elements t ∈ F ( m ) with Tr F ( m ) /F (1) ( t ) = 1 is 2 m − . Thenorm map N F (2 m ) /F ( m ) : F (2 m ) × −→ F ( m ) × is surjective ([33, Ch. X, § x ∈ F (2 m ) with Q ( x ) = 1is equal to 2 m − · [ F (2 m ) × : F ( m ) × ] = 2 m − (2 m + 1) , and the Arf invariant of Q is 1.Let O( Q ) be the group of F -linear automorphisms of F (2 m ) preserving Q . We shallconstruct two elements in O( Q ) satisfying all of the conditions of this lemma. Since F (2 m ) × is a cyclic group, the kernel of the norm mapN F (2 m ) /F ( m ) : F (2 m ) × −→ F ( m ) × is also cyclic. We take a generator s ∈ F (2 m ) × of the kernel of N F (2 m ) /F ( m ) . We define g ∈ O( Q ) by g ( x ) := sx . We define h ∈ O( Q ) by h ( x ) := x .We shall prove that the elements g, h ∈ O( Q ) satisfy the required conditions for σ, τ . The conditions (a), (b) are obvious because s = 1 and x = x for x ∈ F (2 m ). Thecondition (c) is satisfied because we have( h ◦ g ◦ h )( x ) = sx = s − x = g − ( x ) . If g i is not the identity element, we see that s i = 1 and the map x g i ( x ) = s i x has no non-zero fixed vector. Hence the condition (d) is satisfied. Since s − = s , wehave ( h ◦ g i )( x ) = x ⇐⇒ s − i x = x for x ∈ F (2 m ). Since F (2 m ) /F ( m ) is a quadratic Galois extension andN F (2 m ) /F ( m ) ( s − i ) = 1 , there is an element y ∈ F (2 m ) × satisfying s − i y = y by Hilbert’s Theorem 90 ([33, Ch.X, § s − i y = y ” is satisfied if we replace y by ty for t ∈ F ( m ) × . Themap F ( m ) × −→ F ( m ) × , t t is surjective because F ( m ) × is a finite abelian group of odd order. For an element t ∈ F ( m ) × , we have Q ( ty ) := Tr F ( m ) /F (1) (cid:0) N F (2 m ) /F ( m ) ( ty ) (cid:1) := Tr F ( m ) /F (1) (cid:0) t N F (2 m ) /F ( m ) ( y ) (cid:1) . The bilinear form F ( m ) × F ( m ) −→ F (1) , ( u, v ) Tr F ( m ) /F (1) (cid:0) uv (cid:1) is non-degenerate because F ( m ) /F (1) is a separable extension. Therefore, after replac-ing y by ty for some t ∈ F ( m ) × , we have Q ( y ) = 1. The condition (e) is satisfied. (cid:3) Lemma 6.6.
For m ≥
3, there is a subgroup G ⊂ Sp m ( F ) satisfying both of thefollowing conditions:(a) there does not exist a quadratic form of Arf invariant 0 with polar form h , i fixed by all elements of G , and(b) for each g ∈ G , there is a quadratic form of Arf invariant 0 with polar form h , i fixed by g . Proof . Let us decompose F ⊕ m as F ⊕ m ∼ = F ⊕ m − ⊕ F ⊕ ⊕ F ⊕ ∼ = V ⊕ V ⊕ V . We put the standard alternating bilinear form on each V i .We shall define a quadratic form on each V i as follows. Let Q be a quadratic formon V of Arf invariant 1. Let { e , f } (resp. { e , f } ) be the standard symplectic basisof V (resp. V ). For i = 2 ,
3, we define a quadratic form Q i on V i by Q i ( ae i + bf i ) := a + ab + b for a, b ∈ F . The Arf invariant of Q i is 1. Hence the Arf invariant of the direct sum Q := Q ⊕ Q ⊕ Q is 1. We denote the orthogonal group associated with the quadratic form Q (resp. Q ⊕ Q , Q ) by O( Q ) (resp. O( Q ⊕ Q ) , O( Q )). The product O( Q ) × O( Q ⊕ Q )is a subgroup of O( Q ).Now we shall apply Lemma 6.4 to Q . We get two elements σ, τ ∈ O( Q ). We define η ∈ O( Q ⊕ Q ) by η ( e ) := e , η ( f ) := f , η ( e ) := e , η ( f ) := f . Let G := h ( σ, id) , ( τ, η ) i be the subgroup of O( Q ) generated by ( σ, id) and ( τ, η ).We shall show that G satisfies the conditions (a), (b) of this lemma. The followingare well known ([11, Lemma 1]): • For each vector v ∈ F ⊕ m , the map Q v : F ⊕ m −→ F , x Q v ( x ) := Q ( x ) + h x, v i is a quadratic form on F ⊕ m with polar form h , i . • Every quadratic form on F ⊕ m with polar form h , i can be written as Q v for aunique vector v ∈ F ⊕ m . OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 13 • The Arf invariant of Q v is 0 (resp. 1) if and only if Q ( v ) = 1 (resp. 0). • For g ∈ O( Q ), we see that g · Q v = Q v if and only if gv = v because( g · Q v )( x ) = Q v ( g − x ) = Q ( g − x ) + h g − x, v i = Q ( x ) + h x, gv i . In order to check the condition (a), it is enough to prove that there does not exista vector v ∈ F ⊕ m with Q ( v ) = 1 which is fixed by all elements of G . Since σ has nonon-zero fixed vector in V , a vector v ∈ F ⊕ m fixed by all elements of G is necessarilyof the form v = a ( e + e ) + b ( f + f )for some a, b ∈ F . Since( Q ⊕ Q )( v ) = ( a + ab + b ) + ( a + ab + b ) = 0 , there does not exist a vector v ∈ F ⊕ m with Q ( v ) = 1 fixed by all elements of G .We shall check the condition (b). Any element g ∈ G is either of the form ( σ i , id)or ( τ σ i , η ). If g = ( σ i , id), all vectors in V ⊕ V are fixed by g . Hence g fixes e ,which satisfies Q ( e ) = 1. If g = ( τ σ i , η ), by Lemma 6.4 (e), there is a vector v ∈ V satisfying Q ( v ) = 1 and gv = v . Hence G satisfies the condition (b). (cid:3) Remark 6.7.
There are many subgroups G ⊂ Sp m ( F ) satisfying the conditions (a),(b) in Lemma 6.6. When m = 3, a quick search using GAP (version 4.7.5) shows thatthere are 1369 subgroups of Sp ( F ), up to conjugacy. Among them, 411 subgroups, upto conjugacy, satisfy the conditions (a), (b) in Lemma 6.6. The subgroup G constructedin the proof of Lemma 6.6 is a unique subgroup of Sp ( F ) of order 6, up to conjugacy,satisfying the conditions (a), (b) in Lemma 6.6. Remark 6.8.
Lemma 6.6 does not hold for m = 1. On the F -vector space F ⊕ with a standard alternating form, there are three quadratic forms of Arf invariant 0.There is a unique quadratic form of Arf invariant 1. We denote it by Q . Quadraticforms on F ⊕ are written as Q v for v ∈ F ⊕ . Quadratic forms of Arf invariant 0correspond to non-zero vectors in F ⊕ . By Lemma 5.4, we see that no subgroup G ⊂ Sp ( F ) = SL ( F ) = GL ( F ) satisfies the conditions (a), (b) in Lemma 6.6. Thisexplains why the local-global principle for the existence of symmetric determinantalrepresentations holds true for cubics (cf. Theorem 5.1 (2)), but it does not hold truefor quartics. Remark 6.9.
Lemma 6.6 holds true for m = 2. Using GAP (version 4.7.5), we see thatthere are 56 subgroups of Sp ( F ) ∼ = S , up to conjugacy. Among them, 12 subgroups,up to conjugacy, satisfy the conditions (a), (b) in Lemma 6.6. However, the caseof m = 2 is not related to the problem of symmetric determinantal representationsbecause the genus of a smooth plane curve cannot be equal to 2.6.4. A sufficient condition to violate the local-global principle.
We give asufficient condition for a smooth plane curve to violate the local-global principle interms of the associated mod 2 Galois representation.Let K be a global field of characteristic different from two, and C ⊂ P K be asmooth plane curve of degree n ≥
4. The Jacobian variety Jac( C ) is an abelianvariety of dimension ( n − n − /
2. Since char K = 2, the multiplication-by-2isogeny [2] : Jac( C ) −→ Jac( C ) is ´etale, and all K -rational 2-torsion points on Jac( C )are defined over K sep . Hence Jac( C )[2]( K sep ) is an F -vector space of dimension ( n − n − K sep /K ) on Jac( C )[2]( K sep ) preserves the Weil pairing e , by choosing a symplectic F -basis, we have the associated mod 2 Galoisrepresentation ρ C, : Gal( K sep /K ) −→ Sp ( n − n − ( F ) . We fix an embedding ι v : K sep ֒ → K sep v for each place v of K . We consider Gal( K sep v /K v )as a closed subgroup of Gal( K sep /K ). The embeddingGal( K sep v /K v ) ֒ → Gal( K sep /K )is unique up to conjugation. Proposition 6.10.
Assume that at least one of the following conditions is satisfied:(a) C has a K v -rational point for each place v of K , or(b) the degree n is odd.Moreover, assume that all of the following conditions are satisfied:(c) the image of ρ C, satisfies the conditions (a), (b) in Lemma 6.6,(d) the image ρ C, (cid:0) Gal( K sep v /K v ) (cid:1) is a cyclic group for each place v of K , and(e) all even theta characteristics on C K sep are non-effective.Then C admits a symmetric determinantal representation over K v for each place v of K , and C does not admit a symmetric determinantal representation over K . Proof . Let v be a place of K . Note thatJac( C )[2]( K sep ) = Jac( C )[2]( K sep v )because Jac( C )[2] is a finite ´etale group scheme over K .By the conditions (c), (d), there is a quadratic form on Jac( C )[2]( K sep v ) with Arfinvariant 0 fixed by Gal( K sep v /K v ) (see the condition (b) in Lemma 6.6). Proposition6.2 applied to C K v shows that there is an even theta characteristic L K sep v on C K sep v suchthat [ L K sep v ] ∈ Pic
C/K ( K sep v )is fixed by Gal( K sep v /K v ).If the condition (a) is satisfied, C K v satisfies the condition (b) of Corollary 6.3. Ifthe condition (b) is satisfied, there is a finite extension M v /K v of odd degree such that C has an M v -rational point. Hence C satisfies the condition (c) of Corollary 6.3. Inboth cases, L K sep v comes from an even theta characteristic on C K v by Corollary 6.3. Itis non-effective by the condition (e). Therefore, C admits a symmetric determinantalrepresentation over K v by Theorem 2.2.Finally, by the condition (c), there does not exist a quadratic form on Jac( C )[2]( K sep )with Arf invariant 0 fixed by Gal( K sep /K ) (see the condition (a) in Lemma 6.6). Hencethere does not exist an even theta characteristic on C . By Theorem 2.2, C does notadmit a symmetric determinantal representation over K . (cid:3) Remark 6.11.
When n = 4, the condition (e) is always satisfied because even thetacharacteristics on smooth plane quartics are non-effective. (See Remark 2.3.)6.5. Counterexamples to the local-global principle for quartics. of Theorem 1.2.
Recall that K is a global field of characteristic different from two,and C ⊂ P K a smooth plane quartic over K such that the associated mod 2 Galoisrepresentation ρ C, : Gal( K sep /K ) −→ Sp ( F )is surjective. OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 15
We first claim that there is a finite separable extension K ′ /K such that C has a K ′ -rational point and the restriction of ρ C, to Gal( K sep /K ′ ) is still surjective. In someexamples (e.g., Remark 6.14), we can find a K -rational point explicitly, and we cansimply take K = K ′ . But C might not have a K -rational point in general.There is a line ℓ ⊂ P K such that ℓ ∩ C is smooth over K by Bertini’s theorem ([25,Corollaire 6.11 (2)]). Since ℓ ∩ C is defined by a separable polynomial of degree 4,there is a separable extension K ′ /K of degree ≤ C has a K ′ -rational point.Since Gal( K sep /K ′ ) is a closed subgroup of Gal( K sep /K ) of index ≤
4, its image ρ C, (cid:0) Gal( K sep /K ′ ) (cid:1) ⊂ Sp ( F )is a subgroup of index ≤
4. Since Sp ( F ) has no proper subgroup of index ≤ ρ C, to Gal( K sep /K ′ ) is surjective. (A proper subgroup of Sp ( F )of smallest index is conjugate to the orthogonal group of a quadratic form with Arfinvariant 1, which has index 28.) Replacing K by K ′ , we may assume that C has a K -rational point.Next, we shall replace K by a finite extension of it as follows. We choose a subgroup G ⊂ ρ C, (cid:0) Gal( K sep /K ) (cid:1) = Sp ( F )satisfying the conditions in Lemma 6.6. There are 411 subgroups of Sp ( F ) with thisproperty, up to conjugacy (Remark 6.7). We may choose any one of them. We take afinite separable extension M/K such that Gal( K sep /M ) = ρ − C, ( G ).By Remark 6.11, the smooth plane quartic C M over M satisfies the conditions (a),(c), (e) of Proposition 6.10, but it might not satisfy the condition (d). Note that, if v is an archimedean place or a place where ρ C, is unramified, the image ρ C, (cid:0) Gal( M sep v /M v ) (cid:1) ⊂ Sp ( F )is a cyclic group generated by the image of the complex conjugation or the geometricFrobenius element, and the condition (d) of Proposition 6.10 is satisfied at v . Hencethe number of places v where the condition (d) of Proposition 6.10 is not satisfiedis finite. We denote them by v , . . . , v r . Since ρ C, (cid:0) Gal( M sep v i /M v i ) (cid:1) is a finite non-cyclic group, there is a finite extension M ′ v i of M v i such that ρ C, (cid:0) Gal( M sep v i /M ′ v i ) (cid:1) iscyclic. Take a finite separable extension M ′ /K such that M ′ /K is linearly disjointfrom ( K sep ) Ker ρ C, and ( M ′ M ) w i contains all conjugates of M ′ v i for all i , where w i is aplace of the composite M ′ M above v i . The existence of such M ′ /K is a consequenceof Krasner’s lemma and the weak approximation theorem.We put L := M ′ M . The smooth plane quartic C L over L satisfies the conditions(a), (c), (d), (e) of Proposition 6.10. Hence C L is a desired counterexample to thelocal-global principle by Proposition 6.10.The proof of Theorem 1.2 is complete. (cid:3) The above proof shows that the assumption on the surjectivity of the associatedmod 2 Galois representation can be weakened if C has a K -rational point. Corollary 6.12.
Let C ⊂ P K be a smooth plane quartic over a global field K ofcharacteristic different from two satisfying both of the following conditions:(a) C has a K -rational point, and(b) the image of the associated mod 2 Galois representation ρ C, contains a sub-group of Sp ( F ) satisfying the conditions in Lemma 6.6. Then there is a finite separable extension
L/K such that C admits a symmetric de-terminantal representation over L w for each place w of L , and C does not admit asymmetric determinantal representation over L .We shall prove Corollary 1.3 by combining Theorem 1.2 and the results of Harris,and Shioda. of Corollary 1.3. It is known that the associated mod 2 Galois representation of thegeneric family of plane quartics over a prime field of characteristic different from2 , , , , , , Q or F p ( T ) (for p = 2 , , , , , , (cid:3) Remark 6.13.
The condition “ p = 2 , , , , , , p ∈ { , , , , , } , Corollary 1.3 also holds in that characteristic. The case ofcharacteristic two is completely different ([23]).6.6. Concluding remarks.Remark 6.14.
It is possible to take explicit smooth plane quartics satisfying theconditions of Theorem 1.2. When K = Q , we may take C to be a smooth planequartic defined by the equation X X + X ( X + X X + X ) + X + X X + X X + X = 0 , or the equation X X − X X − X X − X X + X X − X X + X X = 0 . The first equation is taken from [34], and the second one is taken from [6], [5]. Note thatthe quartics defined by the above equations have Q -rational points such as (0 , , Remark 6.15.
In principle, the extension
L/K in the proof of Theorem 1.2 can betaken explicitly if we can calculate the associated mod 2 Galois representations. Therequired condition on M ′ /K can be seen from the ramification of ρ C, . For example,let us consider the smooth plane quartic C over Q defined by the following equation([6], [5]): X X − X X − X X − X X + X X − X X + X X = 0 . The extension Q Ker ρ C, / Q is a Galois extension with Galois group Sp ( F ) of degree1451520 ramified only at 2 , ,
347 ([6, 12.9.2]). If we choose G ⊂ Sp ( F ) to be thesubgroup of order 6 constructed in the proof of Lemma 6.6, M/ Q is a subextensionof Q Ker ρ C, / Q of degree 241920 with Gal (cid:0) Q Ker ρ C, /M (cid:1) = G . If we take an extension M ′ / Q such that M ′ is linearly disjoint from Q Ker ρ C, and (cid:0) M ′ Q Ker ρ C, (cid:1) / ( M ′ M ) isunramified at all places above 2 , , L = M ′ M satisfies all therequired conditions, and C L is a counterexample to the local-global principle. It ispossible to reduce the degree [ M : Q ] if you prefer. According to calculations done by GAP (version 4.7.5), the largest subgroup of Sp ( F ) satisfying the conditions in Lemma6.6 is of order 1440. (There are exactly two subgroups of Sp ( F ) of order 1440, up OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 17 to conjugacy. Both are isomorphic to Z / Z × S . Only one of them has the requiredproperties.) If we take this group as G , we reduce the extension degree [ M : Q ] to1008. Remark 6.16.
The degree [ L : K ] of the extension L/K in the proof of Theorem 1.2is rather large. It is an interesting problem to construct counterexamples to the local-global principle over smaller global fields such as Q or F p ( T ) for p = 2. We expect thatcounterexamples to the local-global principle exist over Q or F p ( T ) for p = 2 in anydegree n ≥ m ≥ n ≥
4. Finding counterexamples to the local-global principle in higher degree seems a challenging computational problem becausean algorithm to calculate the associated mod 2 Galois representation of smooth planecurves of higher degree is yet to be developed. (See [6], [5] for recent results on explicitcalculations for smooth plane quartics.)
Acknowledgements.
The work of the first author was supported by JSPS KAK-ENHI Grant Number 13J01450. The work of the second author was supported byJSPS KAKENHI Grant Number 20674001 and 26800013. Some calculations done by
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OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 19
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
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