On the symmetric determinantal representations of the Fermat curves of prime degree
aa r X i v : . [ m a t h . N T ] J u l ON THE SYMMETRIC DETERMINANTAL REPRESENTATIONSOF THE FERMAT CURVES OF PRIME DEGREE
YASUHIRO ISHITSUKA AND TETSUSHI ITO
Abstract.
We prove that the defining equations of the Fermat curves of primedegree cannot be written as the determinant of symmetric matrices with entries inlinear forms in three variables with rational coefficients. In the proof, we use arelation between symmetric matrices with entries in linear forms and non-effectivetheta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curvesof prime degree. Introduction
Let us consider the following Diophantine problem motivated by the ArithmeticInvariant Theory of symmetrized 3 × n × n boxes ([3], [14]): for a smooth plane curve C ⊂ P K of degree n ≥ K , does there exist a triple of symmetricmatrices ( M , M , M ) of size n with entries in K such that C is defined by the equationof the form det (cid:0) X M + X M + X M (cid:1) = 0 ?If C is defined by the above equation, we say C admits a symmetric determinantalrepresentation over K . We say two symmetric determinantal representations of C defined by triples ( M , M , M ), ( M ′ , M ′ , M ′ ) are equivalent if there are P ∈ GL n ( K )and a ∈ K × with M ′ i = a t P M i P for i = 0 , ,
2, where t P is the transpose of the matrix P . Note that equivalent triples give the same plane curve because we havedet (cid:0) X M ′ + X M ′ + X M ′ (cid:1) = det (cid:0) X ( a t P M P ) + X ( a t P M P ) + X ( a t P M P ) (cid:1) = a n · (det P ) · det (cid:0) X M + X M + X M (cid:1) . Finding symmetric determinantal representations of plane curves is a classical prob-lem in algebraic geometry. For the history of this problem and known results, see[6], [8], [5], [28], [2], [7, Ch 4]. When K is algebraically closed of characteristic zero,all plane curves (including singular ones) admit symmetric determinantal representa-tions ([2, Remark 4.4]). When K is not algebraically closed, many plane curves donot admit symmetric determinantal representations over K . In [16], [17], we studiedthe local-global principle for the existence of symmetric determinantal representationsover global fields.In this paper, we study symmetric determinantal representations of the Fermatcurves of prime degree and the Klein quartic over the field Q of rational numbers. Date : July 28, 2018.2010
Mathematics Subject Classification.
Primary 11D41; Secondary 14H50, 14K15, 14K30.
Key words and phrases. plane curve, Fermat curves, determinantal representation, thetacharacteristic.
We prove that the Fermat curves of prime degree do not admit symmetric determi-nantal representations over Q . Theorem 1.1.
Let p ≥ pF p := (cid:0) X p + X p + X p = 0 (cid:1) ⊂ P Q does not admit a symmetric determinantal representation over Q . Remark 1.2.
Theorem 1.1 can be rephrased in concrete terms as follows: there do not exist symmetric matrices M , M , M of size p with entries in Q and a ∈ Q × satisfying X p + X p + X p = a · det (cid:0) X M + X M + X M (cid:1) . Since it is a Diophantine problem with 3 p + 1 variables, it seems difficult to provethe non-existence of solutions in Q directly. We shall prove Theorem 1.1 using themethods and the results from algebraic geometry. Remark 1.3.
The Fermat curves are sometimes defined by the equation X p + X p = X p instead of X p + X p + X p = 0. There is no essential difference when p is odd. Butthere is a difference when p = 2. In fact, the smooth conic ( X + X = X ) admits asymmetric determinantal representation over Q . (See Remark 2.7.)The strategy of the proof of Theorem 1.1 is as follows. The case of p = 2 iseasy and treated separately. Let p ≥ C ⊂ P Q over Q , triples of symmetric matrices giving rise to symmetricdeterminantal representations correspond to certain line bundles on C called non-effective theta characteristics (Proposition 2.2). Therefore, in order to prove Theorem1.1, we have to prove the non-existence of non-effective theta characteristics on F p over Q . If L is a non-effective theta characteristic on F p over Q , the line bundle L ⊗ O F p (cid:0) ( − p + 3) / (cid:1) gives a non-trivial Q -rational 2-torsion point on the Jacobianvariety Jac( F p ). For an integer s with 1 ≤ s ≤ p −
2, let C s be the projective smoothmodel of the affine curve V p = U (1 − U ) s . Gross-Rohrlich calculated the Q -rational torsion points on Jac( C s ) ([12]). There is anisogeny Y ≤ s ≤ p − Jac( C s ) −→ Jac( F p )defined over Q whose degree is a power of p . We can calculate the Q -rational 2-torsionpoints on Jac( F p ) (Corollary 3.4). When p = 7, we have the non-existence of non-effective theta characteristics on F p over Q . The case of p = 7 requires a little carebecause there are three non-trivial Q -rational 2-torsion points on Jac( F ). In fact, wehave Jac( F )[2]( Q ) ∼ = ( Z / Z ) . Fortunately, by explicit calculation, we can prove these 2-torsion points correspond to effective theta characteristics on F over Q . Hence non-effective theta characteristicson F over Q do not exist. (The calculation of Q -rational p -torsion points on Jac( F p )is more subtle ([24], [25], [26], [27]). We do not need these results in this paper.)Along the proof of Theorem 1.1, we also study symmetric determinantal represen-tations of the Klein quartic C Kl := (cid:0) X X + X X + X X = 0 (cid:1) ⊂ P Q YMMETRIC DETERMINANTAL REPRESENTATIONS OF THE FERMAT CURVES 3 over Q . It was already known to Klein that C Kl admits a symmetric determinantalrepresentation over Q . In fact, it is easy to confirm the following equality (cf. [9,p. 161]):(1.1) X X + X X + X X = − det X − X X − X X − X − X − X − X We prove that the above expression gives a unique equivalence class of symmetricdeterminantal representations of C Kl over Q . Theorem 1.4.
There is a unique equivalence class of symmetric determinantal rep-resentations of the Klein quartic C Kl over Q .The proof of Theorem 1.4 is as follows. It is classically known that C Kl is birationalover Q to the curve C (or C ) for p = 7 studied by Gross-Rohrlich ([10, p. 67]). Usingthe results of Gross-Rohrlich, we see thatJac( C Kl )[2]( Q ) ∼ = Z / Z . Hence there are exactly two theta characteristics on C Kl over Q , up to isomorphism.There is at least one effective theta characteristic on C Kl because the line ( X + X + X = 0) is a bitangent to C Kl over Q , which corresponds to an effective thetacharacteristic on C Kl over Q ([7, Ch. 6]). Hence there is exactly one non-effective thetacharacteristic on C Kl over Q , up to isomorphism. Remark 1.5.
There is a long history on the study of the Klein quartic. There isno surprise if some geometers have expected (or presumed) that Theorem 1.4 couldbe true. Note that a rigorous proof of Theorem 1.4 requires the study of the fieldof definition of equivalence classes of symmetric determinantal representations, whichis a slightly delicate arithmetic problem. (See Corollary 2.3 (4). See also [14], [15].)These days, we become more interested in the arithmetic properties of linear orbitsrelated to symmetric determinantal representations thanks to the recent developmentsof Arithmetic Invariant Theory ([14], [3]).In Section 2, we recall a relation between symmetric determinantal representationsand non-effective theta characteristics. Theorem 1.1 for p = 2 is a consequence of thefact that the conic ( X + X + X = 0) has no Q -rational points. In Section 3, werecall some results of Gross-Rohrlich, and prove Theorem 1.1 for p = 2 ,
7. The proofof Theorem 1.1 for p = 7 is given in Section 4. Finally, the proof of Theorem 1.4 isgiven in Section 5.2. Theta characteristics and symmetric determinantalrepresentations
Let K be a field, and C ⊂ P K a smooth plane curve of degree n ≥
1. The genus of C is equal to g ( C ) := ( n − n − / Definition 2.1 ([20]) . (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 .(2) A theta characteristic L on C is effective (resp. non-effective ) if H ( C, L ) = 0(resp. H ( C, L ) = 0). YASUHIRO ISHITSUKA AND TETSUSHI ITO
Proposition 2.2.
There is a bijection between the set of isomorphism classes of non-effective theta characteristics on C and the set of equivalence classes of symmetricdeterminantal representations of C over K . Proof . This proposition is well-known when char K = 2 ([1, Proposition 6.23], [2,Proposition 4.2], [7, Ch 4], [14, Theorem 4.12]). It is not difficult to modify thearguments in [2] to cover the case of characteristic two ([2, Remark 2.2]). For a proofof this proposition which works over arbitrary fields, see also [15]. (cid:3) In order to study the field of definition of equivalence classes of symmetric deter-minantal representations, we use the Picard scheme and the Jacobian variety of thesmooth plane curve C ([4], [18]). Let us recall their basic properties. The Picard group
Pic( C ) := H ( C, O × C )is the group of isomorphism classes of line bundles on C . Let Pic C/K be the
Picardscheme of C representing the relative Picard functor ([4, Theorem 3 in § / / Pic( C ) / / Pic
C/K ( K ) / / Br( K ) , where Br( K ) is the Brauer group of K .The equality Pic( C ) = Pic C/K ( K ) holds if C has a K -rational point ([4, Proposition4 in § C/K is denoted by Jac( C ) called the Jacobianvariety of C . It is known that Jac( C ) is an abelian variety over K of dimension g ( C ) = ( n − n − / § C )[2] be the group of 2-torsion points on Pic( C ), which is the group ofisomorphism classes of line bundles L with L ⊗ L ∼ = O C . The group of K -rational2-torsion points on Jac( C ) is denoted by Jac( C )[2]( K ). It is always true that Pic( C )[2]is a subgroup of Jac( C )[2]( K ). These two groups are not necessarily equal if C has no K -rational points.Using Proposition 2.2 and the following corollary, we give an upper bound of thenumber of equivalence classes of symmetric determinantal representations. Corollary 2.3. (1) The number of isomorphism classes of theta characteristics on C is less than or equal to the order of Jac( C )[2]( K ).(2) If C has a K -rational point, the number of isomorphism classes of theta char-acteristics on C is zero or equal to the order of Jac( C )[2]( K ).(3) The number of equivalence classes of symmetric determinantal representationsof C over K is finite.(4) Let L/K be an extension of fields. Two symmetric determinantal representa-tions of C over K are equivalent over K if and only if they are equivalent over L . Proof . (1) Assume that there is a theta characteristic L on C . The other thetacharacteristics on C are of the form L ⊗ L ′ , where L ′ is a line bundle on C with L ′ ⊗ L ′ ∼ = O C . Hence the number of isomorphism classes of theta characteristicson C is equal to the order of Pic( C )[2], which is less than or equal to the order ofJac( C )[2]( K ) because Pic( C )[2] is a subgroup of Jac( C )[2]( K ).(2) If C has a K -rational point, we have Pic( C )[2] = Jac( C )[2]( K ), and the assertion(2) follows. YMMETRIC DETERMINANTAL REPRESENTATIONS OF THE FERMAT CURVES 5 (3) Since Jac( C )[2]( K ) is a finite group, the assertion (3) follows.(4) Since (Pic C/K ) ⊗ K L = Pic C ⊗ K L/L , we have Pic
C/K ( L ) = Pic C ⊗ K L/L ( L ). Themap Pic C/K ( K ) −→ Pic C ⊗ K L/L ( L ) is injective. Hence Pic( C ) −→ Pic( C ⊗ K L ) is alsoinjective, and the assertion (4) follows. (cid:3) Remark 2.4.
If we define the notion of theta characteristics on singular plane curvesas in [21], Proposition 2.2 can be generalized to the case of singular plane curves.There is a natural bijection between equivalence classes of symmetric determinantalrepresentations of C over K and isomorphism classes of non-effective theta charac-teristics L on C equipped with a certain isomorphism between L and the dual of it([15]). However, when C has singularities, Corollary 2.3 (3),(4) are not true in general.(See [15] for details.) Corollary 2.5.
Assume that n ≥ n is odd . If Jac( C )[2]( K ) = 0, the smoothplane curve C does not admit a symmetric determinantal representation over K . Proof . By Corollary 2.3 (1), there is at most one isomorphism class of theta char-acteristics on C . By Proposition 2.2, we have only to show that there is an effective theta characteristic on C . Since C ⊂ P K is a smooth plane curve of degree n , thecanonical sheaf Ω C is isomorphic to the restriction O C ( n −
3) := O P K ( n − | C ([13, II., 8.20.3], [19, Exercise 6.4.11]). Hence O C (cid:0) ( n − / (cid:1) := O P K (cid:0) ( n − / (cid:1) | C is a theta characteristic on C . It is effective because homogeneous polynomials in X , X , X of degree ( n − / (cid:3) In the following proposition, we consider symmetric determinantal representationsof smooth conics. (See also [17, Proposition 5.1])
Proposition 2.6.
Assume that n = 2. For a smooth conic C ⊂ P K over K , thefollowing are equivalent.(1) C is isomorphic to P K over K .(2) C has a K -rational point.(3) C has a line bundle of odd degree over K .(4) C admits a symmetric determinantal representation over K .If the above conditions are satisfied, there is a unique equivalence class of symmetricdeterminantal representations of C over K . Proof . (3) ⇒ (4) Since the smooth conic C is a projective smooth curve of genus 0, wehave Jac( C ) = 0 and Pic( C ) ⊂ Pic
C/K ( K ) ∼ = Z . The isomorphism Pic C/K ( K ) ∼ = −→ Z is given by the degree of line bundles. Since deg Ω C = −
2, Pic( C ) is a subgroup ofPic C/K ( K ) of index less than or equal to 2. If C has a line bundle of odd degreeover K , we have Pic( C ) = Pic C/K ( K ). There is a line bundle L of degree −
1, whichis a non-effective theta characteristic on C over K . By Proposition 2.2, C admits asymmetric determinantal representation over K .(4) ⇒ (3) If C admits a symmetric determinantal representation over K , there is anon-effective theta characteristic L on C by Proposition 2.2. We see that deg L = − YASUHIRO ISHITSUKA AND TETSUSHI ITO (1) ⇔ (2) ⇔ (3) These implications are well-known. We briefly recall the proof.The implications (1) ⇒ (2) ⇒ (3) are obvious. Assume that C has a line bundle ofodd degree. We have Pic( C ) = Pic C/K ( K ) ∼ = Z , and there is a divisor D on C ofdegree 1. Since the complete linear system | D | is one-dimensional and very ample, C is isomorphic to P K over K ([13, IV., 3.3.1], [19, Proposition 7.4.1]).The last assertion follows from Corollary 2.3 (2). (cid:3) Proof of Theorem 1.1 (for p = 2 ). The smooth conic ( X + X + X = 0) has no Q -rational points. It does not admit a symmetric determinantal representation over Q by Proposition 2.6. (cid:3) Remark 2.7.
The Fermat curves are sometimes defined by the equation X p + X p = X p instead of X p + X p + X p = 0. There is no essential difference when p is odd. However,when p = 2, the conic ( X + X = X ) is not isomorphic over Q to the conic ( X + X + X = 0). Since the conic ( X + X = X ) has a Q -rational point such as (1 , , Q by Proposition 2.6. For example, we have X + X − X = − det (cid:18) X + X X X − X + X (cid:19) . Results of Gross-Rohrlich and the non-existence of symmetricdeterminantal representations of the Fermat curve of degree p = 7We recall some results of Gross-Rohrlich on the Q -rational torsion points on theJacobian varieties of the Fermat curve of prime degree ([12]).Let p ≥ F p := (cid:0) X p + X p + X p = 0 (cid:1) ⊂ P Q the Fermat curve of degree p over Q . For an integer s with 1 ≤ s ≤ p −
2, let C s bethe projective smooth model of the affine curve V p = U (1 − U ) s . Theorem 3.1 (Theorem 1.1 in [12]) . Let Jac( C s )( Q ) tors be the group of Q -rationaltorsion points on Jac( C s ). Then we haveJac( C s )( Q ) tors ∼ = ( Z /p Z p = 7 or ( p, s ) = (7 , , (7 , , (7 , Z / Z × Z / Z ( p, s ) = (7 , , (7 , . Let us calculate the Q -rational prime-to- p torsion points on Jac( F p ) using Theorem3.1. There is a finite morphism ϕ s : F p −→ C s of degree p defined by ϕ s ( X , X , X ) = (cid:0) − ( X /X ) p , ( − s +1 X X s /X s +12 (cid:1) . Remark 3.2.
The above expression may be slightly confusing. The coordinates( X , X , X ) of F p are homogeneous coordinates in the projective plane P Q , whereasthe coordinates ( U, V ) of C s are affine coordinates. There is a difference of signs from[12, p. 207] because the defining equation of the Fermat curve in [12] is X p + Y p = 1. YMMETRIC DETERMINANTAL REPRESENTATIONS OF THE FERMAT CURVES 7
The morphism ϕ s induces morphisms between Jacobian varieties ϕ s ∗ : Jac( F p ) −→ Jac( C s ) , ϕ ∗ s : Jac( C s ) −→ Jac( F p )corresponding to the pushfoward and the pullback of divisor classes. Let ϕ ∗ , ϕ ∗ be theproduct of ϕ s ∗ , ϕ ∗ s (1 ≤ s ≤ p − ϕ ∗ induces an isogeny ([11]): ϕ ∗ : Jac( F p ) −→ Y ≤ s ≤ p − Jac( C s ) . Hence ϕ ∗ is an isogeny from Q ≤ s ≤ p − Jac( C s ) to Jac( F p ). Lemma 3.3.
The composite ϕ ∗ ◦ ϕ ∗ : Jac( F p ) −→ Jac( F p ) is equal to the multiplication-by- p isogeny. Proof . This result is presumably well-known ([24, p. 2]). Since it is not stated explic-itly in [12], we briefly sketch how to deduce it from the results in [22], [12]. It is enoughto prove Lemma 3.3 over Q . So we shall work over Q in the following argument. Let ζ ∈ Q be a primitive p -th root of unity. Define the automorphisms A, B of F p by A ( X , X , X ) := ( ζ X , X , X ) , B ( X , X , X ) := ( X , ζ X , X ) . Since ϕ s : F p −→ C s is a Galois covering of degree p whose Galois group is generatedby A − s B , we have ϕ ∗ ◦ ϕ ∗ = p − X s =1 p − X j =0 ( A − s B ) j on Jac( F p ) ([12, p. 208]). We have ϕ ∗ ◦ ϕ ∗ − p = (cid:18) p − X j =0 A j (cid:19)(cid:18) p − X k =0 B k (cid:19) − p − X j =0 A j − p − X k =0 B k − p − X j =0 ( AB ) j . on Jac( F p ). Therefore, we have ϕ ∗ ◦ ϕ ∗ − p = 0 on Jac( F p ) because the operators P p − j =0 A j , P p − k =0 B k , P p − j =0 ( AB ) j annihilate Jac( F p ) by [22, Remark in p. 121]. (cid:3) By Lemma 3.3, we have an isomorphismJac( F p )( Q ) p ′ -tors ∼ = Y ≤ s ≤ p − Jac( C s )( Q ) p ′ -tors , where Jac( F p )( Q ) p ′ -tors (resp. Jac( C s )( Q ) p ′ -tors ) denotes the group of Q -rational torsionpoints on Jac( F p ) (resp. Jac( C s )) whose orders are prime to p . Corollary 3.4.
Jac( F p )( Q ) p ′ -tors ∼ = ( p = 7( Z / Z ) p = 7 Proof of Theorem 1.1 (for p = 2 , ). There does not exist non-trivial Q -rational2-torsion points on Jac( F p ) by Corollary 3.4. Hence F p does not admit a symmetricdeterminantal representation over Q by Corollary 2.5. (cid:3) YASUHIRO ISHITSUKA AND TETSUSHI ITO The non-existence of symmetric determinantal representations ofthe Fermat curve of degree F )[2]( Q ) ∼ = ( Z / Z ) by Corollary 3.4. We shall prove these 2-torsion points correspond to effective thetacharacteristics on F . Proof of Theorem 1.1 (for p = 7 ). We recall the results of Gross-Rohrlich on di-visors on F representing elements of Jac( F )[2]( Q ) ∼ = ( Z / Z ) . (See [12, p. 209, (3)]for details.)Let ε ∈ Q be a primitive 14-th root of unity, and we put ζ := ε . Let η ∈ Q be aprimitive 6-th root of unity. Then η is conjugate to η − over Q . We put P := ( η, η − , − , Q := ( η − , η, − , R j := ( εζ j , ,
0) (0 ≤ j ≤ D of degree 0 on F by D := X j =0 (cid:0) ( A B ) j ( P ) + ( A B ) j ( Q ) − R j (cid:1) , where A, B are the automorphisms of F defined in the proof of Lemma 3.3. Thedivisor D is invariant under the action of Gal( Q / Q ). Hence the line bundle O F ( D ) isdefined over Q .It is straightforward to confirm that the divisor of the rational function f := X − (cid:0) X X + X X − X X (cid:1) on F is equal to 2 D . Hence 2 D is a principal divisor, and D gives a Q -rational2-torsion point [ D ] ∈ Jac( F )[2]( Q ) . Gross-Rohrlich proved that the divisor class [ D ] is non-trivial, andJac( F )[2]( Q ) = (cid:8) , [ D ] , [ σ ( D )] , [ σ ( D )] (cid:9) . Here σ : F −→ F is the automorphism of order 3 defined by σ ( X , X , X ) := ( X , X , X ) . Recall that O F (2) := O P Q (2) | F is an effective theta characteristic on F . (See theproof of Corollary 2.5.) Since F has a Q -rational point such as (1 , − , F by Corollary 2.3 (2). Theyare represented by the following line bundles O F (2) , O F (2) ⊗ O F ( D ) , O F (2) ⊗ O F (cid:0) σ ( D ) (cid:1) , O F (2) ⊗ O F (cid:0) σ ( D ) (cid:1) . By Proposition 2.2, we need to prove that all of these theta characteristics areeffective. We have already seen that O F (2) is effective. Since the automorphism σ permutes the three theta characteristics on F except O F (2), it is enough to show thatone of them is effective.We shall show O F (2) ⊗ O F ( D ) is effective. The line bundle O F (2) ⊗ O F ( D ) iseffective if and only if the divisor D + 2 H is linearly equivalent to an effective divisor,where H is a hyperplane section (i.e. H is a divisor on F cut out by a line in P Q ).Consider the line at infinity ℓ ∞ ⊂ P Q defined by X = 0. The line ℓ ∞ intersects with F at the 7 points R , R , . . . , R . All of the intersection points have multiplicity one. YMMETRIC DETERMINANTAL REPRESENTATIONS OF THE FERMAT CURVES 9 (Since F ⊂ P Q is a plane curve of degree 7, there are exactly 7 intersections countedwith their multiplicities by B´ezout’s theorem ([13, I., 7.8], [19, Corollary 9.1.20]).) Thedivisor ℓ ∞ ∩ F on F cut out by ℓ ∞ is equal to P j =0 R j . Since D + 2( ℓ ∞ ∩ F ) = X j =0 (cid:0) ( A B ) j ( P ) + ( A B ) j ( Q ) (cid:1) is effective, we conclude that O F (2) ⊗ O F ( D ) is an effective theta characteristic on F .The proof of Theorem 1.1 for p = 7 is complete. (cid:3) Symmetric determinantal representations of the Klein quarticover Q The Klein quartic is a smooth plane curve of degree 4 over Q defined by the equation C Kl := (cid:0) X X + X X + X X = 0 (cid:1) ⊂ P Q . In this section, we study symmetric determinantal representations of C Kl over Q . Foran excellent exposition of the arithmetic and the geometry of the Klein quartic, see[10].The following arithmetic results on the Klein quartic must be well-known. Lemma 5.1. (1) Jac( C Kl )[2]( Q ) ∼ = Z / Z .(2) There is an effective theta characteristic on C Kl defined over Q . Proof . (1) It seems possible to work with the defining equation of C Kl directly. Herewe shall deduce it from the results of Gross-Rohrlich ([12]). The Klein quartic C Kl isisomorphic to the projective smooth model C of the affine curve V = U (1 − U ) ([10,p. 67]). This can be seen as follows. We have a rational map C Kl C , ( a, b, c ) ( s, t ) := ( − a b/c , − b/c )Since t = s (1 − s ) is satisfied, ( s, t ) lies on the affine curve V = U (1 − U ) . Conversely,since c/b = − /t, a/b = t / ( s − a, b, c ) arerecovered from ( s, t ) when s = 1 , t = 0. Hence C Kl is birational to C over Q . Since C Kl , C are projective smooth curves over Q , they are isomorphic over Q ([13, I., 6.12],[19, Proposition 7.3.13 (b)]). Hence we have C Kl ∼ = C , andJac( C Kl )[2]( Q ) ∼ = Jac( C )[2]( Q ) ∼ = Z / Z by Theorem 3.1.(2) Let ζ ∈ Q be a primitive cube root of unity. Consider Q -rational points P :=( ζ , ζ ,
1) and Q := ( ζ , ζ ,
1) on C Kl . The line X + X + X = 0is a bitangent to C Kl at P, Q . Since P + Q is a Gal( Q / Q )-invariant divisor on C Kl ⊗ Q Q ,the line bundle O C Kl ( P + Q ) is defined over Q . The canonical sheaf Ω C Kl is isomorphicto the restriction O C Kl (1) := O P Q (1) | C Kl . Since the divisor 2 P + 2 Q on C Kl is cut outby a line, we have O C Kl (2 P + 2 Q ) ∼ = O C Kl (1). Hence O C Kl ( P + Q ) is an effective thetacharacteristic on C Kl defined over Q . (cid:3) Remark 5.2.
There are 28 bitangents to a smooth plane quartic defined over analgebraically closed field of characteristic different from two. There is a bijectionbetween bitangents and isomorphism classes of effective theta characteristics ([7, Ch.6]). The defining equations of the 28 bitangents to the Klein quartic C Kl ⊗ Q Q over Q can be found in [23, Proposition 9]. In fact, Shioda calculated the defining equationsover any algebraically closed fields of characteristic different from 7. One can verifythat X + X + X = 0 is a unique bitangent defined over Q . Proof of Theorem 1.4.
Since C Kl has a Q -rational point such as (1 , , C Kl , up to isomorphism, by Corollary 2.3 (2) andLemma 5.1. Since C Kl admits a symmetric determinantal representation over Q (cf.the equation (1.1) in Introduction), there is a non-effective theta characteristic on C Kl by Proposition 2.2. On the other hand, there is an effective theta characteristic on C Kl by Lemma 5.1 (2). Therefore, there is a unique isomorphism class of non-effective thetacharacteristics on C Kl over Q . The assertion of Theorem 1.4 follows from Proposition2.2. (cid:3) 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. The authors would like tothank an anonymous referee for simplifying the proof of Lemma 3.3.
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Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
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