The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two
aa r X i v : . [ m a t h . N T ] S e p THE LOCAL-GLOBAL PRINCIPLE FOR SYMMETRICDETERMINANTAL REPRESENTATIONS OF SMOOTH PLANECURVES IN CHARACTERISTIC TWO
YASUHIRO ISHITSUKA AND TETSUSHI ITO
Abstract.
We give an application of Mumford’s theory of canonical theta charac-teristics to a Diophantine problem in characteristic two. We prove that a smoothplane curve over a global field of characteristic two is defined by the determinantof a symmetric matrix with entries in linear forms in three variables if and onlyif such a symmetric determinantal representation exists everywhere locally. It is aspecial feature in characteristic two because analogous results are not true in othercharacteristics. Introduction
Let C ⊂ P be a smooth plane curve of degree d ≥ K . The planecurve C is said to admit a symmetric determinantal representation over K if thereis a symmetric matrix M of size d with entries in K -linear forms in three variables X, Y, Z such that C is defined by the equation det( M ) = 0. Two symmetric determi-nantal representations M, M ′ of C are said to be equivalent if M ′ = λ t SM S for some S ∈ GL d ( K ) and λ ∈ K × , where t S is the transpose of S . Studying symmetric deter-minantal representations is a classical topic in algebraic geometry and linear algebra,which goes back to Hesse’s work on plane cubics and quartics; [7], [8, Ch 4]. Recently,arithmetic properties of symmetric determinantal representations and related linearorbits are studied by several people; [12], [3], [16], [17].In this paper, we prove the local-global principle for the existence of symmetricdeterminantal representations in characteristic two. Theorem 1.1 (see Theorem 5.1) . Let K be a global field of characteristic two , and C ⊂ P a smooth plane curve of degree d ≥ K . Then the following areequivalent:(1) C admits a symmetric determinantal representation over K .(2) C admits a symmetric determinantal representation over K v for every place v of K .In our previous paper [17], we studied the local-global principle for the existenceof symmetric determinantal representations over a global field of characteristic = 2.We proved the local-global principle for conics and cubics; see [17, Theorem 1.1].For quartics, we constructed examples failing the local-global principle under mildconditions on the characteristic of the global field; see [17, Corollary 1.3]. According Date : September 24, 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, characteristic two. to some group theoretic results in [17, Section 6], we expect that similar examplesfailing the local-global principle exist in any degree ≥ = 2; see[17, Problem 1.6].In this paper, we study the remaining case of characteristic two . Rather interest-ingly, the story is completely different. The local-global principle for the existenceof symmetric determinantal representations holds in any degree in characteristic two.The key is Mumford’s theory of canonical theta characteristics on smooth projectivecurves which exist only in characteristic two.The outline of this paper is as follows. After recalling a relation between sym-metric determinantal representations and non-effective theta characteristics in Section2, we recall Mumford’s theory of canonical theta characteristics in Section 3. Thenwe examine the case of conics and cubics in Section 4. For conics and cubics, theexistence of symmetric determinantal representations is related to other (more famil-iar) Diophantine problems. We also give some examples of symmetric determinantalrepresentations for conics and cubics. Then the main theorem is proved in Section5. Finally, in Section 6, we consider an analogous problem for linear determinantalrepresentations (i.e. the matrix M is not assumed to be symmetric). It turns outthat, concerning the existence of linear determinantal representations, the local-globalprinciple does not hold even for cubics. Hence Theorem 1.1 cannot be generalized tothe linear case. Notation.
An algebraic closure of a field K is denoted by K , and a separable closureof K is denoted by K sep . A global field of characteristic two is a finite extension of F ( T ), where F is the finite field with two elements and T is an indeterminate. For aplace v of K , the completion of K at v is denoted by K v .2. Symmetric determinantal representations and thetacharacteristics
In this section, let K be an arbitrary field. Definition 2.1.
Let C be a projective smooth geometrically connected curve over K .A theta characteristic on C is a line bundle L satisfying L ⊗ L ∼ = Ω C , where Ω C is thecanonical sheaf on C . A theta characteristic L on C is effective (resp. non-effective )if H ( C, L ) = 0 (resp. H ( C, L ) = 0).The following result is classical and well-known at least when the base field is al-gebraically closed of characteristic zero; see [7], [8, Ch 4]. In fact, it is valid overarbitrary fields. Proposition 2.2 (Beauville) . Let C ⊂ P be a smooth plane curve over K . There isa natural bijection between the following sets: • the set of equivalence classes of symmetric determinantal representations of C over K , and • the set of isomorphism classes of non-effective theta characteristics on C definedover K . Proof . See [2, Proposition 4.2]. See also [16, Proposition 2.2, Corollary 2.3], [17,Theorem 2.2], (cid:3)
OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 3 Canonical theta characteristics in characteristic two
In this section, let K be a field of characteristic two .In his foundational paper on theta characteristics, Mumford observed the following“strange” fact; see [20, p.191]. Let C be a projective smooth geometrically connectedcurve over K , and K ( C ) the function field of C . Take a rational function f ∈ K ( C )which is not a square in K ( C ). An easy local calculation shows that the order ofthe divisor div( df ) at every geometric point x ∈ C ( K ) is an even integer. Hencediv( df ) is always divisible by 2 as a divisor on C ⊗ K K . The associated line bundle L can := O C ⊗ K K (cid:0) div( df ) (cid:1) is called the canonical theta characteristic on C ⊗ K K . Theisomorphism class of L can does not depend on the choice of f .We are interested in non-effective theta characteristics (cf. Proposition 2.2). Usingthe Cartier operator and logarithmic differential forms, St¨ohr and Voloch proved thefollowing beautiful results. Theorem 3.1 (St¨ohr-Voloch) . (1) Any theta characteristic on C ⊗ K K not iso-morphic to the canonical theta characteristic is effective.(2) The canonical theta characteristic L can is non-effective if and only if the Jaco-bian variety Jac( C ) is ordinary. Proof . See [25, § (cid:3) Combining Proposition 2.2 and Theorem 3.1, we have the following corollary.
Corollary 3.2.
Let K be a field of characteristic two, and C ⊂ P a smooth planecurve of degree d ≥ K .(1) There is at most one equivalence class of symmetric determinantal representa-tions of C over K .(2) C admits a symmetric determinantal representation over K if and only ifJac( C ) is ordinary.(3) C admits a symmetric determinantal representation over K if and only ifJac( C ) is ordinary and the canonical theta characteristic L can is defined over K . Remark 3.3.
The canonical theta characteristic L can is defined as a line bundleon C ⊗ K K , not on C . Since “dividing the divisor div( df ) by 2” is problematic incharacteristic two, L can is not defined over K in general. For conics and cubics,we can give necessary and sufficient conditions for L can to be defined over K ; seeProposition 4.1 and Proposition 4.4. Remark 3.4.
By Corollary 3.2, in characteristic two, smooth plane curves with non-ordinary Jacobian varieties do not admit symmetric determinantal representationseven over an algebraic closure of the base field. In contrast, in characteristic zero, allplane curves (including singular ones) admit symmetric determinantal representationsover an algebraic closure of the base field; see [7], [2, Remark 4.4].4.
Examples: conics and cubics
In order to illustrate the problem of finding symmetric determinantal representationsin characteristic two, we examine the case of smooth plane conics and cubics in somedetail. It turns out that the existence of the canonical theta characteristic is relatedto other Diophantine problems. For analogous results in characteristic = 2, see [17,Section 4]. YASUHIRO ISHITSUKA AND TETSUSHI ITO
Conics.Proposition 4.1.
Let K be a field of characteristic two, and C ⊂ P a smooth planeconic over K . The following are equivalent:(1) The canonical theta characteristic L can is defined over K .(2) C admits a symmetric determinantal representation over K .(3) C is isomorphic to P over K .(4) C has a K -rational point.(5) C has a K -rational divisor of odd degree. Proof . The equivalences (2) ⇔ (3) ⇔ (4) ⇔ (5) are proved in [17, Proposition 4.1].Since the genus of C is zero, Jac( C ) is ordinary. Hence the equivalence (1) ⇔ (2)follows from Corollary 3.2 (3). We can prove it directly without using the results ofSt¨ohr-Voloch as follows. (1) ⇒ (5): If L can is defined over K , we have deg L can = deg Ω C = −
1. (4) ⇒ (1): For a K -rational point P ∈ C ( K ), the line bundle L = O C ( − P ) is the canonical theta characteristic defined over K because it is a linebundle of degree −
1; see [17, Proposition 3.3]. (cid:3)
Example 4.2.
Let B be a quaternion division algebra over a field K of characteristictwo. Such a B exists when K is a global or local field. Let C B be the Severi-Brauervariety associated with B ; see [23, Ch X, § C B has no K -rational point, C B does not admit a symmetric determinantal representation over K by Proposition4.1. The conic C B admits a symmetric determinantal representation over a separable quadratic extension of K because C B has a rational point over a separable quadraticextension by Bertini’s theorem; see [19, Corollaire 6.11 (2)]. (We can use Bertini’stheorem because K is infinite. The Brauer group of a finite field is trivial; see [23, ChX, § C B also admits a symmetric determinantal representation over a purely inseparable quadratic extension of K . To see this, write the defining equationof C B ⊂ P as aX + bY + cZ + dXY + eY Z + f XZ = 0 ( a, b, c, d, e, f ∈ K ) . Since C B is smooth, at least one of d, e, f is not zero. We may assume d = 0 bychanging coordinates. Since C B has no K -rational point, [1 : 0 : 0] does not lie on C B . Hence a = 0. We put t := a − ( bf + cd + ef d ). By direct calculation, the point[ √ t : f : d ] lies on C B . By Proposition 4.1, C B admits a symmetric determinantalrepresentation over K ( √ t ), which is a purely inseparable quadratic extension of K . Example 4.3.
In this example, let K be a field of arbitrary characteristic . Anysmooth plane conic C ⊂ P over K admits at most one equivalence class of symmet-ric determinantal representations over K because Pic( C ) has no non-trivial 2-torsionelement. (In fact, we have Pic( C ) ⊂ Pic( C ⊗ K K ) ∼ = Z ; see [17, Section 4.2].) When C admits one, it is possible to calculate a symmetric determinantal representationexplicitly as follows. Let aX + bY + cZ + dXY + eY Z + f XZ = 0 ( a, b, c, d, e, f ∈ K )be the defining equation of C . By Proposition 4.1, C has a K -rational point. By alinear change of variables, we may assume that [1 : 0 : 0] is a K -rational point on C ,and the tangent line to C at P is ( Z = 0). Then we have a = d = 0 and f = 0. Sincethe conic C intersects with the line ( Z = 0) at [1 : 0 : 0] with multiplicity two, [0 : 1 : 0] OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 5 does not lie on C . Hence we have b = 0. A symmetric determinantal representationof C over K is given by the symmetric matrix M = (cid:18) Z bYbY − bf X − beY − bcZ (cid:19) because det( M ) = − b · ( bY + cZ + eY Z + f XZ ).4.2. Cubics.
By Corollary 3.2 (3), a smooth plane cubic in characteristic two doesnot admit a symmetric determinantal representation when its Jacobian variety is su-persingular. We shall only consider the ordinary case.
Proposition 4.4.
Let K be a field of characteristic two, and C ⊂ P a smooth planecubic over K . Assume that the Jacobian variety Jac( C ) is ordinary. The following areequivalent:(1) The canonical theta characteristic L can is defined over K .(2) C admits a symmetric determinantal representation over K .(3) Jac( C ) has a non-trivial K -rational 2-torsion point. Proof . (1) ⇔ (2) follows from Corollary 3.2 (3). (2) ⇔ (3) is proved in [17, Proposition4.2]. (cid:3) Remark 4.5.
Since Jac( C ) is an ordinary elliptic curve in characteristic two, its j -invariant is not zero. We may assume that the Weierstrass equation of Jac( C ) is givenby the following form: Y Z + XY Z = X + a X Z + a Z ( a , a ∈ K, a = 0) . See [24, Appendix A]. A direct calculation using [24, Ch III, Group Law Algorithm 2.3]shows that the non-trivial K -rational 2-torsion point on Jac( C ) is [0 : √ a : 1]. Hencethe conditions in Proposition 4.4 are satisfied if and only if √ a ∈ K . Therefore, if C does not admit a symmetric determinantal representation over K , it does not admitone over K sep . But, it admits one over K ( √ a ), which is purely inseparable over K .Compare these results with the results in Example 4.2. Remark 4.6.
In principle, it should be possible to calculate symmetric determinan-tal representations of smooth plane cubics corresponding to non-trivial K -rational2-torsion points. An explicit calculation for general cubics seems a computationallyhard problem. The situation becomes much simpler when C has a K -rational point. Inthat case, there is no obstruction accounted by the Brauer group Br( K ) (or the relativeBrauer group Br( C/K )). In [14], [15], the first author gave an algorithm to calculatelinear (i.e. not necessarily symmetric) determinantal representations of smooth planecubics with rational points.
Example 4.7.
The following example over F is calculated in [14, Table 6]: M = Y X Z YX Y X + Y + Z The cubic C : det( M ) = X Z + XY Z + Y + Y Z + Y Z = 0 YASUHIRO ISHITSUKA AND TETSUSHI ITO is a unique smooth plane cubic over F , up to projective equivalence, admitting onlyone equivalence class of linear (i.e. not necessarily symmetric) determinantal represen-tations over F . Similar cubics exist over F , F , F . But there are no such cubics over F q ( q ≥ Example 4.8.
Here we give examples of symmetric determinantal representations oftwisted (or generalized) Hesse cubics C a,b,c,m : aX + bY + cZ + mXY Z = 0 ( a, b, c, m ∈ K, abc ( m + 27 abc ) = 0)over a field K of characteristic two . These curves were studied by Desboves in 19thcentury; [6], [5, p.130]. A formula given by Artin, Rodriguez-Villegas and Tate ([1,(1.5), (1.6)]) shows that the Jacobian variety Jac( C a,b,c,m ) is defined by the followingWeierstrass equation:Jac( C a,b,c,m ) : Y Z + mXY Z + 9 abcY Z = X + ( − a b c + m abc ) Z . Then Jac( C a,b,c,m ) is ordinary if and only if m = 0. Using [24, Ch III, Group LawAlgorithm 2.3], it is not difficult to show that Jac( C a,b,c,m ) has a non-trivial K -rational2-torsion point if and only if √ m − abc ∈ K . When m − abc is a square in K , theunique equivalence class of symmetric determinantal representations of C a,b,c,m over K is represented by M a,b,c,m = aX √ m − abc Z √ m − abc Y √ m − abc Z bY √ m − abc X √ m − abc Y √ m − abc X cZ . In fact, we can directly checkdet( M a,b,c,m ) = m − abc ( aX + bY + cZ + mXY Z ) . Details on the calculation of symmetric and linear determinantal representations oftwisted Hesse cubics will appear elsewhere. Some examples over Q are given in [15,Section 6] for m = 0 (i.e. the case of “twisted Fermat cubics”).5. Proof of the main theorem
The following Theorem 5.1 is stronger than Theorem 1.1.For conics and cubics, Theorem 5.1 can be proved directly using Proposition 4.1 andProposition 4.4. However, it seems difficult to generalize these propositions because,as you see from the proof of Theorem 5.1 below, giving such results amount to giving adefining equation of the “locus of canonical theta characteristics” in the Picard schemeand a splitting of a 2-torsion element in the Brauer group. It seems an infeasible taskin higher degree. Instead, our approach to prove Theorem 5.1 is to use Greenberg’sapproximation theorem ([9, Theorem 1]) and some cohomological results on Brauergroups.
Theorem 5.1.
Let K be a global field of characteristic two, and C ⊂ P a smoothplane curve of degree d ≥
1. Assume that C admits a symmetric determinantalrepresentation over K v for a place v of K . Moreover, assume that at least one of thefollowing conditions is satisfied: • C has a K -rational point, • d is odd, or OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 7 • there is a place w of K such that C admits a symmetric determinantal repre-sentation over K v for every place v = w of K .Then C admits a symmetric determinantal representation over K . Proof . Since C admits a symmetric determinantal representation over K v , by Corol-lary 3.2 (3), Jac( C ) is ordinary and L can is defined over K v . Hence it is enough toprove that L can is defined over K .We shall first prove that L can gives a K -rational point on the relative Picard schemePic C/K ([4, Ch 8], [17, Section 3]). The canonical sheaf Ω C gives a K -rational point[Ω C ] ∈ Pic
C/K ( K ). Let T ⊂ Pic
C/K be the fiber over [Ω C ] under the multiplication-by-2 map [2] : Pic C/K −→ Pic
C/K . The finite K -scheme T is called the theta characteristic torsor ([22]). Since L can isdefined over K v , we have a K v -rational point [ L can ] ∈ T ( K v ). Let O v be the ringof integers of K v . Then R := O v ∩ K is an excellent discrete valuation ring whosecompletion is O v because it is the localization of a finitely generated F -algebra; see[10, (7.8.3) (ii),(iii)]. Let R h be the Henselization of R . Let T be the integral closure of R in T . Then T is a finite flat R -scheme with generic fiber T . By the valuative criterionof properness, the K v -rational point [ L can ] ∈ T ( K v ) extends to an O v -rational point x ∈ T ( O v ).By Greenberg’s approximation theorem ([9, Theorem 1]), for any N ≥
1, there isan R h -rational point y N ∈ T ( R h ) such that x, y N have the same image in T ( R/m N ),where m is the maximal ideal of R . Since T is a finite R -scheme, if we take N tobe sufficiently large, the image of y N in T ( O v ) coincides with x . Therefore, x is an R h -rational point, and [ L can ] is a (Frac R h )-rational point.Since the extension (Frac R h ) /K is separable and algebraic, [ L can ] is a K sep -rationalpoint. Moreover, it is easy to see from the construction of L can that [ L can ] ∈ Pic
C/K ( K sep )is fixed by the action of Gal( K sep /K ). Hence it is a K -rational point, i.e. we have[ L can ] ∈ Pic
C/K ( K ) . We have the exact sequence of low-degree terms coming from the Leray spectralsequence (cf. [4, Ch 8], [17, Section 3]):0 / / Pic( C ) / / Pic
C/K ( K ) / / Br( K ) / / Br( C ) . It remains to prove that [ L can ] ∈ Pic
C/K ( K ) comes from a line bundle on C . We shallprove it using one of the assumptions of the theorem. Let α ∈ Br( K ) be the image of[ L can ] in the Brauer group Br( K ). • If C has a K -rational point, we have Pic( C ) ∼ = Pic C/K ( K ) ([4, Ch 8, Section1, Proposition 4]). Hence [ L can ] comes from a line bundle on C . • If d is odd, there is a finite extension L/K of odd degree such that C has an L -rational point. By a standard cohomological argument using Res and Cor(cf. [23, Ch VII, Proposition 6]), we see that α is killed by [ L : K ]. On theother hand, since 2[ L can ] = [ L can ⊗ L can ] = [Ω C ] comes from the canonicalsheaf on C , we see that α is also killed by 2. Hence α is trivial, and [ L can ]comes from a line bundle on C . YASUHIRO ISHITSUKA AND TETSUSHI ITO • If C admits a symmetric determinantal representation over K v for every place v = w of K , the element α lies in the kernel of the homomorphism ψ : Br( K ) −→ M v : place of K, v = w Br( K v ) . The exact sequence0 −→ Br( K ) −→ M v : place of K Br( K v ) P v inv v −→ Q / Z −→ § § ψ isinjective. Hence α is trivial, and [ L can ] comes from a line bundle on C .The proof of Theorem 5.1 is complete. (cid:3) Remark 5.2.
In higher degree, we do not know how to explicitly calculate a symmetricdeterminantal representation of a smooth plane curve once we know it exists. By [2,Proposition 4.2], it amounts to giving an explicit minimal free resolution of the gradedmodule associated with ι ∗ L can , where ι : C ֒ → P is a fixed embedding.6. The case of linear determinantal representations
It is a natural question to study determinantal representations which are not nec-essarily symmetric. A smooth plane curve C ⊂ P of degree d over a field K is said toadmit a linear determinantal representation over K if there is a square (not necessarilysymmetric) matrix M of size d with entries in K -linear forms in three variables X, Y, Z such that det( M ) = 0 is the defining equation of C .It turns out that, concerning the existence of linear determinantal representations,the local-global principle holds for d = 2; see [13, Corollary 4.2]. But, it does not holdfor d = 3. The following result is proved by the first author. Proposition 6.1.
Let K be a global field of arbitrary characteristic, and C ⊂ P a smooth plane cubic over K . Assume that the Mordell-Weil group Jac( C )( K ) istrivial. Then C does not admit a linear determinantal representation over K , whereas C admits a linear determinantal representation over K v for every place v of K . Proof . See [13, Corollary 4.2]. (cid:3)
It is a simple matter to find smooth plane cubics whose Jacobian varieties havetrivial Mordell-Weil group. For example, the following elliptic curves E : Y Z + T Y Z = X + T Z E : Y Z + T XY Z = X + T Z over F ( T ) have trivial Mordell-Weil group. These are taken from [18, Table 3 in p.182].By Proposition 6.1, E and E do not admit linear determinantal representations over F ( T ), but they admit linear determinantal representations everywhere locally.Since Proposition 6.1 is valid over any global field, we may find similar examplesin other characteristics. In [13], the proportion of smooth plane cubics over Q failingthe local-global principle for the existence of linear determinantal representations isstudied. OCAL-GLOBAL PRINCIPLE FOR SYMMETRIC DETERMINANTAL REPRESENTATIONS 9
Acknowledgements.
When we studied the canonical theta characteristics on alge-braic curves in characteristic two, the posts at mathoverflow were very helpful (
Ef-fectiveness of the distinguished theta characteristic in characteristic 2 ( http://mathoverflow.net/questions/139698/ )). We would like to take this occa-sion to thank the contributors to this wonderful website. The work of the first authorwas supported by JSPS KAKENHI Grant Number 13J01450 and 16K17572. The workof the second author was supported by JSPS KAKENHI Grant Number 20674001 and26800013. References [1] Artin, M., Rodriguez-Villegas, F., Tate, J.,
On the Jacobians of plane cubics , Adv. Math. (2005), no. 1, 366-382.[2] Beauville, A.,
Determinantal hypersurfaces , Michigan Math. J. 48 (2000), 39-64.[3] Bhargava, M., Gross, B. H., Wang, X.,
Arithmetic invariant theory II: Pure inner forms andobstructions to the existence of orbits , Representations of Lie Groups, in Honor of the 60thBirthday of David A. Vogan, Jr., Progress in Mathematics, 312. Birkh¨auser/Springer, 2015,139-171.[4] Bosch, S., L¨utkebohmert, W., Raynaud, M.,
N´eron models , Ergebnisse der Mathematik undihrer Grenzgebiete (3), 21. Springer-Verlag, Berlin, 1990.[5] Cassels, J. W. S.,
Lectures on elliptic curves , London Mathematical Society Student Texts, 24.Cambridge University Press, Cambridge, 1991.[6] Desboves, A.,
R´esolution en nombres entiers et sous sa forme de la plus g´en´erale, de l’´equationcubique, homog`ene `a trois inconnues , Nouv. Ann. de la Math,. S´er. III, vol. (1886), 545-579.[7] Dixon, A. C., Note on the reduction of a ternary quantic to a symmetric determinant , Proc.Cambridge Phil. Soc. 11 (1902), 350-351.[8] Dolgachev, I. V.,
Classical algebraic geometry - A modern view , Cambridge University Press,Cambridge, 2012.[9] Greenberg, M. J.,
Rational points in Henselian discrete valuation rings , Inst. Hautes ´Etudes Sci.Publ. Math. No. 31 (1966), 59-64.[10] Grothendieck, A. ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et des mor-phismes de sch´emas. II. , Inst. Hautes ´Etudes Sci. Publ. Math. No. 24 (1965), 231 pp.[11] Grothendieck, A.,
Le groupe de Brauer III , Dix Expos´es sur la Cohomologie des Sch´emas, 88-188North-Holland, Amsterdam; Masson, Paris, 1968.[12] Ho, W.,
Orbit parametrizations of curves , Ph.D Thesis, Princeton University, 2009.[13] Ishitsuka, Y.,
A positive proportion of cubic curves over Q admit linear determinantal represen-tations , preprint, arXiv:1512.05167 [14] Ishitsuka, Y., Linear determinantal representations of smooth plane cubics over finite fields ,preprint, arXiv:1604.00115 [15] Ishitsuka, Y.,
An algorithm to obtain linear determinantal representations of smooth plane cubicsover finite fields , preprint, to appear in JSIAM Letters, arXiv:1605.06628 [16] Ishitsuka, Y., Ito, T.,
On the symmetric determinantal representations of the Fermat curves ofprime degree , Int. J. Number Theory (2016), 955-967.[17] Ishitsuka, Y., Ito, T., The local-global principle for symmetric determinantal representations ofsmooth plane curves , published online in The Ramanujan Journal.DOI: http://dx.doi.org/10.1007/s11139-016-9775-3[18] Ito, H.,
On extremal elliptic surfaces in characteristic 2 and 3 , Hiroshima Math. J. 32 (2002),no. 2, 179-188.[19] Jouanolou, J.-P.,
Th´eor`emes de Bertini et applications , Progress in Mathematics, 42. Birkh¨auserBoston, Inc., Boston, MA, 1983.[20] Mumford, D.,
Theta characteristics of an algebraic curve , Ann. Sci. ´Ecole Norm. Sup. (4) 4(1971), 181-192.[21] Neukirch, J., Schmidt, A., Wingberg, K.,
Cohomology of number fields , Second edition.Grundlehren der Mathematischen Wissenschaften, 323. Springer-Verlag, Berlin, 2008. [22] Poonen, B., Rains, E.,
Self cup products and the theta characteristic torsor , Math. Res. Lett. 18(2011), no. 6, 1305-1318.[23] Serre, J.-P.,
Local fields , Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin,1979.[24] Silverman, J. H.,
The arithmetic of elliptic curves , Second edition. Graduate Texts in Mathe-matics, 106. Springer, Dordrecht, 2009.[25] St¨ohr, K.-O., Voloch, J. F.,
A formula for the Cartier operator on plane algebraic curves , J.Reine Angew. Math. 377 (1987), 49-64.[26] Tate, J. T.,
Global class field theory , in Algebraic Number Theory (J. W. S. Cassels and A.Fr¨ohlich eds.), Proc. Instructional Conf., Brighton, 1965, 162-203.
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
E-mail address : [email protected] Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
E-mail address ::