An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields
aa r X i v : . [ m a t h . N T ] M a y AN ALGORITHM TO OBTAIN LINEAR DETERMINANTALREPRESENTATIONS OF SMOOTH PLANE CUBICS OVER FINITEFIELDS
YASUHIRO ISHITSUKA
Abstract.
We give a brief report on our computations of linear determinantal represen-tations of smooth plane cubics over finite fields. After recalling a classical interpretationof linear determinantal representations as rational points on the affine part of Jacobianvarieties, we give an algorithm to obtain all linear determinantal representations up toequivalence. We also report our recent study on computations of linear determinantal rep-resentations of twisted Fermat cubics defined over the field of rational numbers. This paperis a summary of the author’s talk at the JSIAM JANT workshop on algorithmic numbertheory in March, 2016. Details will appear elsewhere. Introduction
Let k be a field, and let F ( X, Y, Z ) = a X + a X Y + a X Z + a XY + a XY Z + a XZ + a Y + a Y Z + a Y Z + a Z be a ternary cubic form with coefficients in k defining a smooth plane cubic C ⊂ P . Thecubic C is said to admit a linear determinantal representation over k if there are a nonzeroconstant = λ ∈ k and three square matrices M , M , M ∈ Mat ( k ) of size 3 satisfying F ( X, Y, Z ) = λ · det( M ) , where we put M := XM + Y M + ZM . Two linear determinantal representations M, M ′ of C are said to be equivalent if there are invertible matrices A, B ∈ GL ( k ) satisfying M ′ = AM B.
Studying linear determinantal representations of smooth plane cubics is a classical topic inlinear algebra and algebraic geometry ([Vin89], [Dol12]). Recently, they appear in the studyof the derived category of smooth plane cubics ([Gal14]) and the theory of space-time codes([DG08]). They have been studied from arithmetic viewpoints ([FN14], [II16a], [Ish15]).In this note, we study linear determinantal representations over finite fields. We give analgorithm to obtain all linear determinantal representations of smooth plane cubics up toequivalence. This paper is a summary of the author’s talk at the JSIAM JANT workshop onalgorithmic number theory in March, 2016. Details will appear elsewhere.2.
Linear determinantal representations and rational points
Let k be a field, and F ( X, Y, Z ) ∈ k [ X, Y, Z ] a ternary cubic form with coefficients in k defining a smooth plane cubic C ⊂ P . We fix projective coordinates X, Y, Z of P . Thefollowing theorem gives an interpretation of linear determinantal representations of C in termsof non-effective line bundles on C . It is well-known at least when k is an algebraically closedfield of characteristic zero. For the proof valid for arbitrary fields, see [Bea00, Proposition3.1], [Ish15, Proposition 2.2]. Date : October 31, 2018.2010
Mathematics Subject Classification.
Primary 14H50; Secondary 11D25, 12Y05, 14G15, 15A33.
Key words and phrases.
Linear determinantal representations, Plane cubics, twisted Fermat cubics.
Theorem 2.1.
There is a natural bijection between the following two sets: • the set of equivalence classes of linear determinantal representations of C over k , and • the set of isomorphism classes of non-effective line bundles on C of degree . The bijection is obtained as follows: we take a non-effective line bundle L of degree 0 on C . Let ι : C ֒ → P be the given embedding. We denote the homogeneous coordinate ring of P by R := Γ ∗ ( P , O P )= M n ∈ Z H (cid:0) P , O P ( n ) (cid:1) ∼ = k [ X, Y, Z ] . The graded R -module N = Γ ∗ ( P , ι ∗ L ) ∼ = Γ ∗ ( C, L ) has a minimal free resolution of the form(1) / / R ( − ⊗ k W f M / / R ( − ⊗ k W / / N / / , where W , W are 3-dimensional k -vector spaces [Bea00, Proposition 3.1]. The homomor-phism f M can be expressed by a square matrix M of size with coefficients in k -linear formsin three variables X, Y, Z . We can check M gives a linear determinantal representation of C ,and its equivalence class depends only on the isomorphism class of the line bundle L .When k is an algebraically closed field, the set Pic ( C ) of isomorphism classes of linebundles on C of degree 0 is parametrized by the group Jac( C )( k ) of k -rational points on theJacobian variety Jac( C ) of C , and the only effective line bundle of degree corresponds to theorigin O of Jac( C )( k ) . In general, there can be a difference between Pic ( C ) and Jac( C )( k ) which is measured by the relative Brauer group ([CK12, Theorem 2.1], [Ish15, Example 6.9]).When C has a k -rational point P , the relative Brauer group vanishes, and two sets Pic ( C ) and Jac( C )( k ) are identified. We have a bijection C ( k ) → Pic ( C ) = Jac( C )( k ) ; P
7→ O C ( P − P ) . Hence we obtain the following corollary.
Corollary 2.2.
Let C be a smooth plane cubic over k with a k -rational point P ∈ C ( k ) .There is a natural bijection between the following two sets: • the set of equivalence classes of linear determinantal representations of C over k , and • the set C ( k ) \ { P } of k -rational points on C different from P . An algorithm to obtain linear determinantal representations
Let us make the bijection in Theorem 2.1 explicit. In this section, we give an algorithm toobtain linear determinantal representations of smooth plane cubics over an arbitrary field k .In this algorithm, we do not assume that C has a k -rational point. Algorithm 3.1.Input:: a ternary cubic form F ( X, Y, Z ) with coefficients in k defining a smooth planecubic C ⊂ P with respect to a fixed projective coordinates X, Y, Z , and a k -rationalnon-effective line bundle L on C of degree . Output:: a linear determinantal representation of C over k corresponding to L . Step 1 (Global Section):
Compute a k -basis { v , v , v } of the 3-dimensional k -vector space H ( C, L (1)) . Step 2 (First Syzygy):
Compute a k -basis { e , e , e } of the kernel of the mul-tiplication map H ( C, L (1)) ⊗ H ( C, O C (1)) → H ( C, L (2)) . OMPUTATION OF LINEAR DETERMINANTAL REPRESENTATIONS 3
Step 3 (Output Matrix):
Write the k -basis { e , e , e } as e i = X j =0 v j ⊗ l i,j ( X, Y, Z ) , where l i,j ( X, Y, Z ) ∈ H ( C, O C (1)) are k -linear forms. Output the matrix M = ( l i,j ( X, Y, Z )) ≤ i,j ≤ . By the sequence (1), we have W ∼ = H ( C, L (1)) and W ∼ = Ker (cid:0) H ( C, L (1)) ⊗ H ( C, O C (1)) → H ( C, L (2)) (cid:1) . Using k -bases of W and W , we obtain an explicit matrix representation M of the map f M in(1). This M gives a linear determinantal representation corresponding to L in the bijectionof Theorem 2.1.4. An explicit formula on linear determinantal representations of smoothplane cubics with rational points
We apply Algorithm 3.1 to a smooth plane cubic C with a k -rational point P . By changingthe projective coordinates, we may assume that P = [1 : 0 : 0] and the tangent line of C at P is the line ( Z = 0) . The following theorem gives an explicit formula of the bijection inCorollary 2.2. For the proof, see [Ish16, Theorem 4.1]. Theorem 4.1.
Let C ⊂ P be a smooth plane cubic over an arbitrary field k with a k -rationalpoint P = [1 : 0 : 0] . Assume that the tangent line of C at P is the line ( Z = 0) . We havethe following formula for a linear determinantal representation M P of C over k correspondingto a point P = [ s : t : u ] ∈ C ( k ) \ { P } via Corollary 2.2. Case 1: If u = 0 , the equivalence class of linear determinantal representations of C corresponding to P is given by M P = Z − YuY − tZ L ( X, Y, Z ) uX − sZ L ( X, Y, Z ) L ( X, Y, Z ) , (2) where we denote L ( X, Y, Z ) := − u X − ( a t + a tu + a u + su ) Z,L ( X, Y, Z ) := u a X + u a Y + u ( a t + a u ) Z,L ( X, Y, Z ) := u ( a t + a u ) X + ( a t + a tu + a u ) Z. Case 2: If u = 0 , the equivalence class of linear determinantal representations of C corresponding to P is given by M P = Z − YZ a Y e L ( X, Y, Z ) e L ( X, Y, Z ) e L ( X, Y, Z ) e L ( X, Y, Z ) , (3) YASUHIRO ISHITSUKA where we denote e L ( X, Y, Z ) := X + a Y + a Z, e L ( X, Y, Z ) := a X + a Y, e L ( X, Y, Z ) := a X + ( a a − a a ) Y, e L ( X, Y, Z ) := ( a a − a a ) Y − a a Z. Remark 4.2.
Let k be a field of characteristic not equal to 2 nor 3, and E : ( Y Z − X − aXZ − bZ = 0) ⊂ P (4) an elliptic curve over k with origin P = [0 : 1 : 0] defined by a Weierstrass equation. Let P = [ λ : µ : 1] ∈ E ( k ) \ { P } be a k -rational point on E . Galinat gave in [Gal14, Lemma2.9] a representative of linear determinantal representations of E over k corresponding to thedivisor P − P as M ′ P := X − λZ − Y − µZµZ − Y X + λZ ( a + λ ) Z Z − X . When C has a k -rational flex, Theorem 4.1 is equivalent to Galinat’s formula. However,Theorem 4.1 is also applicable when C has no k -rational flex.When k is algebraically closed of characteristic not equal to 2 nor 3, Vinnikov [Vin89] gaveother representatives. Applications to linear determinantal representations over finite fields
Let p be a prime number, and m ≥ a positive integer. Let F q be a finite field with q = p m elements. It is well-known that any smooth plane cubic C over F q has an F q -rational point ([Lan55]), hence we can freely use Corollary 2.2 and Theorem 4.1 (at least after some changesof coordinates).In [Ish16], we determine projective equivalence classes of smooth plane cubics over F q with 0, 1 or 2 equivalence classes of linear determinantal representations. We denote by Cub q ( n ) the set of projective equivalence classes of smooth plane cubics over F q with n equivalence classes of linear determinantal representations. By Corollary 2.2, the number ofelements q ( n ) coincides with the number of projective equivalence classes of smoothplane cubics over F q with n + 1 F q -rational points. The latter can be determined by Schoof’sformula [Sch87]. The following table summarizes the results of our computations of q ( n ) when ≤ n ≤ . Table 1. F F F F F F q ( q ≥ q (0) q (1) q (2) Cub q (0) . They do not admit lineardeterminantal representations. Each of them has only one rational point [1 : 0 : 0] . • X Z + XZ + Y + Y Z + Z over F . • X Z + Y − Y Z + Z over F . • X Z + XZ + Y + ωZ over F , where ω ∈ F satisfies ω + ω + 1 = 0 .The following ternary cubic forms are representatives of Cub q (1) . Each of them admitsa unique equivalence class of linear determinantal representations. Their rational points are [1 : 0 : 0] and [0 : 0 : 1] . • X Z + XY Z + Y + Y Z + Y Z over F . OMPUTATION OF LINEAR DETERMINANTAL REPRESENTATIONS 5 • X Z − Y + Y Z + Y Z over F . • X Z + ωXY Z + Y + Y Z + ωY Z over F = F [ ω ] . • X Z + Y + 2 Y Z over F .The following ternary cubic forms are representatives of Cub q (2) . Each of them admitstwo equivalence classes of linear determinantal representations. • X Z + XY + Y Z over F . The rational points are [1 : 0 : 0] , [0 : 1 : 0] , [0 : 0 : 1] . • X Z + XZ + Y over F . The rational points are [1 : 0 : 0] , [1 : 0 : 1] , [0 : 0 : 1] . • X Z + XY + Y Z + 2 XY Z over F . The rational points are [1 : 0 : 0] , [0 : 1 : 0] , [0 :0 : 1] . • X Z − XZ − XY Z − Y over F . The rational points are [1 : 0 : 0] , [1 : 0 : 1] , [0 : 0 : 1] . • X Z + XY + ωY Z over F = F [ ω ] . The rational points are [1 : 0 : 0] , [0 : 1 : 0] , [0 :0 : 1] . • X Z + XY + ( ω + 1) Y Z over F = F [ ω ] . The rational points are [1 : 0 : 0] , [0 : 1 :0] , [0 : 0 : 1] . • X Z + XZ + ωY over F = F [ ω ] . The rational points are [1 : 0 : 0] , [1 : 0 : 1] , [0 :0 : 1] . • X Z + XZ + ( ω + 1) Y over F = F [ ω ] . The rational points are [1 : 0 : 0] , [1 : 0 :1] , [0 : 0 : 1] . • X Z + XY + Y Z − XY Z over F . The rational points are [1 : 0 : 0] , [0 : 1 : 0] , [0 :0 : 1] . • X Z − XZ − XY Z − Y over F . The rational points are [1 : 0 : 0] , [1 : 0 : 1] , [0 :0 : 1] . • X Z + XY + 3 Y Z over F . The rational points are [1 : 0 : 0] , [0 : 1 : 0] , [0 : 0 : 1] . • X Z − XZ + 3 Y over F . The rational points are [1 : 0 : 0] , [1 : 0 : 1] , [0 : 0 : 1] .We give some examples of linear determinantal representations of the cubics in the abovelist without k -rational flexes. Example 5.1.
Consider the smooth plane cubic over F defined by X Z + XY + Y Z = 0 . This cubic has three F -rational points; P = [1 : 0 : 0] , P = [0 : 1 : 0] , P = [0 : 0 : 1] . By Theorem 4.1, linear determinantal representations corresponding to P , P are Z YZ Y XX Y , Z YY XX X Z . Example 5.2.
Consider the smooth plane cubic over F defined by X Z + XY + Y Z − XY Z = 0 . This cubic has three F -rational points; P = [1 : 0 : 0] , P = [0 : 1 : 0] , P = [0 : 0 : 1] . By Theorem 4.1, linear determinantal representations corresponding to P , P are Z − YZ Y X − YX − Y , Z − YY − XX X − X + Z . YASUHIRO ISHITSUKA twisted Fermat cubics over the field of rational numbers In this final section, we report our recent study on computations of linear determinantalrepresentations of twisted Fermat cubics defined over the field Q of rational numbers.Over the field Q of rational numbers, some problems arise. The main problem is that a linebundle L P on C is usually given by the corresponding k -rational point P on Jac( C ) , not on C . This causes some problems in Step 1; the calculation of the Q -vector space H ( C, L P (1)) . Using the generalized Clifford algebra and the norm equation, we overcome these problemsfor twisted Fermat cubics. We give simple examples. Details will appear elsewhere.
Example 6.1.
Consider the smooth plane cubic over Q defined by X + Y + Z = 0 . Its Jacobian variety is an elliptic curve whose Weierstrass equation is given by (5) Y Z − Y Z − X + 27 Z = 0 (cf. [AR-VT05] ). Its Q -rational points are O , [3 : 0 : 1] , [3 : 9 : 1] . Let us take P = [3 : 0 : 1] . The equivalence class of linear determinantal representationscorresponding to P is represented by − X + 2 Y + Z − X + Y X + YX − Y X + Z − YX X + 3 Y − Y + Z . The other point [3 : 9 : 1] corresponds to the transpose of the above matrix. This cubic doesnot admit a symmetric determinantal representation over Q (cf. [II16b] ). Example 6.2.
Consider the smooth plane cubic over Q defined by X + 2 Y + Z = 0 . Its Jacobian variety is an elliptic curve whose Weierstrass equation is given by Y Z − · Y Z − X + 27 · Z = 0 (cf. [AR-VT05] ). It is isomorphic to (5) , and its Q -rational points are O , [3 · : 0 : 1] , [3 · : 9 · : 1] . However, in [Ish15] , we prove that these points do not correspond to linear determinantalrepresentations over Q due to non-vanishing obstruction in the relative Brauer group of thiscubic. This cubic does not admit a linear determinantal representation over Q . Example 6.3.
Consider the smooth plane cubic over Q defined by X + 17 Y + Z = 0 . Its Jacobian variety is an elliptic curve whose Weierstrass equation is given by Y Z − · Y Z − X + 27 · Z = 0 (cf. [AR-VT05] ). It is isomorphic to (5) , and its rational points are O , [3 · : 0 : 1] , [3 · : 9 · : 1] . Let us take P = [3 · : 0 : 1] . The equivalence class of linear determinantal representationscorresponding to P is represented by X − Y + Z − X + 153 Y X − Y X − Y − X − Y + Z X + 7 Y X + Y − X − Y Y + Z . OMPUTATION OF LINEAR DETERMINANTAL REPRESENTATIONS 7
The other point [3 · : 9 · : 1] corresponds to the transpose of the above matrix. Acknowledgements
The author would like to thank sincerely to Professor Tetsushi Ito for various and inspiringcomments. The work of the author was supported by JSPS KAKENHI Grant Number16K17572.
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