Tetsuji Shioda

English

立教大学学術リポジトリ

COMMENTARII MATHEMATICIUNIVERSITATIS SANCTI PAULIVol. 67, No. 2 2019ed. RIKKYO UNIV/MATHIKEBUKURO TOKYO171–8501 JAPANThe MWL-Algorithms for Constructing Cubic Surfaces withPreassigned 27 Lines, IIbyTetsuji SHIODA(Received July 12, 2019)In memory of late Professor Goro ShimuraAbstract. We introduce a new defining equation of the multiplicative excellent fam-ily of type E6 (cf. [11], [12]). This will improve the previous ones in the characterisitic-freenature of the key formula. Thus the new algorithm based on it will work in every charac-terisitic.1. The new defining equationLet K be a field of arbitrary characteristic. We consider the Weierstrass equation ofthe formy2 + txy + k0y = x3 + m0x2 + p0x + q0 + q1t + q2t2 + t3 , (1)where t, x, y are variables over K while the other coefficients k0,m0, p0, q0, q1, q2 belongto K . We callλ = (k0,m0, p0, q0, q1, q2) (2)the Weierstrass parameter of (1).We follow below the same argument as in [11], [12] or [7] to establish a new (slightlyimproved) algorithm.Let K¯ denote an algebraic closure of K .First observe that a linear solution P of the formP : x = a t + b, y = d t + e (a, b, d, e ∈ K¯) (3)satisfies (1) (identically in t) if and only ifad = a3 + 1 (4)ae + d2 + bd = 3a2b + a2m0 + q2 (5)be + 2de + dk0 = 3ab2 + 2abm0 + ap0 + q1 (6)e2 + ek0 = b3 + b2m0 + bp0 + q0 . (7)Now P is uniquely determined by a and b, since (4), (5) imply that a = 0 andd = a2 + 1a, e = (3a2 − d)ab + am0 + q2a− 1ad2 . (8)203204 T. SHIODAThen, substituting d, e above into (6) and (7), and eliminating b from them, we get a monicpolynomial, say Ψ (a), of degree 27 in a with coefficients in Z[k0,m0, p0, q0, q1, q2]. Re-placing a by an indeterminate X, Ψ (X) is given as follows:Ψ (X) = X27 + m0X26 + (p0 − q2)X25 + (3k0 + q0 − 9)X24+(−2q22 + p0q2 + k0m0 − 8m0 − k0q1 + 7q1)X23+ · · · (9)+(2m20 − q1m0 + k0p0 − 7p0 − k0q2 + 8q2)X4+ (−3k0 − q0 + 9)X3 + (m0 − q1)X2 − q2X − 1This will be called a universal polynomial of multiplicative type E6. (The middle terms . . .are omitted due to its length, and as we do not need them.)For each zero a of the equation Ψ (a) = 0, there is at least one Pa = (x, y) such that ais equal to the t-coefficient of x = at + b. Thus, for a general λ, there are exactly 27 linearsolutions Pa satisfying the given Weierstrass equation.2. The key formulaBy comparing (9) with the "split" expression of the universal polynomial (see [11, eq.(35)]), we obtain the key formula connecting the Weierstrass parameter and the six basicelementsδ1, 1, 2, 3, −1, −2 ∈ R (10)in the Laurent polynomial ring R (see Remark below).⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩k0 = −δ1m0= −1q2 = −−1p0 = 2 − −1q1 = −2 − 1q0 = −3 + 3δ1 + 9 .(11)2.1. NotationLet us recall the relevant notation. Let T denote the algebraic torus of dimension 6which is defined as a toric hypersurface in G7m:T = {(s1, s2, . . . , s6; s0)|s1s2 · · · s6 = s30} (12)and take any element ξ = (s1, s2, . . . , s6; s0) ∈ T, which is called a data in what follows.Given such a data ξ , defines′i :=sis0(1 ≤ i ≤ 6) , s′′ij :=s0sisj(i = j) (13)and consider the ordered set with 27 elements (a 27-set in short):Ω = Ωξ := {s1, s2, . . . , s6, s′1, s′2, . . . , s′6, s′′12, s′′13, . . . , s′′56}The MWL-Algorithms for Constructing Cubic Surfaces with Preassigned 27 Lines, II 205= {s1, . . . , s27} (14)with the fixed ordering. We also need the following set of 36 elements (a 36-set):Π = Πξ :={1s0,sisj(i < j ≤ 6), s0sisj sk(i < j < k ≤ 6)}. (15)The 36-set corresponds to the set of positive roots of the root lattice E6 and the 27-setcorresponds to half of the 54 minimal vectors of the dual lattice E∗6 on which the Weylgroup W(E6) acts transitively (cf. [10]).Next we denote byn (or −n) (16)the n-th elementary symmetric polynomial of {s1, . . . , s27} (or of {s−11 , . . . , s−127 }). Notethat −n = 27−n. For the algorithm in the next section, we need n and −n only forn = 1, 2, 3. Further we setδ1 = s0 + 1s0+∑i =jsisj+∑i<j<k(s0sisj sk+ sisj sks0). (17)This is the sum of 36 elements of the set Π above and their inverses, corresponding to the72 roots of E6.REMARK. For a moment, suppose ξ is generic over a prime field κ . Then the affinecoordinate ring of the torus T, R = κ[T], can be identified with the ring of Laurent poly-nomials in si :R = κ[s±1i | 0 ≤ i ≤ 6 ]. (18)There is a natural action of the Weyl group W(E6) (as the Galois group) on R (cf. [10])and the ring of invariants contains the six basic elements (10), which forms a set of freegenerators of RW(E6) over κ if its characteristic is relatively prime to the order of the Weylgroup, i.e. if it is greater than 5. (cf. [1], [4], [7], [11], [12]).3. A new algorithmConsider the morphism defined by the key formula (11):ψ : T → A6, ψ(ξ) = (k0,m0, p0, q0, q1.q2) . (19)Fix a data ξ = (s1, s2, . . . , s6; s0) ∈ T, and setλ = ψ(ξ). (20)Namely we consider the Weierstrass equation (1) with the Weierstrass parameter λ = ψ(ξ),which is defined by the key formula (11), plus (16) and (17), all evaluated at the data ξ .In this way, we can construct a 6-parameter family, parametrized by T, of the followinggeometric objects:• Eλ : an elliptic curve over the rational function field k(t) (k: an algebraically closedfield containing κ(ξ) ⊃ κ(λ)),• Sλ : an elliptic surface over P1 (the t-line),206 T. SHIODA• Xλ : an affine surface in A3 with coordinates (t, x, y),• Vλ : a cubic surface in P3 with homogeneous coordinates (T : X : Y : Z) =(t : x : y : 1).Next, for the fixed data ξ , we consider the 27-set Ωξ = {sn|1 ≤ n ≤ 27}, (14), andwe can give an explicit definition of the 27 linear solutions Pn (cf. [12]): the coefficientsa, b, d, e of linear solutions Pn are defined by the following rule: First, we have a = − 1snfor all n. More precisely, we have:(i) For n = i ≤ 6, Pn = Pi has:a = − 1si, b = si + 1s2i+∑j =i(s′j + s′′ij ) (21)(ii) For n = 6 + i, we have sn = s′i , and Pn = P ′i has:a = − 1s′i, b = s′i +1s′2i+∑j =i(sj + s′′ij ) (22)(iii) For n > 12, we have sn = s′′ij for some i = j , and Pn = P ′′ij hasa = − 1s′′ij, b = s′′ij +1s′′2ij+ (si + s′i + sj + s′j ) +∑k<l∑{kl}∩{ij}=∅s′′kl . (23)(iv) For all Pn(1 ≤ n ≤ 27), the coefficients d, e are determined by a, b given as above,using the formula (8). It is straightforward to verify that each of Pn so defined is indeed alinear solution of the Weierstrass equation (1) specified by (20).The 27 linear solutions Pn then translate to the following objects:• 27 k(t)-rational points of the Mordell-Weil group Eλ(k(t)),• 27 linear sections on the elliptic surface Sλ , containing a set of generators of theMordell-Weil group ∼= Eλ(k(t));• 27 affine lines on the affine surface Xλ,• 27 lines ln : X = aT + bZ, Y = dT + eZ on the cubic surface Vλ ⊂ P3.REMARK. There is a handy non-degeneracy condition on the data ξ , saying that the36-set Πξ is free from 1. It guarantees in the above statements that “27” can be replaced by“27 distinct”, in addition to the smoothness of the elliptic (or affine or cubic) surface Sλ (orXλ or Vλ) where λ = ψ(ξ) is the Weierstrass parameter corresponding to the data ξ . SeeTheorem 8.1 of the paper [12].Furthermore, counting the number of times 1 appears in the 36-set Πξ , one can mea-sure how degenerate the surface Sλ is, etc. cf. [12, Th. 8.4] or [7]. We will study the"totally degenerate" case such that Πξ = {136} in the last section.4. Examples4.1. Example, with the data ξ = (1, 2, 3, 4,−1,− 13; 2)We let K = Q here. This is a non-degenerate example. The Mordell-Weil lattice isisomorphic to E∗6 . Cf. [12, 9.2], which are computed by the previous algorithm.The MWL-Algorithms for Constructing Cubic Surfaces with Preassigned 27 Lines, II 207For the given data ξ , the new algorithm gives:λ =(1369,−214,−1753,322712,−1753,−214). (24)The Weierstrass equation reads:y2 + txy + 136y9= x3 − 21x24− 175x3+ 322712− 175t3− 21t24+ t3 . (25)The 27 linear solutions are given as follows. The first six:P(−t, 212), P(196− t2,4112− 7t4), P(409− t3,−26t9− 2354),P(8516− t4,−63t16− 21364), P(t + 203, 2t − 476), P(3t + 203,28t3− 13118).The second six:P(193− 2t, 7t2− 233), P(254− t,−334), P(559− 2t3,−19t18− 23927),P(356− t2,−7t4− 11912), P(2t + 15, 9t2+ 33), P(6t + 1153,217t6+ 18779)The remaining 15:P(8 − t,−272), P(556− 3t2,19t12− 66736), P(353− 2t, 7t2− 913),P(t2− 152,9t4− 34), P(t6− 11518,217t36− 2431108), P(403− 3t, 26t3− 70118),P(854− 4t, 63t4− 87), P(t − 203, 2t − 1276), P(t3− 209,28t9− 151354),P(1253− 6t, 215t6− 22939), P(3t2− 256,35t12− 99736), P(t2+ 12,9t4− 994),P(2t − 1, 9t2− 27), P(2t3+ 259,35t18− 52927), P(12518− t6,−215t36− 641108).The first six Q(t)-rational points P1, . . . , P6 are independent and they generate a subgroupof index 3 of the Mordell-Weil group Eλ(Q(t)). The first 7 elements P1, . . . , P7 generatethe full group ∼= E∗6 , and dropping P6, one gets free genarators.These facts hold true also if Q is replaced by any finite field Fp with p > 13.5. Totally degenerate caseNext we study the most degenerate case; cf. [12, 9.5]. For ξ = (1, 1, 1, 1, 1, 1; 1), wehaveλ = (−72,−27, 324,−2700, 324,−27) (26)208 T. SHIODAandy2 + txy − 72y = x3 − 27x2 + 324x − 2700 + 324t − 27t2 + t3 . (27)The discriminant and the j -invariant are given as follows:Δ = −(t − 6)8(t + 21) , j = − (t − 6)(t + 18)3(t + 21) . (28)It follows, by [3] or [13], that the elliptic surface S has a new reducible fibre of Kodairatype IV ∗ at t = 6, at least if characteristic = 2, 3.Finally, let us study the case of exceptional characteristic p = 3 or p = 2 in detail.The equations (27) and (28) reduce modulo p to the following:(p = 3) y2 + txy = x3 + t3 , Δ = −t9, j = −t3 , (29)(p = 2) y2 + txy = x3 + x2 + t2 + t3 , Δ = t8(t + 1), j = t4(t + 1) . (30)Note that the last one (30) is new, since it was not accessible with the previous algorithmsin the papers [11], [12].PROPOSITION 5.1. In both cases, the singular fibre at t = 0 remains to be of Ko-daira type IV ∗. The wild ramification index is equal to 1 in case p = 3, while it is equalto 0 in case p = 2.Proof. For the equation (29) for p = 3, x3+t3 is equal to (x+t)3. Setting X = x+t ,we can rewrite it as follows:y2 + tXy − t2y = X3 . (31)On the other hand, for the equation (30) for p = 2, we transpose the terms x2 + t2 tothe left-hand side to gety2 + x2 + t2 + txy = x3 + t3 .Then, setting Y = y + x + t , it is rewritten asY 2 + tx(Y − x − t) = x3 + t3, i.e. Y 2 + txY = x3 + t3 + tx(x + t) .Now the right-hand side is equal to (x + t)3 as p = 2. Hence setting X = x + t , we haveY 2 + t (X − t)Y = X3, i.e. Y 2 + tXY − t2Y = X3 . (32)Thus both (29) and (30) are transformed by a change of coordinates to the same equa-tion (up to notation). So let’s consider the equation:y2 + txy − t2y = x3 (33)over any field of arbitarary characteristic. Now, by using notation [13, §8], x = xntn(n =1, 2, . . .), etc., we gety22 + tx2y2 − y2 = t2x32 . (34)It reduces modulo (t) to the conic y22 − y2 = 0 with two distinct roots y2 = 0, 1, and thisimplies, by [13, §8, 8)] (cf. [7, §5.8.2]), that the Kodaira type is IV ∗.Finally the wild ramification index, defined as v(Δ) − 1 − m (m: the number ofirreducible components in the singular fibre), is equal respectively to9 − 1 − 7 = 1 (p = 3) or 8 − 1 − 7 = 0 (p = 2) ,The MWL-Algorithms for Constructing Cubic Surfaces with Preassigned 27 Lines, II 209as claimed. q.e.d.As an afterthought looking at the above proof for p = 2, 3, a natural question arises:QUESTION 5.2. Is it possible to transform the equation (27) to the equation (33)over Z?The answer is: Yes!PROPOSITION 5.3. The equation (27) can be transformed over Z to the equation(33) by the following coordinate change:x = x ′ − t ′ + 6 , t = t ′ + 6 , y = y ′ − 3x ′ + 18 . (35)Proof. Straightforward verification.COROLLARY 5.4. For every prime number p, the reduction mod p, S(p), of theelliptic surface over P1 defined by (27) has a singular fibre of Kodaira type IV ∗ at t = 6.In other words, for every p, S(p) is a rational elliptic surface of OS-type No.69 (cf. [6] or[7]). It follows:• The Mordell-Weil group E(Fp(t)) is a torsion group of order 3, generated by P1 =(12 − t, 18). One checks−P1 = (12 − t, 54 − 12t + t2) = 2 · P1 .• For every p, the affine surface X(p) has an E6-singularity at (t, x, y) = (6, 6, 18).• For every p, the cubic surface V (p) contains a unique linel1 : X = −T + 12Z , Y = 18Z ,which passes through the E6-singularity (T : X : Y : 1) = (6 : 6 : 18 : 1).References[ 1 ] Bourbaki, N.: Groupes et Algèbres de Lie, Chap. 4,5 et 6, Hermann, Paris (1968); Masson (1981).[ 2 ] Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups, Springer-Verlag (1988).[ 3 ] Kodaira, K.: On compact analytic surfaces II–III. Ann. Math. 77 (1963) 563-626; 78 (1963) 1–40; Col-lected Works, vol III. Iwanami and Princeton Univ. Press, 1975, pp. 1269–1372[ 4 ] Kumar, A., Shioda, T.: Multiplicative excellent families of elliptic surfaces of type E7 or E8, Algebraand Number Theory 7-7, 1613–1639 (2013); http://arXiv.org/pdf/1204.1531[ 5 ] Manin, Ju.: Cubic Forms, 2nd edn., North-Holland (1986).[ 6 ] Oguiso, K., Shioda, T.: The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St.Pauli 40, 83–99 (1991).[ 7 ] Schütt, M., Shioda, T.: Mordell-Weil Lattices, Egebnisse 70, Springer, (to appear).[ 8 ] Shioda, T.: On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39, 211–240 (1990).[ 9 ] Shioda, T.: Theory of Mordell-Weil lattices, Proc. ICM Kyoto-1990, vol. I, 473–489 (1991).[ 10 ] Shioda, T.: A uniform construction of the root lattices E6, E7, E8 and their dual lattices, Proc. JapanAcad.71A, 140–143 (1995).[ 11 ] Shioda, T.: Multiplicative excellent family of type E6, Proc. Japan Acad. 86A, 21–26 (2012).[ 12 ] Shioda, T.: The MWL-Algorithms for constructing cubic surfaces with preassigned 27 lines, Comment.Math. Univ. St. Pauli 64, 157–186 (2015).[ 13 ] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. Lecture Notes inMathematics, vol. 476, 33–52, Springer, (1975).210 T. SHIODA[ 14 ] Shimura, G.: Reduction of algebraic varieties with respect to a discrete valuation ring of the basic field,Amer. J. Math. 77 (1955), 134–176.Department of Mathematics,Rikkyo University,Tokyo 171, Japanshioda@rikkyo.ac.jp

The MWL-Algorithms for Constructing Cubic Surfaces with Preassigned 27 Lines, II

Tetsuji, Shioda

Institutional Repositories DataBase (IRDB)

https://rikkyo.repo.nii.ac.jp/?action=repository_action_common_download&item_id=18750&item_no=1&attribute_id=18&file_no=1

The MWL-Algorithms for Constructing Cubic Surfaces with Preassigned 27 Lines, II

Abstract

Similar works

Full text

Available Versions

立教大学学術リポジトリ

Institutional Repositories DataBase (IRDB)