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8
УДК 514.2
Studies on Systems of Six Lines on a Projective Plane over a Prime Field
Jiro Sekiguchi*
Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan
Received 11.01.2008, received in revised form 20.02.2008, accepted 05.03.2008
A simple six-line arrangement on a projective plane is obtained by a system of six labelled lines Li, L2,... , L6 with the conditions; (1) they are mutually different and (2) no three of them intersect at a point. We add the condition that (3) there is no conic tangent to all the lines. The main subject of this paper is to treat such arrangements on a projective plane over a finite prime field.
Keywords: projective plane, finite prime field, quadratic residue
Introduction
A simple six-line arrangement on a projective plane is obtained by a system of six labelled lines Li, L2,..., ¿6 with the conditions; (1) they are mutually different and (2) no three of them intersect at a point. We add the condition that (3) there is no conic tangent to all the lines. The main subject of this paper is to treat such arrangements on a projective plane over a finite prime field.
Before entering into the main subject, we now explain some results on the real case. There are four types of simple six-line arrangements on a real projective plane (cf. B. Grunbaum [1]). Among the four types, one is characterized by the existence of a hexagon and one is characterized by the condition that the conic tangent to any five lines of the six lines does not intersect the remaining line. The totality of systems of six labelled lines with conditions (1), (2) admits the action of the sixth symmetric group by permutations among six lines. The advantage of the condition (3) is that the action of the sixth symmetric group on the totality of systems of six labelled lines with conditions (1), (2), (3) naturally extends to that of the Weyl group W(E6) of type E6. It is shown in J. Sekiguchi and M. Yoshida [2] that W(E6) acts transitively on the set of systems of six labelled lines fixed by a group isomorphic to a fifth symmetric group and that this is decomposed into four orbits by the sixth symmetric group action. These four S^-orbits are in a one to one correspondence with the four types of simple-six line arrangements mentioned above.
The purpose of this paper is to study what happens when we replace a real projective plane by a projective plane over a finite prime field. Let p be a prime number, Fp the field consisting of p points and P2(Fp) the projective plane over Fp. Let L\,L2,... ,L6 be six lines on P2(Fp) with the conditions (1), (2), (3). Then we shall show the following theorems.
* e-mail: sekiguti@cc.tuat.ac.jp
© Siberian Federal University. All rights reserved
Theorem 1. If 5 is a quadratic residue mod p, there is a system of six labelled-line arrangement on P2(Fp) fixed by a fifth symmetric group.
It is easy to determine systems of six labelled lines fixed by a fifth symmetric group. They are related with the diagonal surface of Clebsch. In fact, if 5 is a quadratic residue mod p, the twenty-seven lines on it are defined over Fp and any system of six labelled lines fixed by a fifth symmetric group is obtained by blowing down the diagonal surface.
Theorem 2. Now assume that there is n G Fp such that n2 = 5 (p). Then there is a system of six labelled lines fixed by a fifth symmetric group such that the conic tangent to any five lines of the six lines does not intersect the remaining line if and only if ±2n — 5 is a non-quadratic residue mod p.
It is well-known that for a prime p, 5 is a quadratic residue mod p if and only if p = 10k +1 or p = 10k — 1 for a positive integer k. The following theorem was conjectured by the author and later proved by T.Ibukiyama.
Theorem 3. For a prime p with 5 < p, there is n G Fp such that n2 = 5 (p) and there is no m G Fp such that m2 = 2n — 5 (p) if and only if p = 10k — 1 for a positive integer k.
We are going to explain the contents of this paper. In §1, we review geometry of six lines on a real projective plane and in §2, we do systems of six labelled lines fixed by S5-action. The results of both sections are contained in [2]. In §3, we start to study six lines on a projective plane over a finite prime field.
1. Review on Geometry of Six Lines on a Real Projective Plane
In this section, we collect some results given in [2] and its references, necessary to our present study. A system of six labelled lines on a real projective plane consists of six labelled lines L\, L2,..., L6j on a real projective plane P2(R). It defines an arrangement of six-lines (cf. [1]). An arrangement of six-lines is called simple if (C1) they are mutually different and (C2) no three of them intersect at a point.
In terms of a system of homogeneous coordinates t1 : t2 : t3 on P2(R), the six lines Li, L2,... ,L66 are expressed by linear equations:
Lj : xji ti + Xj2t2 + Xjsts = 0 (j = 1, 2,..., 6).
Thus the system S of six labelled lines L1,L2,...,L6 is represented by a 3 x 6 matrix X = (xj ). Then for any ai G R — {0} (i = 1, 2,..., 6), X = (xj ) and X' = (a^Xij) define the same system of six labelled lines. Two systems of six labelled lines are equivalent if they are transformed into each other by a projective linear transformation. Since we are interested in the space of systems of six labelled lines with conditions (1), (2), we are led to define the configuration space
P(2,6)= G\M */H, M * = M *(3,6), G = GL(3, R), H = H6,
where M*(3, 6) is the set of real 3 x 6 matrices where no 3-minor vanishes, and H6 is the subgroup of GL(6, R) consisting of diagonal matrices. Any system of six labelled lines S is represented by a matrix of the form
i 10 0 11 1 \
X = I 0 10 1 xi x2 I (1)
V 0 0 1 1 yi y2 j
This implies that P(2, 6) is identified with an affine open subset of R4.
The symmetric group Sq is generated by transpositions sj (1 < i < j < 6). We may identify sij with the transposition between the lines Lj and Lj. This induces an S6-action on the space P(2, 6). Let X G M* be a matrix representing a system S of six labelled lines. Regarding X as a linear map of R6 to R3, we choose a basis {y1,y2,y3} of its kernel. Then srX = (yiy2y3) G M* defines a system srS. The map sr induces a biregular involution on P(2, 6). The following lemma is an easy consequence of the definition of sr:
Lemma 1. (i) The action sr commutes with that of Sq on P(2, 6).
(ii) A system of six labelled lines is fixed by sr if and only if there is a conic tangent to all the six lines of the system.
Noting this lemma, we add a condition on systems of six labelled lines; (C3) There is no conic tangent to all the six lines.
We define a subspace Po(2, 6) of P(2, 6) consisting of systems of six labelled lines which are not fixed by the operation sr. The space Po(2, 6) admits an action si23 which is not contained in Sq. Take a representative X G Po(2, 6) defined in (1). Then si23 is defined by
/10011 1 \ / 1001 1 1 \
X = I 0 1 0 1 xi x2 I —► I 0 1 0 1 1/xi 1/x2 I = si23X V 0 0 1 1 yi y2 ) V 0 0 1 1 1/yi 1/y2 J
By the condition (C3), s^X is also contained in Po(2, 6). The group generated by Sq and si23 is nothing but the Weyl group W(Eq) of type Eq and the operation sr is contained in W(E6). In this manner, the space P0(2, 6) admits the action of W(E6).
Let P6(R) be the totality of connected components of P0(2, 6). Then the action of W(E6) on P0(2, 6) naturally extends to that on P6(R).
We define p-gons for the system of six labelled lines Li, L2,..., L6. Each connected component of P2(R) — U®=i Lj is called a polygon. If it is surrounded by p lines, it is called a p-gon. It is known (cf. [1]) that there are four types of simple six-line arrangements. They are characterized by numbers of p-gons and referred to as O, I, II, III.
Types hexagon pentagons rectangles triangles
O 1 0 9 6
I 0 2 8 6
II 0 3 6 7
III 0 6 0 10
In the case of systems of six labelled lines on a projective plane over a finite prime field, it is hard to define p-gons of a system. As a consequence, the characterizations of systems of types O, I, II, III above lose their meanings if we consider systems over a finite prime field. Instead, it is possible to characterize systems of types I and III in a different manner.
Lemma 2. Let S be a system of six labelled lines Li, L2,. .., Lq on a real projective plane. Let Ci be the conic tangent to five lines Lk (k = 1, 2, . . . , 6, k = i).
(i) S is of type III if and only if the conic tangent to any five lines of Li, L2,. .. ,Lq does not intersect the remaining line.
(ii) Suppose that S is of type O and that Li, L2,. .., Lq bounds a hexagon in this order. Then one of the following holds:
(a) Ci n Li = 0 (i = 1, 3, 5) and Ci n L, = 0 (i = 2, 4, 6).
(b) Ci n Li = 0 (i = 1, 3, 5) and Ci n Li. = 0 (i = 2, 4, 6).
(iii) Suppose that S is of type I and that Li, L2, .. ., L5 (resp. Li,. .., L4, Lq) bounds a pentagon. Then Ci n Li = 0 (i =1, 2, 3, 4) and Ci n Li = 0 (i = 5, 6).
(iv) Suppose that S is of type II and that Li, L2,. .., L5 (resp. Li,. .., L4, Lq, and Li, L2, L3, L5, Lq)) bounds a pentagon. Then C\ n Li = 0 (i = 1, 2, 3) and C, n Li = 0 (i = 4, 5, 6).
Remark 1. Unfortunately, systems of types O and II are not distinguished by the lemma above.
2. Systems of Six Labelled Lines Fixed by S5-action
We begin with this section by defining an outer automorphism t of Sq defined as follows:
Permutation Image by t
(12) - (12)(34)(56)
(23) - (16)(24)(35)
(34) - (12)(36)(45)
(45) - (16)(25)(34)
(56) - (12)(35)(46)
Here we identify sij with the permutation (ij). Since (i i + 1) (i = 1, 2, 3,4, 5) generate Sq, t is actually an automorphism of Sq. As in [2], we put t(ij') = t((ij)) o sr. Define the matrix X(±^5) G M* by
i 10 0 1 1 1 \
X (±V5)= I 0 10 1 ( —1 T^5)/2 (1 T^5)/2 I \0 0 1 1 (1 T^5)/2 (3 tV5)/2/
Then the following holds:
Proposition 1. Let H be the subgroup of W(E6) generated by t'((i i + 1)) (i = 2, 3, 4, 5) (which is isomorphic to S5). Then X(v^) is fixed by H as an element of Po(2, 6).
Remark 2 (cf. [2]). (i) By blowing up P2(C) at the six points
(1:0:0), (0:1:0), (0 : 0 : 1), (1 : 1 : 1),
(1 : (-1 - V5)/2 : (1 - V5)/2), (1 : (1 - V5)/2 : (3 - V5)/2),
we obtain Clebsch diagonal surface in P3(C).
(ii) We consider a regular icosidodecahedron in R3 whose center is the origin. There are six hyperplanes containing the origin in R3 which cut the edges. From these hyperplanes, we obtain a system S of labelled six lines on a real projective plane. Let X G M * be the matrix obtained from S. Then X is equivalent to X(v^5) by choosing a label appropriately.
Proposition 2. The system of six labelled lines defined by X(±\/5) is of type III. Moreover, the systems of six labelled lines defined by S123X(V5), s 145S123X(V5), and S123X(-\/5) are of types O, I, II, repsectively, where S145 = S35S24S123S24S35.
3. Systems of Six Labelled Lines on a Projective Plane over a Prime Field
Let p be a prime number and let Fp be the prime field consisting of p elements. In this section, we study systems of six labelled lines on a projective plane defined over Fp. Let P2(Fp) be a projective plane defined over Fp. As before let t1 : t2 : t3 be its homogeneous coordinate system.
First of all, we consider the conic tangent to five lines on P2(Fp). Let
t1 = 0, t2 = 0, t3 = 0, «1Î1 + a,2Î2 + 03Î3 = 0, M1 + M2 + 63Î3 = 0 (2)
be equations of five lines. We assume that no three of them intersect at a point. Then it is easy to show that there is a unique conic tangent to the five lines (2) and it is defined by
p^2 + p^2 + P3Î3 - 2P2P3Î2Î3 - 2P3P1Î3Î1 - 2P1P2Î1Î2 = 0, (3)
where
P1 = 0461(0,263 - 0362), P2 = 0262(0361 - 0163), P3 = 0363(0162 - 0261).
A system S of six labelled lines on P2(Fp) is defined similarly to the real case. We consider conditions (C1), (C2), (C3). From the systems S with conditions (C1), (C2), (C3), we are naturally led to define the configuration space Po(2, 6)Fp over Fp. The matrices of the form (1) with x1,x2,y1,y2 G Fp are regarded as representatives of P0(2, 6)Fp. Noting this, we may identify P0(2, 6)Fp with an affine open subset SFp of F44. In order to define Sfp definitely, we introduce the fifteen polynomials fj (j = 1, 2,..., 15) by
/1 = X1, f2 = X2, f3 = y1, f4 = y2, f5 = y1 - X1, f6 = y2 - ^2,
fr = 1 - X1, fs = 1 - X2, fg = 1 - y1, f10 = 1 - y2, f11 = X1 - X2, f12 = y1 - y2, f13 = ^1^2 - X2y1, f14 = X1^2 - X2^1 - X1 + X2 + y1 - y2,
f15 = X1^1^2 - X2^1^2 + X1X2y2 - X1X2^1 - X1^2 + X2^1.
Then
Spp = {(xi,x2,yi,y2) G Fp ; fj = 0 (j = 1, 2,..., 15)}.
Remark 3 (cf. [2], p.315). The fourteen polynomials fj (j < 15) are obtained as determinants of 3 x 3 minors of the matrix (1) and fi5 = 0 corresponds to the condition that the system of six labelled lines defined by the matrix (1) does not satisfy (C3).
It is easy to show that the Weyl group W(E6) acts on the space SFp ~ P0(2, 6)Fp by the same manner as in the real case.
Let S be a system of six labelled lines Li,... ,L6 in P2(Fp). Then as mentioned before, for each j (j = 1, 2,..., 6), there is a unique conic Cj in P2(Fp) tangent to the five lines Lk (k = 1,..., 6, k = j). Since systems of six labelled lines of type III play an important role in the study [2], we introduce the notion of systems of six labelled lines of type III.
Definition 1. A system S is of type III if Cj n Lj = 0 for j = 1, 2,..., 6.
Then it is interesting to study the following problems.
Problem 1. (i) Find a condition for the prime p which implies the existence of a system of six labelled lines of type III.
(ii) Fix a prime p for which there is a system of six labelled lines of type III. For any (xi,X2,yi,y2) G S, does there exist w G W(Eq) satisfying the condition that w transforms (xi,X2,yi,y2) to (ui,u2,vi,V2) G S so that the system of six labelled lines corresponding to (ui ,U2,vi, V2) is of type III?
Problem 2. Find a condition for the prime p which implies the existence of a system of six labelled lines fixed by a subgroup H of W(Eq) isomorphic to the symmetric group of degree five.
In the next section, we shall study topics related to these problems.
4. Systems of Six Labelled Lines Fixed by S5-action over Fp
In this section, we restrict our attention to such systems of six labelled lines that they are fixed by subgroups of W(E6) isomorphic to S5.
We begin with this section with defining a subgroup H(5) of W(E6) generated by t(ii + 1)' (i = 2, 3,4, 5). Clearly H(5) is isomorphic to S5. It is shown in [3] that there are forty five involutions in W(E6) conjugate to t(i i + 1)' (i =1, 2,..., 5). As actions on SFp, the explicit forms of t(i i +1)' (i = 2,..., 5) are given in [2], Lemma 2. For example,
x2 x2yi x2
t(23) : (xi, x2,yi, y2) —M —,x2,-,—
y2 xiy2 xi
Noting this, we conclude that for (xi,x2,yi,y2) G Sfp, (xi,x2,yi,y2) is fixed by t(23)' if and only if x2 — xiy2 = 0. More generally we have the following lemma (cf. [2], Corollary 1 to Lemma 2).
Lemma 3. For i = 2, 3, 4, 5, the fixed point set of t(i i + 1)' is given by ka+i = 0, where
k23 = -X2 + xiy^, k34 = (xi - X2 - yi + X2Vl), k45 = X2yi - y2, k56 = Xi - X2 + y2 - Xiy2.
(4)
Theorem 4. Let p be a prime number with p > 5. Then there is (Xi,X2,yi,y2) € SFp fixed by H(5) if and only if there is n € Z such that n2 = 5 (p).
Proof. From the argument before the theorem, there is (xi,X2,yi,y2) € Sfp fixed by H(5) if and only if k23 = = k45 = k^a = 0 hold for (xi, X2, yi, y2). Since (xi, X2, yi, y2) € F4 is contained in SFp if and only if fj = 0 (j = 1, 2,..., 15). Then it is easy to find that (Xi,X2,yi,y2) € Sf fixed by H(5) if and only if
x\ + Xi - 1 = 0, X2
yi = xi + 1, y2 + xi +2.
(5)
If there is n € Z such that n = 5 (p), we take xi as the residue class of (p + 1)/2 • (n - 1) in Fp. Then x2 + xi - 1 = 0 in Fp. On the other hand, if n2 ^ 5 (p) for any n € Z, there is no solution of x2 + x - 1=0 in Fp. Hence the theorem follows. □
Let p be a prime number with p > 5. Then it follows from the reciprocity law for the Legendre symbol that 5 is a quadratic residue mod p if and only if p +1 or p -1 is divisible by 5. Noting that p is odd, this is equivalent to that there is an integer k such that p = 10k + 1 or p = 10k - 1 .
In the rest of this section, we always assume that p is a prime number of the form p = 10k + 1 or p = 10k - 1. Moreover let n be an integer such that n2 = 5 (p) and fix it.
It follows from the computation above that ((-1 -n)/2, (1 -n)/2, (1 -n)/2, (3 -n)/2) is the fixed point of H(5) in W(E6). Let G0 be the subgroup of W(E6) generated by Sj (1 < i < j < 6) and sr. Then G0 ~ S6 x (sr) and the G0-orbit of ((-1 - n)/2, (1 - n)/2, (1 -n)/2, (3 - n)/2) in SFp consists of twelve points defined by
(-1 ± n)/2, (-1 ± n)/2, (1 ± n)/2, (3 ± n)/2, (-1 ± n)/2,
(1 ± n)/2, (-1 ± n)/2, (3 ± n)/2, (1 ± n)/2,
(3 T n)/2,
(1 ± n)/2, (3 ± n)/2, (-1 ± n)/2, (1 ± n)/2,
(3 T n)/2,
(3 ± n)/2 (1 ± n)/2 (1 ± n)/2 (-1 ± n)/2 (-1 ± n)/2
(3 T n)/2, (-1 ± n)/2, (-1 ± n)/2, (3 T n)/2
Put a = (p + 1)/2 • (1 - n). Then we find that
(a - 1,a,a,a + 1) = ((-1 - n)/2, (1 - n)/2, (1 - n)/2, (3 - n)/2) and the corresponding matrix is
1 0 0 1 1 1 Xfp (n) = ( 0 10 1 a - 1 a 0 0 11 a a+1
This matrix is equivalent to
Ypp (n)
1
—a
-1 a — 1
1 0 0 0 1 0
namely, putting
-1 a - 1 -a + 2 0 0 1
( 11 1
U (n) = I 1 a - 1 a
\ 1 a a +1
we find that det(U (n)) = -1 and U (n)-iXFp (n) = YFp (n). Let S and T be the systems of six labelled lines defined by XFp (n) and YFp (n), respectively. Namely, if the lines Li(n), Li(n) (i = 1, 2,..., 6) are defined by
Lj(n)
L4 (n) L5 (n) L6 (n)
tj = 0 (j = 1, 2, 3),
ti + t2 + t3 = 0,
ti + (a - 1)t2 + at3 = 0,
ti + at 2 + (a +1)t3 = 0,
and
L1(n) : ti +12 - t3 = 0,
L2(n) L3(n) Lj (n)
ti - at2 + (a - 1)t3 = 0, -ti + (a - 1)t2 + (-a + 2)t3 tj-3 =0(j =4, 5, 6),
then S is the system of six labelled lines Li(n),... ,L6(n) and T is the system of six labelled lines Li(n),..., L6(n). Let Cj (n) (resp. Cj (n)) be the conic tangent to the five lines Lk(n) (k = 1,..., 6, k = j) (resp. L'k(n) (k = 1,..., 6, k = j)). Then it follows from the definition that Cj(n) n Lj (n) = 0 (resp. Cj(n) n Lj(n) = 0) if and only if Cj (n) n Lj (n) = 0 (resp. Cj (n) n Lj (n) = 0). By the computations in the previous section, the conic C6(n) is defined by
2ti + (7 + 3n)t2 + (3 + n)t2 + 4(2 + n)t2t3 - 2(1 + n)tit3 + 2(3 + n)tit2 = 0. We consider the points of C6(n) n L6(n). Then since ti = - at 2 - (a + 1)t3, we find that
(3 + n)t2 + 2t2t3 + 2t§ = 0,
which is equivalent to
(t2 + 2t3)2 + (5 + 2n)t2 =0.
This implies that C6(n)nL6(n) = 0 if and only if there is m G Fp such that m2 = - 2n-5(p).
2n 5
Therefore C6(n) n L6(n) = 0 if and only if
this, we find that C4(n) n L4(n) = 0 if and only if
C5(n) n L5(n) = 0 if and only if
2n - 5 P
2n 5
P
-1, 1.
-1. By computation similar to
0
Concerning the system T, we find that
2n 5
Ci (n) n Li(n) = 0 if and only if
C2 (n) n L2(n) = 0 if and only if C3 (n) n L3(n) = 0 if and only if
P
'-2n -
P
2n 5
= -1, = -1, -1.
Since n2 = 5(p), it follows that (—5 + 2n)(—5 — 2n) = 5 = n2 (p) and therefore the conditions
2n — 5 P
theorem.
= 1 and
2n 5
= -1 are equivalent. We have thus proved the following
Theorem 5. Let p be a prime number and suppose that p = 10k - 1 or p = 10k + 1 for an integer k. Then the system of six labelled lines defined by the matrix Xfp (n) is of type III if
and only if (- ) = -1.
p
5. Systems of Other Types
Let p be a prime number. In this section, we always assume that (A1) There is n G Fp such that n2 = 5 (p).
(A2) For any m G Fp, m2 ^ 2n - 5 (p), where n is an integer given (A1). It is easy to show that si23(a - 1, a, a, a + 1) = (a, a - 1, a - 1, -a + 2) as elements of Sf . The corresponding matrix is
Xpp (n)s12
10 0 1 0 10 1
1
a — 1
0 0 1 1 a- 1 -a + 2
This matrix is equivalent to
/ (1 - 2a)/5 (1 - 2a)/5 (3 + 4a)/5 1 0 0 Yfp (n)s123 = I (1 - 2a)/5 (1 + 3a)/5 (-2 - a)/5 0 1 0 P \ (3 + 4a)/5 (-2 - a)/5 (-1 - 3a)/5 0 0 1
Let si23S and si23T be the systems of six labelled lines defined by XFp (n)s123 and YFp (n)s123, respectively. Lines Li(n)s123, Li(n)s123 (i = 1, 2,..., 6) and conics C(n)s123, Cj(n)s123 are defined by using XFp(n)s123 and YFp(n)s123 as the lines Li(n), Li(n) (i = 1, 2,..., 6) and conics Ci(n) C?(n) (i = 1, 2,..., 6) by XFp (n) and YFp (n). From XFp (n)s123, YFp (n)s123, we compute the condition for which Ci(n)s123 n Li(n)s123 = 0 (i = 4, 5, 6) and C^(n)8123 n Li(n)s123 = 0 (i = 1, 2, 3). As a consequence, we easily find that
i) ii) iii)
iv)
v)
(ti : t2 (ti : t2
(ti : t2
t3) G C4(n)s123 n L4(n)s123 if and only if t2(t2 +13) = 0.
t3) G C6(n)s123 n L6(n)s123 if and only if t3(2t2 + (3 + n)t3) = 0.
t3) G C6(n)s123 n L6(n)s123 if and only if t3(2t1 + nt2) = 0.
Ci (n)s123 n Li(n)s123 C1 (n)s123 n Lii(n)s123
0 if and only if 2n - 5 = m2 (p) for some m G Fp. 0 if and only if there is (t2 : t3) G Pi(Fp) such that
2t2 + ( —1 + n)t2t3 + 2t3 =0.
( vi) C3(n)s123 n L3(n)s123 ^ 0 if and only if 2n — 5 = m2 (p) for some m G Fp.
P
We continue the computation in the case (v). Since
(n - 1)2(2n + 5) = 10 + 2n (p),
it follows that
8{2t2 + (-1 + n)t2ts + 2t23} = {4t2 + (n - 1)ts}2 + (10 + 2n)t2
= {4t2 + (n - 1)ts}2 - (n - 1)2(-2n - 5)t2 (p).
Then we find from (A2) that there is no point (t2 : ts) G P1(Fp) satisfying the condition 2t'2 + (-1+ n)t2t3 + 2t§ = 0. Summarizing the computation above and noting that Ci(n
)S123n
Lj(n)si23 = 0 if and only if Ci(n)Si23 n Li(n)Si23 = 0, we conclude the following:
(i) Cj(n)si23 n Lj(n)si23 = 0 (i = 1, 2, 3),
(ii) Cj(n)si23 n Lj(n)si23 = 0 (i = 4, 5, 6).
By direct computation, we find that si45si2s(a - 1,a,a,a + 1) = (a - 1, 3a - 4, a, -a + 3) as elements of Sf . The corresponding matrix is
Xfp (n)
This matrix is equivalent to
S 145 S 123 _
YFp (n)s
1 0 0 1 1 1 0 1 0 1 a - 1 3a - 4 0 0 1 1 a- 1 -a + 3
3 + 4a -1 - 2a -1 - 2a 1 0 0 3 + 4a -2 - 3a -1 - a 0 1 0 -5 - 8a 3 + 5a 2 + 3a 0 0 1
Let
Li(n)si45si23, Li(n)si45si23 (i =1, 2,..., 6)
be the lines constructed from XFp(n)si45si23, YFp(n)si45si23, respectively defined similarly to the cases Lj(n), Li(n). Then it follows from direct computation that
(i) (ii) (iii)
(iv)
(ti : t2 (ti : t2 (ti : t2 (ti : t2
S145S123 s 1
t3) G C4(n)S145S123 n L4(n) t3) G C5(n)S145S123 n L5(n)S145S123 t3) G C6(n)S145S123 n L6(n)S145S123 t3) G Ci(n)S145S123 n Li(n)S145S123
(v)
(vi)
)S145S123 ^ S145 S1
t2{2t2 + (n - 1)t3} = 0. t3{2t2 + (n - 1)t3} = 0. (2t2 + t3)2 - (2n - 5)t§ =0.
{2nt2 + (2 - n)t3}2
-(2n - 5)t§ =
t2(t2 - t3) =0. (t2 - t3){2t2 + (n - 3)t3}
0.
0.
= 0. = 0 if
(ti : t2 : ts) G C2(n)si45si23 n L2(n)s (t1 : t2 : ts) G CS(n)si45si23 n L's(n)Si45Si23 By the condition (A2), C6(n)Si45Si23nL6(n)Si45Si23 = 0 and C[(n)Si45Si23nLi(n)Si45Si23 Summarizing the computation above and noting that Cj(n)Si45Si23 n Lj(n)Si45Si23 = and only if Ci(n)Si45Si23 n Li(n)Si45Si23 = 0, we conclude the following:
(i) C,(n)Si23 n Li(n)Si23 = 0 (i = 1, 6),
(ii) Cj(n)Si23 n Lj(n)Si23 = 0 (i = 2, 3, 4, 5).
By direct computation, we have sr (a - 1,a,a,a + 1) = (-a, 1 - a, 1 - a, 2 - a) as elements of Sfp . The corresponding matrix is
i 10 0 1 1 1
Xfp (n)Sr = I 0 10 1 -a 1 - a P \ 0 0 1 1 1- a 2-a
Next we have si23sr(a — 1, a, a, a + 1) corresponding matrix is
Xfp (n)Sl23Sr =
This matrix is equivalent to
(1 — a, —a, —a, 1 + a) as elements of SF . The
10 0 1 0 10 1 0 0 11
1 1
1 — a —a —a 1 + a
YFp (n)S12
— 1 + 2a —1 + 2a 7 — 4a 5 0 0
— 1 + 2a 4 — 3a —3 + a 0 5 0 7 — 4a —3 + a —4 + 3a 0 0 5
As before, let Lj(n)
Sl23Sr S123 Sr
Li(n)s
(i = 1, 2,..., 6) be the lines constructed from
XF (n)Sl23Sr, Yf (n)Sl23Sr, respectively defined similarly to the cases Lj(n), L,(n). Then
it follows from direct computation that
(i) (ti t2 ts) G C4(n)S123Sr n L4(n)
(ii) (ti t2 ts) G Cs(n)S123Sr n L5(n)
(iii) (ti t2 ts) G Ce(n)S123Sr n Le(n)
(iv) (ti t2 ts) G Ci (n)S123Sr n Li(n)
(v) (ti t2 ts) G C2 (n)S123Sr n L2(n)
(vi) (ti t2 ts) G C(n)S123Sr n LS (n)
S123 Sr S123 Sr S123 Sr S123 Sr
0.
By the condition (A2), Q(n)sl23s' 0, we conclude the following:
(i) Ci(n)Sl23Sr n Li(n)Sl23S
(ii) Cj(n)Sl23Sr n Li(n)sl23s
n Li(n)
S 123 Sr S 123 Sr
i2(i2 + is)=0.
ts{(3 + n)t2 + is}
Î2ÎS = 0.
{2t2 — (2 + n)ts}2
— (—2n — 5)t§ = 0. {4t2 — (n +1)ts}2
— (2n — 5)(n + 1)2tS =0. (2t2 — ts)2 — (—2n — 5)i§ = 0. = 0 if and only if C|(n)S123Sr n Li(n)
S123Sr
= 0 (i = 1, 2, 3), = 0 (i = 4, 5, 6).
S123S
6. Some Results on Prime Numbers
In this section, we study prime numbers satisfying the conditions (A1), (A2) introduced in
§5.
Let p be a prime integer. If p satisfies (A1), then p = 10k +1 or p = 10k — 1 for an integer k. In the sequel, we always assume that p is a prime satisfying (A1) and let n G Z be so taken that n2 = 5 (p).
It is interesting to determine such prime numbers satisfying the condition in Theorem 5. The following theorem answers this question.
Theorem 6. Let p be a prime number with 5 < p. Then there is n G Fp such that n2 = 5(p) and there is no m G Fp such that m2 = 2n — 5 (p) if and only if p = 10k — 1 for a positive integer k.
The outline of his proof is as follow. By the assumption of the theorem, we get an equation of degree 4 over Q. The field generated by one of the roots of this equation is nothing but the cyclotomic field Q(Z) generated by the fifth root Z of unity. It is well known that a rational
prime splits completely if and only if p = 1(5). This is rougly the condition that there exists the number m in Theorem 6. But p = ±1(5) by the existence of n (if p is not 2). So we have p = -1(5). Since we assumed the p is odd, we have p = -1(10), too.
Remark 4. The author proved Theorem 6 for such primes that p < 1000 by direct computation. Later T. Ibukiyama proved for an arbitrary prime p (5 < p).
7. Concluding Remarks
(1) The condition for a prime number p that there is n G Z such that n2 = 5 (p) is equivalent to the condition that the twenty seven lines on the Clebsch diagonal surface xi + x2 + x'S + X4 + X5 = 0, xi + X2 + xs + X4 + X5 =0 are defined over the prime number field Fp.
(2) Let p be a prime number such that — = -1. In this case, we consider a field
p
extension Fp(n) over Fp attaching n such that n2 = 5 in Fp. Let P2(Fp(n)) be a projective plane over Fp(n). Then it is possible to define systems of six labelled lines on P2(Fp (n)) with conditions (C1), (C2), (C3). In this case, by direct computation, the condition for the existence of a system of six labelled lines of type III and fixed by the group H(5) is equivalent to that there is no pair (a, b) G Z2 such that (an + b)2 = 2n - 5 in Fp(n), which is also equivalent to the condition that there is no pair (a, b) G Z2 such that 5a2 + b2 = -5, ab = 1 (p). It is easy to show that if p ^ ±1 (5), then there is no pair (a,b) of integers satisfying the conditions 5a2 + b2 = -5, ab = 1 (p). As a consequence, we conclude that there is a system of six labelled lines of type III and fixed by the group H(5) on P2(Fp(n)).
The author thanks his colleague Professor H. Maeda for the discussion between them on number theory being very useful to formulate Theorem 5 in §4 and Professor T. Ibukiyama for proving it and kindly explaining the proof to the author. The author was partially supported by Grand-in-Aid for Scientific Research (No. 17540013), Japan Society for the Promotion of Science.
References
[1] B.Grunbaum, Convex Polytopes. Interscience (1967).
[2] J.Sekiguchi, M.Yoshida, W(E6)-action on the configuration space of 6 points of the real projective plane. Kyushu J. Math., 51(1997), 297-354.
[3] I.Naruki, Cross ratio variety as a moduli space of cubic surfaces. Proc. London Math. Soc. 45(1982), 1-30.
