- Email: [email protected]
The six semifield planes associated with a semifield flock
ARTICLE IN PRESS
Advances in Mathematics 189 (2004) 68–87
http://www.elsevier.com/locate/aim
The six semifield planes associated with a semifield flock Simeon Balla,,1 and Matthew R. Brownb,2 a
Queen Mary, University of London, London E1 4NS, UK b University of Ghent, Galglaan 2, Gent 9000, Belgium Received 8 April 2003; accepted 13 November 2003 Communicated by Laszlo Lovasz
Abstract In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array ðaijk Þ and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions ð f ; gÞ where f and g are functions from GF ðqÞ to GF ðqÞ with the property that f and g are linear over some subfield and gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ ; q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Qð4; qÞ: We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six nonisomorphic semifield planes, unless ð f ; gÞ are of linear type or of Dickson–Kantor–Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations. r 2003 Elsevier Inc. All rights reserved. MSC: 12K10; 51E15; 51A40 Keywords: Semifields; Division rings; Translation planes; Flocks
Corresponding author. Fax: +34-934-01-5981. E-mail address: [email protected] (S. Ball). 1 Supported by British EPSRC Fellowship No. AF/990-480. 2 Acknowledges the support of Ghent University Grant GOA 12050300. 0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.11.006
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
69
1. Introduction and definitions A finite semifield S is a finite algebraic system that possesses two binary operations, addition and multiplication, which satisfy the following axioms: (S1) (S2) (S3) (S4)
Addition is a group with identity 0. aðb þ cÞ ¼ ab þ ac and ða þ bÞc ¼ ac þ bc for all a; b; cAS: There exists an element 1a0 such that 1a ¼ a ¼ a1 for all aAS: If ab ¼ 0 then at least one of a or b is zero.
Throughout this article the term semifield will refer to a finite semifield. Some authors refer to a semifield as a division ring. A finite field is a semifield. If the multiplication is associative then the semifield is a finite field. A system S is a finite pre-semifield if it satisfies all the axioms of a finite semifield except possibly (S3), i.e. it need not have a multiplicative identity. Throughout this article the term pre-semifield will refer to a finite pre-semifield. For any uAS; we can obtain a semifield from a pre-semifield S with multiplication denoted by by defining a new multiplication 3 by the rule ða uÞ3ðu bÞ ¼ a b: This semifield has an identity element u u: It is not difficult to show that the additive group of a pre-semifield is an elementary abelian p-group. The left nucleus of a pre-semifield S is defined to be L ¼ fx j xðabÞ ¼ ðxaÞb for all a; bASg: The middle nucleus is defined to be M ¼ fx j aðxbÞ ¼ ðaxÞb for all a; bASg: The right nucleus is defined to be R ¼ fx j aðbxÞ ¼ ðabÞx for all a; bASg: The left, middle and right nuclei are all finite fields containing GF ðpÞ and so S is said to have characteristic p: The pre-semifield S can be viewed as a left vector space over the left nucleus, a right vector space over the right nucleus or either a left or right vector space over the middle nucleus. Every semifield determines a projective plane and the projective plane is Desarguesian if and only if the semifield is a field. The plane p constructed from a pre-semifield S with multiplication 3 has Points : ð0; 0; 1Þ Lines : ð0; 1; aÞ ð1; a; bÞ
½0; 0; 1 ½0; 1; a
aAS
½1; a; b a; bAS
such that the point ðx1 ; x2 ; x3 Þ is incident with the line ½y1 ; y2 ; y3 if and only if y1 x3 ¼ x2 3y2 þ x1 y3 : Here, by definition, 1x ¼ x1 ¼ x and 0x ¼ x0 ¼ 0: It is a simple matter to check that any two points of p are incident with a unique line and dually the any two lines of p are incident with a unique point and hence that p is a projective plane. We call p the plane coordinatised by S: We would like to know when two pre-semifields S and S0 determine the same projective plane. Let S and S0 be two pre-semifields of
ARTICLE IN PRESS 70
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
characteristic p with multiplication and 3; respectively. An isotopism from S to S0 is a triple ðF ; G; HÞ of non-singular linear transformations from S to S0 over GF ðpÞ such that F ðaÞ3GðbÞ ¼ Hða bÞ for all a; b; cAS: Two pre-semifields S and S0 are isotopic if there is an isotopism from S to S0 : We have the following theorem due to Albert [1], a proof of which can also be found in [17]. Theorem 1.1. Two pre-semifields coordinatise the same projective plane if and only if they are isotopic. Note that the semifield we constructed from a pre-semifield is isotopic to the presemifield and as a consequence of the above will therefore coordinatise the same projective plane. Let us consider a projective plane p coordinatised by a semifield S with multiplication 3 and define Um ¼ fðx; yÞAS S j y ¼ x3mg: The pair ðx; yÞAUm if and only if the point ð1; x; yÞ is on the line ½1; m; 0 : Let n be the rank (vector space dimension) of the vector space S viewed as a left vector space over its left nucleus L: If ðx1 ; y1 Þ and ðx2 ; y2 ÞAUm and l; mAL then ly1 þ my2 ¼ lðx1 3mÞ þ mðx2 3mÞ ¼ ðlx1 Þ3m þ ðmx2 Þ3m ¼ ðlx1 þ mx2 Þ3m and hence Um is a subspace of the vector space S S: The equation y ¼ x3m has a unique solution y for each xAS and hence Um is of rank n: Moreover, for all m; lAS; either Um ¼ Ul or Um -Ul ¼ fð0; 0Þg: If we put UN ¼ fð0; yÞ j yASg then, fUm j mAS,fNgg is a spread S of rank n subspaces of the vector space S S of rank 2n: This spread has the property that there is a group that acts on the elements of the spread fixing one element point-wise ðUN Þ and acting regularly on the others. A spread with this property is called a semifield spread. The Andre´ construction of a (affine) translation plane from a spread (also referred to as the Bruck–Bose construction) takes as points the vectors of a vector space and as lines the cosets of the elements of a spread. The projective completion of the affine plane constructed via the Andre´ construction from the spread S is the plane p coordinatised by the semifield S: The dual plane p is the semifield plane that can be coordinatised by the semifield S which is the semifield with multiplication defined by a b ¼ b3a: The corresponding semifield spread S is given by the elements Um and UN ; where Um ¼ fðx; yÞ j y ¼ x m ¼ m3xg
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
71
for each mAS: Let r be the rank of the vector space S viewed as a right vector space over its right nucleus R: Then S is a spread of subspaces of rank r of the right vector space S S of rank 2r over the right nucleus R . We have seen that the dual plane of a semifield plane is also a semifield plane. The following theorem is an important characterisation of semifield planes, see [14, Chapter 8]. Theorem 1.2. A plane is a translation plane and its dual plane is also a translation plane if and only if the plane is coordinatised by a semifield. Although the following theorem is well known we have included a short proof since we were unable to find a suitable reference. Theorem 1.3. Let p be a translation plane constructed from a spread S: The spread S is a semifield spread if and only if the plane p is coordinatised by a semifield. Proof. Suppose S is a semifield spread with a group G fixing the element U of S pointwise and acting regularly on S\fUg: Let p be the projective plane constructed from S with ideal points ½X for each X AS: The group generated by G and the translations of p fixing ½U ; fix ½U linewise and act transitively on the lines of p not incident with ½U : Hence the dual projective plane is a translation plane with translation line ½U and so p is coordinatised by a semifield. Now suppose that p is coordinatised by a semifield S with multiplication : The maps ðx; yÞ-ðx; y þ x aÞ fix the line fð0; yÞ j yASg pointwise and act regularly on the other lines incident with ð0; 0Þ: Hence the associated spread is semifield. &
2. Spread sets, dual spreads and the Knuth cubical array A spread set C is a set of qn ðn nÞ-matrices whose entries come from GF ðqÞ; with the property that for all M; NAC with MaN; detðM NÞa0: We can construct a spread of subspaces of rank n in a vector space of rank 2n from a spread set (hence the name) in the following way. Let UM be the subspace spanned by the rows of the matrix ðI j MÞ and let UN be the subspace spanned by the rows of the matrix ð0 j IÞ; where I is the ðn nÞ identity matrix. The set of subspaces fUM j MAC,fNgg
ARTICLE IN PRESS 72
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
is a spread. This can be checked in the following way. One shows that if there is a vector in both UM and UN then there exists a linear combination of the rows of M N which is zero, contradicting the spread set property. Conversely for any spread there is an equivalent spread which can be constructed as above, see [8]. Each column of the ðn 2nÞ-matrix M I considered as a point of the vector space of rank 2n over GF ðqÞ; dualises, with respect to the standard inner product, to a hyperplane which contains all the elements of UM since M ðI j MÞ ¼ 0: I w spanned by the rows In the dual space the subspace UM dualises to the subspace UM of the matrix
ðI j M T Þ where M T denotes the transpose of the matrix M: We can construct the spread Sw which is equivalent to the spread w j MAC,fNgg fUM
from the spread set Cw which we define to be fM T j MACg: We denote the translation plane constructed via the Andre´ construction from the spread Sw as pw : Let us consider again the spread S ¼ fUm j mAS,fNgg constructed from the semifield S as in the previous section. We write the vector x as Pn1 P x ¼ n1 i¼0 xi ei and m ¼ j¼0 mj ej so that ! ! n1 n1 n1 X X X xi ei 3 m j ej ¼ xi mj aijk ek ; y ¼ x3m ¼ i¼0
j¼0
i; j;k¼0
P where ei 3ej ¼ n1 k¼0 aijk ek : If we let Mm be the ðn nÞ matrix whose ikth entry is Pn1 m a then the above equation can be written in matrix form as y ¼ xMm : The j ijk j¼0 vector ðx; yÞAUm satisfies ðx; yÞ ¼ xðI j Mm Þ
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
73
and hence Um is the subspace spanned by the rows of ðI j Mm Þ: The set of matrices fMm j mASg is a spread set whose corresponding spread gives rise via the Andre´ construction to the semifield plane p coordinatised by S: Note that the spread set is closed under addition, which characterises spread sets arising from semifield spreads. The multiplication 3 of a semifield S determines and is determined by the P multiplication of the basis elements ei 3ej ¼ n1 k¼0 aijk ek : The elements ðaijk Þ form what Knuth [17] describes as a 3-cube, an example of a cubical array. The multiplication defined by x y ¼ y3x of the semifield S is described by the cubical array ðajik Þ: Knuth noticed that any permutation of the subscripts would determine a semifield from which one can of course construct a semifield plane. The matrices MmT P have ikth entry n1 j¼0 mi akji and working our way back through the argument in the previous paragraph, these are the matrices of the spread set of the semifield determined by the cubical array ðakji Þ: So the permutation of the indices ð13Þ will give the multiplication of the semifield Sw that coordinatises the plane pw ; which can be constructed from the spread via the Andre´ construction, that is the dual of the spread that we obtain from the semifield S: Now as Knuth noted there are six permutations of the indices giving possibly six non-isotopic semifields. He does not realise the geometric interpretation of the permutations of the indices as he looks at the permutation ð23Þ: However, now we see that this takes the semifield S to Sw and the plane p to pw :
The six semifield planes associated to p:
3. Rank 2 commutative semifields A semifield S with multiplication 3 is commutative if a3b ¼ b3a for all a; bAS: The translation plane p coordinatised by a commutative semifield is self-dual. Note however that commutativity is not an isotopic invariant and therefore that a semifield plane that is self-dual could be coordinatised by a non-commutative semifield. It is a simple matter to check that the left nucleus L is equal to the right nucleus R for a commutative semifield. Indeed xAL if and only if for all a; bAS x3ða3bÞ ¼ ðx3aÞ3b:
ARTICLE IN PRESS 74
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
The commutativity implies that this is if and only if ðb3aÞ3x ¼ b3ða3xÞ and hence if and only if xAR: Cohen and Ganley [7] are concerned with commutative semifields that are of rank 2 over their middle nucleus GF ðqÞ: When q is even they show that all such semifields are GF ðq2 Þ and for q odd they prove the following theorem. The proof is short and so we include it here. Theorem 3.1. A commutative semifield S of rank 2 over its middle nucleus M ¼ GF ðqÞ; q odd, can be represented by the set fðx; yÞ j x; yAGF ðqÞg such that multiplication 3 is determined by two additive functions f and g from GF ðqÞ to GF ðqÞ with the property that gðxÞ2 þ 4xf ðxÞ is a non-square in GF ðqÞ for all xAGF ðqÞ and ðx; yÞ3ðu; vÞ ¼ ðxv þ yu þ gðuxÞ; yv þ f ðuxÞÞ: Proof. Let aAS\M: Then f1; ag forms a basis for S over its middle nucleus. ðx3a þ yÞ3ðu3a þ vÞ ¼ ðx3aÞ3ðu3aÞ þ ðx3aÞ3v þ y3ðu3aÞ þ y3v ¼ ðu3ðx3aÞÞ3a þ ða3xÞ3v þ ðy3uÞ3a þ y3v ¼ ððxu3aÞ3aÞ þ ðxv þ yuÞ3a þ yv: Now let f and g by functions from GF ðqÞ to GF ðqÞ such that ðx3aÞ3a ¼ gðxÞ3a þ f ðxÞ: The distributive laws, (S2) in the axioms of a semifield, imply that f and g are additive functions, or in other words that f ðx þ yÞ ¼ f ðxÞ þ f ðyÞ and gðx þ yÞ ¼ gðxÞ þ gðyÞ for all x; yAGF ðqÞ: And ðx3a þ yÞ3ðu3a þ vÞ ¼ ðxv þ yu þ gðuxÞÞ3a þ yv þ f ðuxÞ: We have only to check axiom (S4) now so we assume that xv þ yu þ gðuxÞ ¼ 0 and yv þ f ðuxÞ ¼ 0: Then eliminating y and writing U ¼ ux and V ¼ vx we have that V 2 þ gðUÞV f ðUÞU ¼ 0:
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
75
If U or V is 0 then either ðx; yÞ or ðu; vÞ is ð0; 0Þ: Hence (S4) is always satisfied if for every non-zero U this has no solutions which is if and only if gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ : & We call a pair of functions ð f ; gÞ which are additive and have the property that gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ a Cohen–Ganley pair (of functions). We say that two ˆ are equivalent if there Cohen–Ganley pairs of functions ð f ; gÞ and ð f;ˆ gÞ corresponding commutative semifields of rank 2 over their middle nuclei are isotopic. The following are all the known examples of inequivalent Cohen–Ganley pairs of functions. The Cohen–Ganley pairs can be used to construct flocks of the quadratic cone, ovoids of Qð4; qÞ and, as we have seen, translation planes. It is because of this that the known examples are attributed to various people since their discovery occurred in different settings. 1. The linear example where f ðxÞ ¼ mx and gðxÞ ¼ 0: 2. The Dickson [9], Kantor [15], Knuth [17] example where f ðxÞ ¼ mxs ; gðxÞ ¼ 0; m is a non-square in GF ðqÞ and s is a non-trivial automorphism of GF ðqÞ: 3. The Cohen–Ganley [7], Thas–Payne [24] example where q ¼ 3h ; f ðxÞ ¼ m1 x þ mx9 and gðxÞ ¼ x3 with m a non-square in GF ðqÞ: 4. The Penttila–Williams [20] example where q ¼ 35 ; f ðxÞ ¼ x9 and gðxÞ ¼ x27 :
4. Flocks of the quadratic cone Let K be a quadratic cone of PGð3; qÞ with vertex v: A flock F of K is a partition of K\fvg into q conics. Two flocks F and F0 are equivalent if there exists an element of the stabiliser group of K that is a bijection between the planes of F and the planes of F0 : The quadratic cones of PGð3; qÞ are equivalent under an element of PGLð4; qÞ: Therefore, we let v be the point /0; 0; 0; 1S and let the conic C; defined by the equation X0 X1 ¼ X22 in the plane p with equation X3 ¼ 0; be the base of the cone K: The planes determined by the conics can be written as pt : tX0 f ðtÞX1 þ gðtÞX2 þ X3 ¼ 0; where tAGF ðqÞ and f ; g are functions from GF ðqÞ to GF ðqÞ and this flock is denoted Fð f ; gÞ: The line that is incident with both the planes pt and ps is /ð f ðtÞ f ðsÞ; t s; 0; tf ðsÞ sf ðtÞÞ; ðgðtÞ gðsÞ; 0; ðt sÞ; tgðsÞ sgðtÞÞS: A point on this line /lð f ðtÞ f ðsÞÞ þ nðgðtÞ gðsÞÞ; lðt sÞ; nðt sÞ; lðtf ðsÞ sf ðtÞÞ þ nðtgðsÞ sgðtÞÞS
ARTICLE IN PRESS 76
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
is skew from K and therefore for l and nAGF ðqÞ not both zero the equation 0 ¼ ðt sÞn2 ðgðtÞ gðsÞÞln ð f ðtÞ f ðsÞÞl2 has no solutions for sat: Hence for q odd, the pair of functions ð f ; gÞ will give a flock if and only if ðgðtÞ gðsÞÞ2 þ 4ðt sÞð f ðtÞ f ðsÞÞ is a non-square in GF ðqÞ for all distinct t and sAGF ðqÞ: Note that if f and g are additive functions then the pair of functions ð f ; gÞ will give a flock if and only if they are a Cohen–Ganley pair. The following construction of a spread of PGð3; qÞ from a flock of the quadratic cone is due to Thas and Walker [25]. Let Qþ ð5; qÞ denote the hyperbolic quadric. A canonical form for such a quadric is Qþ :¼ X0 X1 þ X2 X3 þ X4 X5 : The quadratic cone K is embedded in Qþ ; it is the intersection of the hyperplanes X4 ¼ 0 and X2 þ X3 ¼ 0 and Qþ : The associated bilinear form to Qþ is bðX ; Y Þ :¼ X0 Y1 þ Y0 X1 þ X2 Y3 þ Y2 X3 þ X4 Y5 þ Y4 X5 : Let p> t denote the plane that is dual to the plane pt dualising with respect to the bilinear form b: Then p> t ¼ /ðf ðtÞ; t; 0; gðtÞ; 1; 0Þ; ð0; 0; 0; 0; 0; 1Þ; ð0; 0; 1; 1; 0; 0ÞS ¼ f/ f ðtÞ; t; u; u þ gðtÞ; 1; sS j s; uAGF ðqÞg,f/0; 0; u; u; 0; sS j s; uAGF ðqÞg: The plane pt meets the quadric Qþ in a conic and hence likewise the plane p> t meets þ þ > the quadric Q in a conic. The points of this intersection Q -pt are f/ f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS j uAGF ðqÞg,f/0; 0; 0; 0; 0; 1Sg: þ > For any point xAQþ -p> t and yAQ -ps with xay;
bðx; yÞ ¼ sf ðtÞ tf ðsÞ þ uw þ ugðsÞ þ uw þ wgðtÞ þ sf ðsÞ w2 wgðsÞ þ tf ðtÞ u2 ugðtÞ ¼ ðs tÞð f ðsÞ f ðtÞÞ þ ðu wÞðgðsÞ gðtÞÞ ðu wÞ2 is non-zero by the flock condition. We could also argue that x and y are not orthogonal in Qþ geometrically. The plane pt and ps are incident with a line l that is skew from Qþ : The space l > meets the quadric Qþ in an elliptic quadric, no two of
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
77
> whose points are orthogonal, and contains p> t and ps : Hence the set
[
Qþ -p> t
tAGF ðqÞ
¼ f/ f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS j t; uAGF ðqÞg,f/0; 0; 0; 0; 0; 1Sg is an ovoid of Qþ : The Klein correspondence takes an ovoid of Qþ ð5; qÞ to a spread of PGð3; qÞ and vice versa and in this case one can check using Plu¨cker coordinates that the lines of the corresponding spread Sð f ; gÞ are f/ð1; 0; f ðtÞ; uÞ; ð0; 1; u þ gðtÞ; tÞS j t; uAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg: Indeed the point / f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS on X0 X1 þ X2 X3 þ X4 X5 ¼ 0 is the point /p12 ; p03 ; p31 ; p02 ; p01 ; p23 S where pij are the Plu¨cker coordinates of the line. Let p be the plane constructed from this spread via the Andre´ construction. This spread Sð f ; gÞ is that constructed from the spread set Cð f ; gÞ :¼
f ðtÞ u u þ gðtÞ t
j u; tAGF ðqÞ :
Theorem 4.1. The spread Sð f ; gÞ of PGð3; qÞ f/ð f ðtÞ; u þ gðtÞ; 0; 1Þ; ðu; t; 1; 0ÞS j t; uAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg; is self-dual. Proof. Under the Klein correspondence a duality of PGð3; qÞ is equivalent to a collineation of PGð5; qÞ that fixes Qþ ð5; qÞ and interchanges the two classes of generators of Qþ ð5; qÞ (see [12, Chapter 15]). Using this we will work in Qþ ð5; qÞ: Following the Thas/Walker construction above we embed the quadratic cone K in S Qþ ð5; qÞ and if F ¼ fpt : tAGF ðqÞg; then the ovoid OðFÞ is tAGF ðqÞ Qþ ð5; qÞ-p> t : Now any collineation fixing Qþ ð5; qÞ and K pointwise must also fix OðFÞ; so we look for such a collineation that also interchanges the two classes of generators of Qþ ð5; qÞ: The polar image of /KS is a line of PGð5; qÞ tangent to Qþ ð5; qÞ at the point v which is the vertex of K: Let u be any other point on this line. The collineation mu (see [13, Chapter 22]) that acts by x/x ðbðx; uÞ=Qþ ðuÞÞu is an involution that fixes Qþ ð5; qÞ and u: Further, if p is a point of Qþ ð5; qÞ; then mu fixes p if up is a tangent to Qþ ð5; qÞ and interchanges p with p0 if pu meets Qþ ð5; qÞ again in p0 : Any generator p of Qþ ð5; qÞ meets u> in a line which is fixed by mu which implies that p and mu ðpÞ intersect in this line. Consequently, p and mu ðpÞ belong to different classes, and mu is of the required form. &
ARTICLE IN PRESS 78
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
Lemma 4.2. Let O be an ovoid of Qþ ð5; qÞ arising from a flock by the Thas/Walker construction, that is, there is a line c tangent to O at a point v such that O is the union of q conics each of whose planes contains c: If there is a second such tangent c0 ac to O; then O is an elliptic quadric. Proof. First suppose that the line c0 also contains the point v: Consider a conic C contained in O; containing v and with tangent c0 at v: Taking the span of c with the points of C\fvg in turn yields the q conic sections of O with tangent c at v: Further, these are all contained in the space spanned by the plane of C and c: Hence O must be an elliptic quadric. Next, suppose that c0 is tangent to O at a point uav and that c and c0 intersect in a point w: The plane /u; v; wS meets O in a conic C; and let C0 be a second conic contained in O with tangent c at v: For each point of xAC0 \fvg the plane /c0 ; xS meets O in a conic. Thus in the three-dimensional subspace P generated by C and C0 there are at least q2 =2 þ q þ 1 points of O and at least q2 =2 points of O\C: Since this is more than half the points of O\C it follows that no other three space containing C may contain a point of O\C: Hence, O is contained in P and is an elliptic quadric. Finally, suppose that c0 is tangent to O at a point uav and that c and c0 are skew. Let C be any conic section of O containing u and with tangent c0 at u: The span of c with any point of C must meet O in a conic section with tangent c at v: Since there are only q such sections it follows that at least one of these planes contains two points of C: Thus the plane of C must contain a point of c and so C is contained in /c; c0 S: It follows that O is contained in /c; c0 S and is hence an elliptic quadric. & Note that Gevaert, Johnson and Thas [11] have a stronger version of this result requiring that O has only one additional conic to those whose planes contain c: Theorem 4.3. Let F and G be flocks of a quadratic cone in PGð3; qÞ; then the following are equivalent: 1. The flocks F and G are equivalent. 2. The spreads SðFÞ and SðGÞ are isomorphic. 3. The planes pðFÞ and pðGÞ that are constructed from SðFÞ and SðGÞ via the Andre´ construction are isomorphic. Proof. These equivalences come from [10, Theorem 7.3], but we shall also provide our own proof. The equivalence of 2 and 3 is found in [2], so we will now prove the equivalence of 1 and 2. Following the Thas/Walker construction let SðFÞ be constructed from the flock F of the cone KF embedded in Qþ with /KF S> ¼ cF ; and similarly for SðGÞ replacing F with G: Firstly, suppose that the spreads SðFÞ and SðGÞ are equivalent. We shall consider the classical and non-classical cases separately. If they are both regular spreads, then they correspond to elliptic quadric ovoids EðFÞ and EðGÞ of Qþ and there is an
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
79
automorphism of Qþ mapping EðFÞ to EðGÞ: Further, since the group of an elliptic quadric ovoid of Qþ is induced by the group of Qþ we may also assume that cF is mapped onto cG : From this it follows that the same automorphism must map KF onto KG and F onto G: Now suppose that SðFÞ and SðGÞ are not classical. The corresponding ovoids OðFÞ and OðGÞ of Qþ are the union of q conics about cF and cG ; respectively. Since the spreads are equivalent it follows that there is an automorphism of Qþ that maps OðFÞ onto OðGÞ: Further, by applying Lemma 4.2 this automorphism must also map cF onto cG and F onto G: Next suppose that F and G are equivalent flocks. Since the group of Qþ is transitive on quadratic cones we may assume that F and G are embedded as flocks of the same quadratic cone of Qþ : Since the group of Qþ induces the full group of the quadratic cone it follows that there is an automorphism of Qþ mapping F to G: We may also assume that this automorphism fixes the generator classes of Qþ : (Since in the proof of Theorem 4.1 we saw an automorphism of Qþ fixing the subspace of a quadratic cone pointwise and swapping the generator classes of Qþ :) Under the Klein correspondence this automorphism of Qþ becomes a collineation of PGð3; qÞ mapping SðFÞ to SðGÞ: & The following theorem on flocks is due to Thas [22]. Theorem 4.4. A flock whose planes are all incident with a common point is either linear (in which case the planes of the flock share a common line) or equivalent to a semifield flock of Dickson, Kantor, Knuth type. If the functions f and g are additive functions then the spread Sð f ; gÞ is a semifield spread. For this reason we call the flock Fð f ; gÞ where the functions f and g are additive a semifield flock. The maximal subfield with the property that f and g are linear over the subfield is called the kernel of the (semifield) flock. Let Fð f ; gÞ be a semifield flock. The functions f and g are a Cohen–Ganley pair and can be used to construct a commutative semifield of rank 2 over its nucleus following Section 3. In Section 2, we saw that for a semifield spread the matrices in the corresponding spread set determine the multiplication in the semifield. The matrices of the spread set Cð f ; gÞ determine the multiplication 3 in the pre-semifield S that coordinatises the plane pð f ; gÞ constructed via the Andre´ construction from the spread Sð f ; gÞ: Indeed f ðtÞ u ðx; yÞ3ðu; tÞ ¼ ðx; yÞ ¼ ðuy þ xf ðtÞ þ ygðtÞ; ux þ tyÞ: u þ gðtÞ t
5. Ovoids of Qð4; qÞ The generalised quadrangle Qð4; qÞ is the structure of totally isotropic points and lines of a non-degenerate quadric of PGð4; qÞ: Let Q be the non-degenerate quadratic
ARTICLE IN PRESS 80
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
form defined by QðX Þ :¼ X0 X4 þ X1 X3 X22 : An ovoid of Qð4; qÞ is equivalent to an ovoid containing /0; 0; 0; 0; 1S and for any such ovoid OðF Þ there is a function F such that OðF Þ ¼ f/1; y; x; F ðx; yÞ; x2 yF ðx; yÞS j x; yAGF ðqÞg,f/0; 0; 0; 0; 1Sg: Thas [23] and later Lunardon [19] showed that it is possible to construct an ovoid of Qð4; qÞ from a semifield flock and vice versa. Lunardon also proves that two such ovoids are equivalent if and only if the corresponding semifields are equivalent. Lavrauw [18] (see also [3]) explicitly calculates the polynomial F ðx; yÞ from the flock Fð f ; gÞ: Let GF ðq0 Þ be the kernel of the semifield flock, that is the largest subfield of GF ðqÞ over which f and g are linear. Then f and g can be written as f ðxÞ ¼
n1 X
i
ci xq0
and
gðxÞ ¼
i¼0
n1 X
i
bi xq0
i¼0
for some ci ; bi AGF ðqÞ: The semifield flock Fð f ; gÞ is in one-to-one correspondence ˆ þ gðxÞ ˆ with the ovoid OðF Þ of Qð4; qÞ given by F ðx; yÞ ¼ fðyÞ where ˆ ¼ fðyÞ
n1 X i¼0
ni
ðci yÞq0
and
ˆ gðxÞ ¼
n1 X
ni
ðbi xÞq0 :
i¼0
This ovoid OðF Þ has the property that there is a distinguished point, namely /0; 0; 0; 0; 1S; such that for each line c of Qð4; qÞ incident with /0; 0; 0; 0; 1S there is an automorphism group of Qð4; qÞ fixing OðF Þ; fixing /0; 0; 0; 0; 1S linewise, fixing c pointwise, and for each PAc\f/0; 0; 0; 0; 1Sg acts transitively on the set of points of OðF Þ\f/0; 0; 0; 0; 1Sg collinear with P: An ovoid with this property is called a translation ovoid with respect to the point P, or sometimes just a translation ovoid, see [6]. Using the Klein correspondence the points of an ovoid of Qð4; qÞ corresponds to # Þ of the symplectic generalised quadrangle W ðqÞ and vice versa. a spread SðF Theorem 5.1 (Lunardon [19], Bloemen [5]). An ovoid OðF Þ of Qð4; qÞ is a translation # Þ of W ðqÞ is a ovoid with respect to a point if and only if the corresponding spread SðF semifield spread. Proof. If OðF Þ is a translation ovoid with respect to the point P; then by [6] there is a group G of collineations of PGð4; qÞ fixing Qð4; qÞ; the point P; the lines of Qð4; qÞ on P; and acting regularly on OðF Þ\fPg: Embedding Qð4; qÞ in the Klein quadric Qþ ð5; qÞ as a hyperplane intersection we can extend G to a group of PGð5; qÞ fixing Qþ ð5; qÞ and the generators (the totally isotropic planes) of Qþ ð5; qÞ on P: Under the # Þ stabilised by a group fixing Klein correspondence this is equivalent to a spread SðF
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
81
one element pointwise and acting regularly on the remaining lines, that is, a semifield spread. & The previous theorem implies that from a translation ovoid of Qð4; qÞ we obtain a semifield. One may expect that this semifield will be that constructed directly from the corresponding semifield flock, however we shall see that in general this is not the # that coordinatises case. Firstly, we calculate the multiplication for the semifield S # Þ the plane pðF # Þ; where pðF # Þ is the plane constructed from the symplectic spread SðF # Þ via the Andre´ construction. One can check that the spread SðF f/ð1; 0; F ðx; yÞ; xÞ; ð0; 1; x; yÞS j x; yAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg corresponds via the Klein correspondence to the ovoid OðF Þ: Indeed the point /1; y; x; F ðx; yÞ; x2 yF S on X0 X4 þ X1 X3 X22 ¼ 0 is /p01 ; p03 ; p02 ; p21 ; p32 S in the Plu¨cker coordinates of the line /ð1; 0; F ðx; yÞ; xÞ; ð0; 1; x; yÞS: This is the spread constructed from the spread set F ðx; yÞ x j x; yAGF ðqÞ : x y # with As before we conclude that the plane p# is coordinatised by the pre-semifield S multiplication 3; F ðu; vÞ u ðx; yÞ3ðu; vÞ ¼ ðx; yÞ ¼ ðxF ðu; vÞ þ uy; xu þ yvÞ; u v ˆ þ xgðuÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ:
Theorem 5.2. The following are equivalent: 1. The translation ovoids OðF Þ and OðGÞ are equivalent. # Þ and SðGÞ # 2. The spreads SðF are equivalent. # Þ and SðGÞ # 3. The planes pðF # Þ and pðGÞ # that are constructed from SðF via the Andre´ construction are isomorphic.
Proof. By the Klein correspondence 1 and 2 are equivalent, while 2 and 3 are equivalent comes from [2]. &
ARTICLE IN PRESS 82
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
We conclude this section with the following theorem which is a straightforward consequence of Theorem 4.4. However, it is important to note that this shows in # constructed general for a Cohen–Ganley pair of functions ð f ; gÞ that the semifield S from the translation ovoid of Qð4; qÞ is not isotopic to the semifield S constructed from the semifield flock. Theorem 5.3. Let ð f ; gÞ be a Cohen–Ganley pair of functions. The pre-semifield S constructed from the semifield flock Fð f ; gÞ with multiplication 3 given by ðx; yÞ3ðu; vÞ ¼ ðuy þ xf ðvÞ þ ygðvÞ; xu þ yvÞ # constructed from the translation ovoid O with multiplication and the pre-semifield S given by ˆ þ xgðuÞ ˆ ðx; yÞ ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ are isotopic if and only if the pair of functions ð f ; gÞ are linear or Dickson–Kantor– Knuth. # are isotopic if and only if the semifield Proof. The two pre-semifields S and S planes they coordinatise are isomorphic. By [2] this is the case if and only if the spreads of PGð3; qÞ giving rise to the planes are equivalent, that is, if and only if the spread arising from the flock F is symplectic. Via the Thas/Walker construction this is equivalent to the planes of F having a common point. By Theorem 4.4 this is the case if and only if F is linear or Dickson–Kantor–Knuth which is if and only if the pair of functions ð f ; gÞ are linear or Dickson–Kantor–Knuth.
6. The six semifields associated with a semifield flock Throughout the remainder of the article ð f ; gÞ will be a Cohen–Ganley pair of functions and Fð f ; gÞ will be the semifield flock. Let GF ðq0 Þ be the kernel of the semifield flock, the maximal subfield such that f and g can be written as f ðxÞ ¼
n1 X i¼0
i
ci xq0
and
gðxÞ ¼
n1 X
i
bi xq0
i¼0
for some ci ; bi AGF ðqÞ: Let fˆ and gˆ be the polynomials defined from f and g as in the ˆ be the translation ovoid OðF Þ where F ðx; yÞ ¼ previous section and let Oð f;ˆ gÞ ˆ ˆ fðyÞ þ gðxÞ: Let S be the pre-semifield constructed from the semifield flock Fð f ; gÞ # be the pre-semifield constructed from the translation ovoid Oð f;ˆ gÞ: ˆ We and let S # shall calculate the pre-semifields associated to S and S via the Knuth cubical array method described in Section 2.
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
83
In Section 4, we saw that the pre-semifield S has multiplication 3; where ðx; yÞ3ðu; vÞ ¼ ðuy þ xf ðvÞ þ ygðvÞ; xu þ yvÞ and that the plane p coordinatised by S is a translation plane whose spread Sð f ; gÞ can be constructed from the spread set f ðvÞ u þ gðvÞ j u; vAGF ðqÞ : u v In Section 1, we saw that the plane p ; the dual plane of p; is the plane coordinatised by S ; the semifield whose multiplication is ðx; yÞ3ðu; vÞ ¼ ðxv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ: In Theorem 4.1, we saw that the spread S is self-dual and therefore that the plane pw constructed from the spread dual to S via the Andre´ construction is isomorphic to p: The plane pw is coordinatised by a semifield Sw isotopic to S; we write SCSw : It follows immediately that the plane pw ; the plane dual to pw ; is isomorphic to the plane p and hence that the semifield Sw CS : Theorem 6.1. Let S be the spread from which the translation plane p is constructed via the Andre´ construction. Let Sw be the spread dual to the spread S : Let the semifield Sw be a semifield which coordinatises the plane pw obtained from the spread Sw via the Andre´ construction. Then the semifield Sw is isotopic to a pre-semifield with multiplication given by ˆ ˆ ðx; yÞ3ðu; vÞ ¼ ðxv þ yu; yv þ fðxuÞ þ gðxvÞÞ: Proof. This proof was significantly shortened using the ideas that appear in [16]. For all u; vAGF ðqÞ fðx; y; xv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ j x; yAGF ðqÞg is an element of the spread S : Following Section 2, we dualise with respect to the inner-product ððx; y; z; wÞ; ða; b; c; dÞÞ ¼ Trðxa þ yb þ zc þ wdÞ; 2
where TrðxÞ ¼ x þ xq0 þ xq0 þ ? þ xq=q0 is the trace function from GF ðqÞ to GF ðq0 Þ: So we want to find ða; b; c; dÞ such that Trðxa þ yb þ cðxv þ uf ðyÞ þ vgðyÞÞ þ dðxu þ yvÞÞ ¼ 0 for all x; yAGF ðqÞ: If we put x ¼ 0 then ˆ ˆ Trðyðb þ fðucÞ þ gðvcÞ þ dvÞÞ ¼ 0
ARTICLE IN PRESS 84
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
ˆ ˆ and hence b ¼ ð fðucÞ þ gðvcÞ þ dvÞ: If we put y ¼ 0 then Trðxða þ cv þ duÞÞ ¼ 0 and hence a ¼ ðcv þ duÞ: The corresponding spread element of Sw is therefore ˆ ˆ fððcv þ duÞ; ð fðucÞ þ gðvcÞ þ dvÞ; c; dÞ j c; dAGF ðqÞg: By a straightforward change of coordinates the multiplication for Sw is as claimed. & # constructed from the ovoid Oð f;ˆ gÞ ˆ of In Section 5, we saw that the pre-semifield S Qð4; qÞ has multiplication 3; where ˆ þ xgðuÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ # is a translation plane whose spread S # can be and that the plane p# coordinatised by S constructed from the spread set ( ! ) ˆ þ gðuÞ ˆ fðvÞ u j u; vAGF ðqÞ : u v The matrices in this spread set are symmetric and so the spread constructed from the spread set will be self-dual. Hence, the planes p and pw are isomorphic and # and S # w are isotopic, SC # S # w: the semifields that coordinatise these planes S # is given by The multiplication in the semifield S ˆ þ ugðxÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðufðyÞ þ xv; xu þ yvÞ # CS # w : and it follows immediately that S Theorem 6.2. Let S# be the spread from which the translation plane p# is constructed # : Let the # w be the spread dual to the spread S via the Andre´ construction. Let S # w be a semifield which coordinatises the plane p# w obtained from the spread semifield S # w is isotopic to a pre-semifield #Sw via the Andre´ construction. Then the semifield S with multiplication given by ðx; yÞ ðu; vÞ ¼ ðxv þ yu þ gðuxÞ; yv þ f ðuxÞÞ:
Proof. The proof of this theorem is similar to that of Theorem 6.1. For all u; vAGF ðqÞ ˆ þ ugðxÞ ˆ fðx; y; ufðyÞ þ xv; xu þ yvÞ j x; yAGF ðqÞg
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
85
# : Again we dualise with respect to the inner-product is an element of the spread S ððx; y; z; wÞ; ða; b; c; dÞÞ ¼ Trðxa þ yb þ zc þ wdÞ: So we want to find ða; b; c; dÞ such that ˆ þ ugðxÞ ˆ Trðxa þ yb þ cðufðyÞ þ xvÞ þ dðxu þ yvÞÞ ¼ 0 for all x; yAGF ðqÞ: If we put x ¼ 0 then Trðyðb þ f ðucÞ þ dvÞÞ ¼ 0 and hence b ¼ ð f ðucÞ þ dvÞ: If we put y ¼ 0 then Trðxða þ gðucÞ þ cv þ duÞÞ ¼ 0 # w is and hence a ¼ ðgðucÞ þ cv þ duÞ: The corresponding spread element of S therefore fððgðucÞ þ cv þ duÞ; ð f ðucÞ þ dvÞ; c; dÞ j c; dAGF ðqÞg: # w is as By a straightforward change of coordinates the multiplication for S claimed. & # w is the commutative semifield of rank 2 over its middle Note that the semifield S nucleus that we saw in Section 3. In this section, we have constructed potentially six non-isotopic semifields from a Cohen–Ganley pair of functions ð f ; gÞ: The following table we list the left, right and middle nuclei of these six semifields. We have determined the multiplication of a presemifield isotopic to each these six semifields. However, to determine the nuclei one must first calculate the multiplication in the semifield itself using the method described in Section 1. In Table 1 we list the multiplication in a pre-semifield isotopic to the relevant semifield. This table does not apply to the linear example in which all nuclei are S itself since associativity holds. In this case all the six semifields are GF ðq2 Þ: # S # ; S # w are pairwise non-isotopic Theorem 6.3. The six semifields S; S ; Sw ; S; # S CS # ; unless ð f ; gÞ; is of Dickson–Kantor–Knuth type in which case SCS; # w are three pairwise non-isotopic, or linear in which case they are all isotopic Sw CS to GF ðq2 Þ: Proof. As mentioned before in the case of the linear example all the six semifields are GF ðq2 Þ so we consider only the so-called proper semifields. If two planes p and p0 coordinatised by semifields S and S0 respectively are isomorphic then the ranks of S and S0 over their left/right nucleus are equal. Hence it is only possible that # S CS # and Sw ¼ S # w : By Theorem 5.3 SCS # if and only if S is a linear SCS;
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
86
Table 1 The nuclei of the six semifields associated with a semifield flock Semifield
ðx; yÞ3ðu; vÞ
L
M
R
S S Sw # S # S # Sw
ðyu þ xf ðvÞ þ ygðvÞ; xu þ yvÞ ðxv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ ˆ ˆ ðxv þ fðyuÞ þ gðyvÞ; xu þ yvÞ ˆ þ xgðuÞ; ˆ ðyu þ xfðvÞ xu þ yvÞ ˆ þ ugðxÞ; ˆ ðxv þ ufðyÞ xu þ yvÞ ðxv þ yu þ gðxuÞ; yv þ f ðuxÞÞ
GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ
GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ
GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ
or Dickson–Kantor–Knuth example and hence in these cases we also have that # w : In the Dickson–Kantor–Knuth examples GF ðq0 ÞaGF ðqÞ S CS and Sw CS and so by the preceding argument SI / S ; SI / Sw and S I / Sw : &
7. Literature The literature concerning this subject is somewhat confusing. As proved in [3] the so-called sporadic examples of Cohen–Ganley pairs of functions in [7] and the examples in [21] are equivalent to Dickson–Kantor–Knuth examples. The fact that the commutative semifield of rank 2 over its middle nucleus in these examples is isotopic to a Dickson–Kantor–Knuth example now follows from Theorems 4.4, 6.1, 5.2 and 6.3. Finally for q odd the only non-existence theorem concerning Cohen–Ganley pairs of functions is the following result which is a consequence of the main theorem in [4]. Theorem 7.1. Let ð f ; gÞ be a Cohen–Ganley pair of functions in GF ðqÞ½X ; q odd, and suppose that GF ðq0 Þ; q ¼ qn0 ; is the largest subfield of GF ðqÞ such that both f and g are linear over GF ðq0 Þ: If ð f ; gÞ is not linear over GF ðqÞ and not of Dickson–Kantor– Knuth type then q0 o4n2 8n þ 2:
Acknowledgments We thank W.M. Kantor for his comments and suggestions.
References [1] A.A. Albert, Finite division algebras and finite planes, Proc. Sympos. Appl. Math. 10 (1960) 53–70. [2] J. Andre´, U¨ber nicht-Desarguessche Ebenen mit transitiver Translationgruppe, Math. Z. 60 (1954) 156–186.
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
87
[3] S. Ball, M. Lavrauw, Commutative semifields of rank 2 over their middle nucleus, in: G.L. Mullen, et al., (Eds.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Springer, Berlin, 2002, pp. 1–21. [4] S. Ball, A. Blokhuis, M. Lavrauw, On the classification of semifield flocks, Adv. Math. 180 (2003) 104–111. [5] I. Bloemen, Substructures and characterizations of finite generalized polygons, Ph.D. Thesis, University of Ghent, 1995. [6] I. Bloemen, J.A. Thas, H. Van Maldeghem, Translation ovoids of generalized quadrangles and hexagons, Geom. Dedicata 72 (1998) 19–62. [7] S.D. Cohen, M.J. Ganley, Commutative semifields, two dimensional over their middle nuclei, J. Algebra 75 (1982) 373–385. [8] P. Dembowski, Finite Geometries, Springer, Berlin, 1968. [9] L.E. Dickson, Linear algebra in which division is always uniquely possible, Trans. Amer. Math. Soc. 7 (1906) 514–527. [10] H. Gevaert, N.L. Johnson, Flocks of quadratic cones, generalized quadrangles and translation planes, Geom. Dedicata 27 (1988) 301–317. [11] H. Gevaert, N.L. Johnson, J.A. Thas, Spreads covered by reguli, Simon Stevin 62 (1) (1988) 51–62. [12] J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985. [13] J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991. [14] D.R. Hughes, F.C. Piper, Projective Planes, Springer, Berlin, 1973. [15] W.M. Kantor, Ovoids and translation planes, Canad. J. Math. 34 (1982) 1195–1207. [16] W.M. Kantor, Commutative semifields and symplectic spreads, preprint. [17] D.E. Knuth, Finite semifields and projective planes, J. Algebra 2 (1965) 182–217. [18] M. Lavrauw, Scattered subspaces with respect to spreads and eggs in finite projective spaces, Ph.D. Thesis, Technical University of Eindhoven, The Netherlands, 2001. [19] G. Lunardon, Flocks, ovoids of Qð4; qÞ and designs, Geom. Dedicata 66 (1997) 163–173. [20] T. Penttila, B. Williams, Ovoids of parabolic spaces, Geom. Dedicata 82 (2000) 1–19. [21] A.R. Prince, Two new families of commutative semifields, Bull. London Math. Soc. 32 (2000) 547–550. [22] J.A. Thas, Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987) 441–452. [23] J.A. Thas, Generalized quadrangles of order ðs; s2 Þ: II, J. Combin. Theory Ser. A 79 (1997) 223–254. [24] J.A. Thas, S.E. Payne, Spreads and ovoids in finite generalized quadrangles, Geom. Dedicata 52 (1994) 227–253. [25] M. Walker, A class of translation planes, Geom. Dedicata 5 (1976) 135–146.
Advances in Mathematics 189 (2004) 68–87
http://www.elsevier.com/locate/aim
The six semifield planes associated with a semifield flock Simeon Balla,,1 and Matthew R. Brownb,2 a
Queen Mary, University of London, London E1 4NS, UK b University of Ghent, Galglaan 2, Gent 9000, Belgium Received 8 April 2003; accepted 13 November 2003 Communicated by Laszlo Lovasz
Abstract In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array ðaijk Þ and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions ð f ; gÞ where f and g are functions from GF ðqÞ to GF ðqÞ with the property that f and g are linear over some subfield and gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ ; q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Qð4; qÞ: We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six nonisomorphic semifield planes, unless ð f ; gÞ are of linear type or of Dickson–Kantor–Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations. r 2003 Elsevier Inc. All rights reserved. MSC: 12K10; 51E15; 51A40 Keywords: Semifields; Division rings; Translation planes; Flocks
Corresponding author. Fax: +34-934-01-5981. E-mail address: [email protected] (S. Ball). 1 Supported by British EPSRC Fellowship No. AF/990-480. 2 Acknowledges the support of Ghent University Grant GOA 12050300. 0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.11.006
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
69
1. Introduction and definitions A finite semifield S is a finite algebraic system that possesses two binary operations, addition and multiplication, which satisfy the following axioms: (S1) (S2) (S3) (S4)
Addition is a group with identity 0. aðb þ cÞ ¼ ab þ ac and ða þ bÞc ¼ ac þ bc for all a; b; cAS: There exists an element 1a0 such that 1a ¼ a ¼ a1 for all aAS: If ab ¼ 0 then at least one of a or b is zero.
Throughout this article the term semifield will refer to a finite semifield. Some authors refer to a semifield as a division ring. A finite field is a semifield. If the multiplication is associative then the semifield is a finite field. A system S is a finite pre-semifield if it satisfies all the axioms of a finite semifield except possibly (S3), i.e. it need not have a multiplicative identity. Throughout this article the term pre-semifield will refer to a finite pre-semifield. For any uAS; we can obtain a semifield from a pre-semifield S with multiplication denoted by by defining a new multiplication 3 by the rule ða uÞ3ðu bÞ ¼ a b: This semifield has an identity element u u: It is not difficult to show that the additive group of a pre-semifield is an elementary abelian p-group. The left nucleus of a pre-semifield S is defined to be L ¼ fx j xðabÞ ¼ ðxaÞb for all a; bASg: The middle nucleus is defined to be M ¼ fx j aðxbÞ ¼ ðaxÞb for all a; bASg: The right nucleus is defined to be R ¼ fx j aðbxÞ ¼ ðabÞx for all a; bASg: The left, middle and right nuclei are all finite fields containing GF ðpÞ and so S is said to have characteristic p: The pre-semifield S can be viewed as a left vector space over the left nucleus, a right vector space over the right nucleus or either a left or right vector space over the middle nucleus. Every semifield determines a projective plane and the projective plane is Desarguesian if and only if the semifield is a field. The plane p constructed from a pre-semifield S with multiplication 3 has Points : ð0; 0; 1Þ Lines : ð0; 1; aÞ ð1; a; bÞ
½0; 0; 1 ½0; 1; a
aAS
½1; a; b a; bAS
such that the point ðx1 ; x2 ; x3 Þ is incident with the line ½y1 ; y2 ; y3 if and only if y1 x3 ¼ x2 3y2 þ x1 y3 : Here, by definition, 1x ¼ x1 ¼ x and 0x ¼ x0 ¼ 0: It is a simple matter to check that any two points of p are incident with a unique line and dually the any two lines of p are incident with a unique point and hence that p is a projective plane. We call p the plane coordinatised by S: We would like to know when two pre-semifields S and S0 determine the same projective plane. Let S and S0 be two pre-semifields of
ARTICLE IN PRESS 70
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
characteristic p with multiplication and 3; respectively. An isotopism from S to S0 is a triple ðF ; G; HÞ of non-singular linear transformations from S to S0 over GF ðpÞ such that F ðaÞ3GðbÞ ¼ Hða bÞ for all a; b; cAS: Two pre-semifields S and S0 are isotopic if there is an isotopism from S to S0 : We have the following theorem due to Albert [1], a proof of which can also be found in [17]. Theorem 1.1. Two pre-semifields coordinatise the same projective plane if and only if they are isotopic. Note that the semifield we constructed from a pre-semifield is isotopic to the presemifield and as a consequence of the above will therefore coordinatise the same projective plane. Let us consider a projective plane p coordinatised by a semifield S with multiplication 3 and define Um ¼ fðx; yÞAS S j y ¼ x3mg: The pair ðx; yÞAUm if and only if the point ð1; x; yÞ is on the line ½1; m; 0 : Let n be the rank (vector space dimension) of the vector space S viewed as a left vector space over its left nucleus L: If ðx1 ; y1 Þ and ðx2 ; y2 ÞAUm and l; mAL then ly1 þ my2 ¼ lðx1 3mÞ þ mðx2 3mÞ ¼ ðlx1 Þ3m þ ðmx2 Þ3m ¼ ðlx1 þ mx2 Þ3m and hence Um is a subspace of the vector space S S: The equation y ¼ x3m has a unique solution y for each xAS and hence Um is of rank n: Moreover, for all m; lAS; either Um ¼ Ul or Um -Ul ¼ fð0; 0Þg: If we put UN ¼ fð0; yÞ j yASg then, fUm j mAS,fNgg is a spread S of rank n subspaces of the vector space S S of rank 2n: This spread has the property that there is a group that acts on the elements of the spread fixing one element point-wise ðUN Þ and acting regularly on the others. A spread with this property is called a semifield spread. The Andre´ construction of a (affine) translation plane from a spread (also referred to as the Bruck–Bose construction) takes as points the vectors of a vector space and as lines the cosets of the elements of a spread. The projective completion of the affine plane constructed via the Andre´ construction from the spread S is the plane p coordinatised by the semifield S: The dual plane p is the semifield plane that can be coordinatised by the semifield S which is the semifield with multiplication defined by a b ¼ b3a: The corresponding semifield spread S is given by the elements Um and UN ; where Um ¼ fðx; yÞ j y ¼ x m ¼ m3xg
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
71
for each mAS: Let r be the rank of the vector space S viewed as a right vector space over its right nucleus R: Then S is a spread of subspaces of rank r of the right vector space S S of rank 2r over the right nucleus R . We have seen that the dual plane of a semifield plane is also a semifield plane. The following theorem is an important characterisation of semifield planes, see [14, Chapter 8]. Theorem 1.2. A plane is a translation plane and its dual plane is also a translation plane if and only if the plane is coordinatised by a semifield. Although the following theorem is well known we have included a short proof since we were unable to find a suitable reference. Theorem 1.3. Let p be a translation plane constructed from a spread S: The spread S is a semifield spread if and only if the plane p is coordinatised by a semifield. Proof. Suppose S is a semifield spread with a group G fixing the element U of S pointwise and acting regularly on S\fUg: Let p be the projective plane constructed from S with ideal points ½X for each X AS: The group generated by G and the translations of p fixing ½U ; fix ½U linewise and act transitively on the lines of p not incident with ½U : Hence the dual projective plane is a translation plane with translation line ½U and so p is coordinatised by a semifield. Now suppose that p is coordinatised by a semifield S with multiplication : The maps ðx; yÞ-ðx; y þ x aÞ fix the line fð0; yÞ j yASg pointwise and act regularly on the other lines incident with ð0; 0Þ: Hence the associated spread is semifield. &
2. Spread sets, dual spreads and the Knuth cubical array A spread set C is a set of qn ðn nÞ-matrices whose entries come from GF ðqÞ; with the property that for all M; NAC with MaN; detðM NÞa0: We can construct a spread of subspaces of rank n in a vector space of rank 2n from a spread set (hence the name) in the following way. Let UM be the subspace spanned by the rows of the matrix ðI j MÞ and let UN be the subspace spanned by the rows of the matrix ð0 j IÞ; where I is the ðn nÞ identity matrix. The set of subspaces fUM j MAC,fNgg
ARTICLE IN PRESS 72
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
is a spread. This can be checked in the following way. One shows that if there is a vector in both UM and UN then there exists a linear combination of the rows of M N which is zero, contradicting the spread set property. Conversely for any spread there is an equivalent spread which can be constructed as above, see [8]. Each column of the ðn 2nÞ-matrix M I considered as a point of the vector space of rank 2n over GF ðqÞ; dualises, with respect to the standard inner product, to a hyperplane which contains all the elements of UM since M ðI j MÞ ¼ 0: I w spanned by the rows In the dual space the subspace UM dualises to the subspace UM of the matrix
ðI j M T Þ where M T denotes the transpose of the matrix M: We can construct the spread Sw which is equivalent to the spread w j MAC,fNgg fUM
from the spread set Cw which we define to be fM T j MACg: We denote the translation plane constructed via the Andre´ construction from the spread Sw as pw : Let us consider again the spread S ¼ fUm j mAS,fNgg constructed from the semifield S as in the previous section. We write the vector x as Pn1 P x ¼ n1 i¼0 xi ei and m ¼ j¼0 mj ej so that ! ! n1 n1 n1 X X X xi ei 3 m j ej ¼ xi mj aijk ek ; y ¼ x3m ¼ i¼0
j¼0
i; j;k¼0
P where ei 3ej ¼ n1 k¼0 aijk ek : If we let Mm be the ðn nÞ matrix whose ikth entry is Pn1 m a then the above equation can be written in matrix form as y ¼ xMm : The j ijk j¼0 vector ðx; yÞAUm satisfies ðx; yÞ ¼ xðI j Mm Þ
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
73
and hence Um is the subspace spanned by the rows of ðI j Mm Þ: The set of matrices fMm j mASg is a spread set whose corresponding spread gives rise via the Andre´ construction to the semifield plane p coordinatised by S: Note that the spread set is closed under addition, which characterises spread sets arising from semifield spreads. The multiplication 3 of a semifield S determines and is determined by the P multiplication of the basis elements ei 3ej ¼ n1 k¼0 aijk ek : The elements ðaijk Þ form what Knuth [17] describes as a 3-cube, an example of a cubical array. The multiplication defined by x y ¼ y3x of the semifield S is described by the cubical array ðajik Þ: Knuth noticed that any permutation of the subscripts would determine a semifield from which one can of course construct a semifield plane. The matrices MmT P have ikth entry n1 j¼0 mi akji and working our way back through the argument in the previous paragraph, these are the matrices of the spread set of the semifield determined by the cubical array ðakji Þ: So the permutation of the indices ð13Þ will give the multiplication of the semifield Sw that coordinatises the plane pw ; which can be constructed from the spread via the Andre´ construction, that is the dual of the spread that we obtain from the semifield S: Now as Knuth noted there are six permutations of the indices giving possibly six non-isotopic semifields. He does not realise the geometric interpretation of the permutations of the indices as he looks at the permutation ð23Þ: However, now we see that this takes the semifield S to Sw and the plane p to pw :
The six semifield planes associated to p:
3. Rank 2 commutative semifields A semifield S with multiplication 3 is commutative if a3b ¼ b3a for all a; bAS: The translation plane p coordinatised by a commutative semifield is self-dual. Note however that commutativity is not an isotopic invariant and therefore that a semifield plane that is self-dual could be coordinatised by a non-commutative semifield. It is a simple matter to check that the left nucleus L is equal to the right nucleus R for a commutative semifield. Indeed xAL if and only if for all a; bAS x3ða3bÞ ¼ ðx3aÞ3b:
ARTICLE IN PRESS 74
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
The commutativity implies that this is if and only if ðb3aÞ3x ¼ b3ða3xÞ and hence if and only if xAR: Cohen and Ganley [7] are concerned with commutative semifields that are of rank 2 over their middle nucleus GF ðqÞ: When q is even they show that all such semifields are GF ðq2 Þ and for q odd they prove the following theorem. The proof is short and so we include it here. Theorem 3.1. A commutative semifield S of rank 2 over its middle nucleus M ¼ GF ðqÞ; q odd, can be represented by the set fðx; yÞ j x; yAGF ðqÞg such that multiplication 3 is determined by two additive functions f and g from GF ðqÞ to GF ðqÞ with the property that gðxÞ2 þ 4xf ðxÞ is a non-square in GF ðqÞ for all xAGF ðqÞ and ðx; yÞ3ðu; vÞ ¼ ðxv þ yu þ gðuxÞ; yv þ f ðuxÞÞ: Proof. Let aAS\M: Then f1; ag forms a basis for S over its middle nucleus. ðx3a þ yÞ3ðu3a þ vÞ ¼ ðx3aÞ3ðu3aÞ þ ðx3aÞ3v þ y3ðu3aÞ þ y3v ¼ ðu3ðx3aÞÞ3a þ ða3xÞ3v þ ðy3uÞ3a þ y3v ¼ ððxu3aÞ3aÞ þ ðxv þ yuÞ3a þ yv: Now let f and g by functions from GF ðqÞ to GF ðqÞ such that ðx3aÞ3a ¼ gðxÞ3a þ f ðxÞ: The distributive laws, (S2) in the axioms of a semifield, imply that f and g are additive functions, or in other words that f ðx þ yÞ ¼ f ðxÞ þ f ðyÞ and gðx þ yÞ ¼ gðxÞ þ gðyÞ for all x; yAGF ðqÞ: And ðx3a þ yÞ3ðu3a þ vÞ ¼ ðxv þ yu þ gðuxÞÞ3a þ yv þ f ðuxÞ: We have only to check axiom (S4) now so we assume that xv þ yu þ gðuxÞ ¼ 0 and yv þ f ðuxÞ ¼ 0: Then eliminating y and writing U ¼ ux and V ¼ vx we have that V 2 þ gðUÞV f ðUÞU ¼ 0:
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
75
If U or V is 0 then either ðx; yÞ or ðu; vÞ is ð0; 0Þ: Hence (S4) is always satisfied if for every non-zero U this has no solutions which is if and only if gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ : & We call a pair of functions ð f ; gÞ which are additive and have the property that gðxÞ2 þ 4xf ðxÞ is a non-square for all xAGF ðqÞ a Cohen–Ganley pair (of functions). We say that two ˆ are equivalent if there Cohen–Ganley pairs of functions ð f ; gÞ and ð f;ˆ gÞ corresponding commutative semifields of rank 2 over their middle nuclei are isotopic. The following are all the known examples of inequivalent Cohen–Ganley pairs of functions. The Cohen–Ganley pairs can be used to construct flocks of the quadratic cone, ovoids of Qð4; qÞ and, as we have seen, translation planes. It is because of this that the known examples are attributed to various people since their discovery occurred in different settings. 1. The linear example where f ðxÞ ¼ mx and gðxÞ ¼ 0: 2. The Dickson [9], Kantor [15], Knuth [17] example where f ðxÞ ¼ mxs ; gðxÞ ¼ 0; m is a non-square in GF ðqÞ and s is a non-trivial automorphism of GF ðqÞ: 3. The Cohen–Ganley [7], Thas–Payne [24] example where q ¼ 3h ; f ðxÞ ¼ m1 x þ mx9 and gðxÞ ¼ x3 with m a non-square in GF ðqÞ: 4. The Penttila–Williams [20] example where q ¼ 35 ; f ðxÞ ¼ x9 and gðxÞ ¼ x27 :
4. Flocks of the quadratic cone Let K be a quadratic cone of PGð3; qÞ with vertex v: A flock F of K is a partition of K\fvg into q conics. Two flocks F and F0 are equivalent if there exists an element of the stabiliser group of K that is a bijection between the planes of F and the planes of F0 : The quadratic cones of PGð3; qÞ are equivalent under an element of PGLð4; qÞ: Therefore, we let v be the point /0; 0; 0; 1S and let the conic C; defined by the equation X0 X1 ¼ X22 in the plane p with equation X3 ¼ 0; be the base of the cone K: The planes determined by the conics can be written as pt : tX0 f ðtÞX1 þ gðtÞX2 þ X3 ¼ 0; where tAGF ðqÞ and f ; g are functions from GF ðqÞ to GF ðqÞ and this flock is denoted Fð f ; gÞ: The line that is incident with both the planes pt and ps is /ð f ðtÞ f ðsÞ; t s; 0; tf ðsÞ sf ðtÞÞ; ðgðtÞ gðsÞ; 0; ðt sÞ; tgðsÞ sgðtÞÞS: A point on this line /lð f ðtÞ f ðsÞÞ þ nðgðtÞ gðsÞÞ; lðt sÞ; nðt sÞ; lðtf ðsÞ sf ðtÞÞ þ nðtgðsÞ sgðtÞÞS
ARTICLE IN PRESS 76
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
is skew from K and therefore for l and nAGF ðqÞ not both zero the equation 0 ¼ ðt sÞn2 ðgðtÞ gðsÞÞln ð f ðtÞ f ðsÞÞl2 has no solutions for sat: Hence for q odd, the pair of functions ð f ; gÞ will give a flock if and only if ðgðtÞ gðsÞÞ2 þ 4ðt sÞð f ðtÞ f ðsÞÞ is a non-square in GF ðqÞ for all distinct t and sAGF ðqÞ: Note that if f and g are additive functions then the pair of functions ð f ; gÞ will give a flock if and only if they are a Cohen–Ganley pair. The following construction of a spread of PGð3; qÞ from a flock of the quadratic cone is due to Thas and Walker [25]. Let Qþ ð5; qÞ denote the hyperbolic quadric. A canonical form for such a quadric is Qþ :¼ X0 X1 þ X2 X3 þ X4 X5 : The quadratic cone K is embedded in Qþ ; it is the intersection of the hyperplanes X4 ¼ 0 and X2 þ X3 ¼ 0 and Qþ : The associated bilinear form to Qþ is bðX ; Y Þ :¼ X0 Y1 þ Y0 X1 þ X2 Y3 þ Y2 X3 þ X4 Y5 þ Y4 X5 : Let p> t denote the plane that is dual to the plane pt dualising with respect to the bilinear form b: Then p> t ¼ /ðf ðtÞ; t; 0; gðtÞ; 1; 0Þ; ð0; 0; 0; 0; 0; 1Þ; ð0; 0; 1; 1; 0; 0ÞS ¼ f/ f ðtÞ; t; u; u þ gðtÞ; 1; sS j s; uAGF ðqÞg,f/0; 0; u; u; 0; sS j s; uAGF ðqÞg: The plane pt meets the quadric Qþ in a conic and hence likewise the plane p> t meets þ þ > the quadric Q in a conic. The points of this intersection Q -pt are f/ f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS j uAGF ðqÞg,f/0; 0; 0; 0; 0; 1Sg: þ > For any point xAQþ -p> t and yAQ -ps with xay;
bðx; yÞ ¼ sf ðtÞ tf ðsÞ þ uw þ ugðsÞ þ uw þ wgðtÞ þ sf ðsÞ w2 wgðsÞ þ tf ðtÞ u2 ugðtÞ ¼ ðs tÞð f ðsÞ f ðtÞÞ þ ðu wÞðgðsÞ gðtÞÞ ðu wÞ2 is non-zero by the flock condition. We could also argue that x and y are not orthogonal in Qþ geometrically. The plane pt and ps are incident with a line l that is skew from Qþ : The space l > meets the quadric Qþ in an elliptic quadric, no two of
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
77
> whose points are orthogonal, and contains p> t and ps : Hence the set
[
Qþ -p> t
tAGF ðqÞ
¼ f/ f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS j t; uAGF ðqÞg,f/0; 0; 0; 0; 0; 1Sg is an ovoid of Qþ : The Klein correspondence takes an ovoid of Qþ ð5; qÞ to a spread of PGð3; qÞ and vice versa and in this case one can check using Plu¨cker coordinates that the lines of the corresponding spread Sð f ; gÞ are f/ð1; 0; f ðtÞ; uÞ; ð0; 1; u þ gðtÞ; tÞS j t; uAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg: Indeed the point / f ðtÞ; t; u; u þ gðtÞ; 1; tf ðtÞ u2 ugðtÞS on X0 X1 þ X2 X3 þ X4 X5 ¼ 0 is the point /p12 ; p03 ; p31 ; p02 ; p01 ; p23 S where pij are the Plu¨cker coordinates of the line. Let p be the plane constructed from this spread via the Andre´ construction. This spread Sð f ; gÞ is that constructed from the spread set Cð f ; gÞ :¼
f ðtÞ u u þ gðtÞ t
j u; tAGF ðqÞ :
Theorem 4.1. The spread Sð f ; gÞ of PGð3; qÞ f/ð f ðtÞ; u þ gðtÞ; 0; 1Þ; ðu; t; 1; 0ÞS j t; uAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg; is self-dual. Proof. Under the Klein correspondence a duality of PGð3; qÞ is equivalent to a collineation of PGð5; qÞ that fixes Qþ ð5; qÞ and interchanges the two classes of generators of Qþ ð5; qÞ (see [12, Chapter 15]). Using this we will work in Qþ ð5; qÞ: Following the Thas/Walker construction above we embed the quadratic cone K in S Qþ ð5; qÞ and if F ¼ fpt : tAGF ðqÞg; then the ovoid OðFÞ is tAGF ðqÞ Qþ ð5; qÞ-p> t : Now any collineation fixing Qþ ð5; qÞ and K pointwise must also fix OðFÞ; so we look for such a collineation that also interchanges the two classes of generators of Qþ ð5; qÞ: The polar image of /KS is a line of PGð5; qÞ tangent to Qþ ð5; qÞ at the point v which is the vertex of K: Let u be any other point on this line. The collineation mu (see [13, Chapter 22]) that acts by x/x ðbðx; uÞ=Qþ ðuÞÞu is an involution that fixes Qþ ð5; qÞ and u: Further, if p is a point of Qþ ð5; qÞ; then mu fixes p if up is a tangent to Qþ ð5; qÞ and interchanges p with p0 if pu meets Qþ ð5; qÞ again in p0 : Any generator p of Qþ ð5; qÞ meets u> in a line which is fixed by mu which implies that p and mu ðpÞ intersect in this line. Consequently, p and mu ðpÞ belong to different classes, and mu is of the required form. &
ARTICLE IN PRESS 78
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
Lemma 4.2. Let O be an ovoid of Qþ ð5; qÞ arising from a flock by the Thas/Walker construction, that is, there is a line c tangent to O at a point v such that O is the union of q conics each of whose planes contains c: If there is a second such tangent c0 ac to O; then O is an elliptic quadric. Proof. First suppose that the line c0 also contains the point v: Consider a conic C contained in O; containing v and with tangent c0 at v: Taking the span of c with the points of C\fvg in turn yields the q conic sections of O with tangent c at v: Further, these are all contained in the space spanned by the plane of C and c: Hence O must be an elliptic quadric. Next, suppose that c0 is tangent to O at a point uav and that c and c0 intersect in a point w: The plane /u; v; wS meets O in a conic C; and let C0 be a second conic contained in O with tangent c at v: For each point of xAC0 \fvg the plane /c0 ; xS meets O in a conic. Thus in the three-dimensional subspace P generated by C and C0 there are at least q2 =2 þ q þ 1 points of O and at least q2 =2 points of O\C: Since this is more than half the points of O\C it follows that no other three space containing C may contain a point of O\C: Hence, O is contained in P and is an elliptic quadric. Finally, suppose that c0 is tangent to O at a point uav and that c and c0 are skew. Let C be any conic section of O containing u and with tangent c0 at u: The span of c with any point of C must meet O in a conic section with tangent c at v: Since there are only q such sections it follows that at least one of these planes contains two points of C: Thus the plane of C must contain a point of c and so C is contained in /c; c0 S: It follows that O is contained in /c; c0 S and is hence an elliptic quadric. & Note that Gevaert, Johnson and Thas [11] have a stronger version of this result requiring that O has only one additional conic to those whose planes contain c: Theorem 4.3. Let F and G be flocks of a quadratic cone in PGð3; qÞ; then the following are equivalent: 1. The flocks F and G are equivalent. 2. The spreads SðFÞ and SðGÞ are isomorphic. 3. The planes pðFÞ and pðGÞ that are constructed from SðFÞ and SðGÞ via the Andre´ construction are isomorphic. Proof. These equivalences come from [10, Theorem 7.3], but we shall also provide our own proof. The equivalence of 2 and 3 is found in [2], so we will now prove the equivalence of 1 and 2. Following the Thas/Walker construction let SðFÞ be constructed from the flock F of the cone KF embedded in Qþ with /KF S> ¼ cF ; and similarly for SðGÞ replacing F with G: Firstly, suppose that the spreads SðFÞ and SðGÞ are equivalent. We shall consider the classical and non-classical cases separately. If they are both regular spreads, then they correspond to elliptic quadric ovoids EðFÞ and EðGÞ of Qþ and there is an
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
79
automorphism of Qþ mapping EðFÞ to EðGÞ: Further, since the group of an elliptic quadric ovoid of Qþ is induced by the group of Qþ we may also assume that cF is mapped onto cG : From this it follows that the same automorphism must map KF onto KG and F onto G: Now suppose that SðFÞ and SðGÞ are not classical. The corresponding ovoids OðFÞ and OðGÞ of Qþ are the union of q conics about cF and cG ; respectively. Since the spreads are equivalent it follows that there is an automorphism of Qþ that maps OðFÞ onto OðGÞ: Further, by applying Lemma 4.2 this automorphism must also map cF onto cG and F onto G: Next suppose that F and G are equivalent flocks. Since the group of Qþ is transitive on quadratic cones we may assume that F and G are embedded as flocks of the same quadratic cone of Qþ : Since the group of Qþ induces the full group of the quadratic cone it follows that there is an automorphism of Qþ mapping F to G: We may also assume that this automorphism fixes the generator classes of Qþ : (Since in the proof of Theorem 4.1 we saw an automorphism of Qþ fixing the subspace of a quadratic cone pointwise and swapping the generator classes of Qþ :) Under the Klein correspondence this automorphism of Qþ becomes a collineation of PGð3; qÞ mapping SðFÞ to SðGÞ: & The following theorem on flocks is due to Thas [22]. Theorem 4.4. A flock whose planes are all incident with a common point is either linear (in which case the planes of the flock share a common line) or equivalent to a semifield flock of Dickson, Kantor, Knuth type. If the functions f and g are additive functions then the spread Sð f ; gÞ is a semifield spread. For this reason we call the flock Fð f ; gÞ where the functions f and g are additive a semifield flock. The maximal subfield with the property that f and g are linear over the subfield is called the kernel of the (semifield) flock. Let Fð f ; gÞ be a semifield flock. The functions f and g are a Cohen–Ganley pair and can be used to construct a commutative semifield of rank 2 over its nucleus following Section 3. In Section 2, we saw that for a semifield spread the matrices in the corresponding spread set determine the multiplication in the semifield. The matrices of the spread set Cð f ; gÞ determine the multiplication 3 in the pre-semifield S that coordinatises the plane pð f ; gÞ constructed via the Andre´ construction from the spread Sð f ; gÞ: Indeed f ðtÞ u ðx; yÞ3ðu; tÞ ¼ ðx; yÞ ¼ ðuy þ xf ðtÞ þ ygðtÞ; ux þ tyÞ: u þ gðtÞ t
5. Ovoids of Qð4; qÞ The generalised quadrangle Qð4; qÞ is the structure of totally isotropic points and lines of a non-degenerate quadric of PGð4; qÞ: Let Q be the non-degenerate quadratic
ARTICLE IN PRESS 80
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
form defined by QðX Þ :¼ X0 X4 þ X1 X3 X22 : An ovoid of Qð4; qÞ is equivalent to an ovoid containing /0; 0; 0; 0; 1S and for any such ovoid OðF Þ there is a function F such that OðF Þ ¼ f/1; y; x; F ðx; yÞ; x2 yF ðx; yÞS j x; yAGF ðqÞg,f/0; 0; 0; 0; 1Sg: Thas [23] and later Lunardon [19] showed that it is possible to construct an ovoid of Qð4; qÞ from a semifield flock and vice versa. Lunardon also proves that two such ovoids are equivalent if and only if the corresponding semifields are equivalent. Lavrauw [18] (see also [3]) explicitly calculates the polynomial F ðx; yÞ from the flock Fð f ; gÞ: Let GF ðq0 Þ be the kernel of the semifield flock, that is the largest subfield of GF ðqÞ over which f and g are linear. Then f and g can be written as f ðxÞ ¼
n1 X
i
ci xq0
and
gðxÞ ¼
i¼0
n1 X
i
bi xq0
i¼0
for some ci ; bi AGF ðqÞ: The semifield flock Fð f ; gÞ is in one-to-one correspondence ˆ þ gðxÞ ˆ with the ovoid OðF Þ of Qð4; qÞ given by F ðx; yÞ ¼ fðyÞ where ˆ ¼ fðyÞ
n1 X i¼0
ni
ðci yÞq0
and
ˆ gðxÞ ¼
n1 X
ni
ðbi xÞq0 :
i¼0
This ovoid OðF Þ has the property that there is a distinguished point, namely /0; 0; 0; 0; 1S; such that for each line c of Qð4; qÞ incident with /0; 0; 0; 0; 1S there is an automorphism group of Qð4; qÞ fixing OðF Þ; fixing /0; 0; 0; 0; 1S linewise, fixing c pointwise, and for each PAc\f/0; 0; 0; 0; 1Sg acts transitively on the set of points of OðF Þ\f/0; 0; 0; 0; 1Sg collinear with P: An ovoid with this property is called a translation ovoid with respect to the point P, or sometimes just a translation ovoid, see [6]. Using the Klein correspondence the points of an ovoid of Qð4; qÞ corresponds to # Þ of the symplectic generalised quadrangle W ðqÞ and vice versa. a spread SðF Theorem 5.1 (Lunardon [19], Bloemen [5]). An ovoid OðF Þ of Qð4; qÞ is a translation # Þ of W ðqÞ is a ovoid with respect to a point if and only if the corresponding spread SðF semifield spread. Proof. If OðF Þ is a translation ovoid with respect to the point P; then by [6] there is a group G of collineations of PGð4; qÞ fixing Qð4; qÞ; the point P; the lines of Qð4; qÞ on P; and acting regularly on OðF Þ\fPg: Embedding Qð4; qÞ in the Klein quadric Qþ ð5; qÞ as a hyperplane intersection we can extend G to a group of PGð5; qÞ fixing Qþ ð5; qÞ and the generators (the totally isotropic planes) of Qþ ð5; qÞ on P: Under the # Þ stabilised by a group fixing Klein correspondence this is equivalent to a spread SðF
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
81
one element pointwise and acting regularly on the remaining lines, that is, a semifield spread. & The previous theorem implies that from a translation ovoid of Qð4; qÞ we obtain a semifield. One may expect that this semifield will be that constructed directly from the corresponding semifield flock, however we shall see that in general this is not the # that coordinatises case. Firstly, we calculate the multiplication for the semifield S # Þ the plane pðF # Þ; where pðF # Þ is the plane constructed from the symplectic spread SðF # Þ via the Andre´ construction. One can check that the spread SðF f/ð1; 0; F ðx; yÞ; xÞ; ð0; 1; x; yÞS j x; yAGF ðqÞg,f/ð1; 0; 0; 0Þ; ð0; 1; 0; 0ÞSg corresponds via the Klein correspondence to the ovoid OðF Þ: Indeed the point /1; y; x; F ðx; yÞ; x2 yF S on X0 X4 þ X1 X3 X22 ¼ 0 is /p01 ; p03 ; p02 ; p21 ; p32 S in the Plu¨cker coordinates of the line /ð1; 0; F ðx; yÞ; xÞ; ð0; 1; x; yÞS: This is the spread constructed from the spread set F ðx; yÞ x j x; yAGF ðqÞ : x y # with As before we conclude that the plane p# is coordinatised by the pre-semifield S multiplication 3; F ðu; vÞ u ðx; yÞ3ðu; vÞ ¼ ðx; yÞ ¼ ðxF ðu; vÞ þ uy; xu þ yvÞ; u v ˆ þ xgðuÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ:
Theorem 5.2. The following are equivalent: 1. The translation ovoids OðF Þ and OðGÞ are equivalent. # Þ and SðGÞ # 2. The spreads SðF are equivalent. # Þ and SðGÞ # 3. The planes pðF # Þ and pðGÞ # that are constructed from SðF via the Andre´ construction are isomorphic.
Proof. By the Klein correspondence 1 and 2 are equivalent, while 2 and 3 are equivalent comes from [2]. &
ARTICLE IN PRESS 82
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
We conclude this section with the following theorem which is a straightforward consequence of Theorem 4.4. However, it is important to note that this shows in # constructed general for a Cohen–Ganley pair of functions ð f ; gÞ that the semifield S from the translation ovoid of Qð4; qÞ is not isotopic to the semifield S constructed from the semifield flock. Theorem 5.3. Let ð f ; gÞ be a Cohen–Ganley pair of functions. The pre-semifield S constructed from the semifield flock Fð f ; gÞ with multiplication 3 given by ðx; yÞ3ðu; vÞ ¼ ðuy þ xf ðvÞ þ ygðvÞ; xu þ yvÞ # constructed from the translation ovoid O with multiplication and the pre-semifield S given by ˆ þ xgðuÞ ˆ ðx; yÞ ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ are isotopic if and only if the pair of functions ð f ; gÞ are linear or Dickson–Kantor– Knuth. # are isotopic if and only if the semifield Proof. The two pre-semifields S and S planes they coordinatise are isomorphic. By [2] this is the case if and only if the spreads of PGð3; qÞ giving rise to the planes are equivalent, that is, if and only if the spread arising from the flock F is symplectic. Via the Thas/Walker construction this is equivalent to the planes of F having a common point. By Theorem 4.4 this is the case if and only if F is linear or Dickson–Kantor–Knuth which is if and only if the pair of functions ð f ; gÞ are linear or Dickson–Kantor–Knuth.
6. The six semifields associated with a semifield flock Throughout the remainder of the article ð f ; gÞ will be a Cohen–Ganley pair of functions and Fð f ; gÞ will be the semifield flock. Let GF ðq0 Þ be the kernel of the semifield flock, the maximal subfield such that f and g can be written as f ðxÞ ¼
n1 X i¼0
i
ci xq0
and
gðxÞ ¼
n1 X
i
bi xq0
i¼0
for some ci ; bi AGF ðqÞ: Let fˆ and gˆ be the polynomials defined from f and g as in the ˆ be the translation ovoid OðF Þ where F ðx; yÞ ¼ previous section and let Oð f;ˆ gÞ ˆ ˆ fðyÞ þ gðxÞ: Let S be the pre-semifield constructed from the semifield flock Fð f ; gÞ # be the pre-semifield constructed from the translation ovoid Oð f;ˆ gÞ: ˆ We and let S # shall calculate the pre-semifields associated to S and S via the Knuth cubical array method described in Section 2.
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
83
In Section 4, we saw that the pre-semifield S has multiplication 3; where ðx; yÞ3ðu; vÞ ¼ ðuy þ xf ðvÞ þ ygðvÞ; xu þ yvÞ and that the plane p coordinatised by S is a translation plane whose spread Sð f ; gÞ can be constructed from the spread set f ðvÞ u þ gðvÞ j u; vAGF ðqÞ : u v In Section 1, we saw that the plane p ; the dual plane of p; is the plane coordinatised by S ; the semifield whose multiplication is ðx; yÞ3ðu; vÞ ¼ ðxv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ: In Theorem 4.1, we saw that the spread S is self-dual and therefore that the plane pw constructed from the spread dual to S via the Andre´ construction is isomorphic to p: The plane pw is coordinatised by a semifield Sw isotopic to S; we write SCSw : It follows immediately that the plane pw ; the plane dual to pw ; is isomorphic to the plane p and hence that the semifield Sw CS : Theorem 6.1. Let S be the spread from which the translation plane p is constructed via the Andre´ construction. Let Sw be the spread dual to the spread S : Let the semifield Sw be a semifield which coordinatises the plane pw obtained from the spread Sw via the Andre´ construction. Then the semifield Sw is isotopic to a pre-semifield with multiplication given by ˆ ˆ ðx; yÞ3ðu; vÞ ¼ ðxv þ yu; yv þ fðxuÞ þ gðxvÞÞ: Proof. This proof was significantly shortened using the ideas that appear in [16]. For all u; vAGF ðqÞ fðx; y; xv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ j x; yAGF ðqÞg is an element of the spread S : Following Section 2, we dualise with respect to the inner-product ððx; y; z; wÞ; ða; b; c; dÞÞ ¼ Trðxa þ yb þ zc þ wdÞ; 2
where TrðxÞ ¼ x þ xq0 þ xq0 þ ? þ xq=q0 is the trace function from GF ðqÞ to GF ðq0 Þ: So we want to find ða; b; c; dÞ such that Trðxa þ yb þ cðxv þ uf ðyÞ þ vgðyÞÞ þ dðxu þ yvÞÞ ¼ 0 for all x; yAGF ðqÞ: If we put x ¼ 0 then ˆ ˆ Trðyðb þ fðucÞ þ gðvcÞ þ dvÞÞ ¼ 0
ARTICLE IN PRESS 84
S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
ˆ ˆ and hence b ¼ ð fðucÞ þ gðvcÞ þ dvÞ: If we put y ¼ 0 then Trðxða þ cv þ duÞÞ ¼ 0 and hence a ¼ ðcv þ duÞ: The corresponding spread element of Sw is therefore ˆ ˆ fððcv þ duÞ; ð fðucÞ þ gðvcÞ þ dvÞ; c; dÞ j c; dAGF ðqÞg: By a straightforward change of coordinates the multiplication for Sw is as claimed. & # constructed from the ovoid Oð f;ˆ gÞ ˆ of In Section 5, we saw that the pre-semifield S Qð4; qÞ has multiplication 3; where ˆ þ xgðuÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðxfðvÞ þ uy; xu þ yvÞ # is a translation plane whose spread S # can be and that the plane p# coordinatised by S constructed from the spread set ( ! ) ˆ þ gðuÞ ˆ fðvÞ u j u; vAGF ðqÞ : u v The matrices in this spread set are symmetric and so the spread constructed from the spread set will be self-dual. Hence, the planes p and pw are isomorphic and # and S # w are isotopic, SC # S # w: the semifields that coordinatise these planes S # is given by The multiplication in the semifield S ˆ þ ugðxÞ ˆ ðx; yÞ3ðu; vÞ ¼ ðufðyÞ þ xv; xu þ yvÞ # CS # w : and it follows immediately that S Theorem 6.2. Let S# be the spread from which the translation plane p# is constructed # : Let the # w be the spread dual to the spread S via the Andre´ construction. Let S # w be a semifield which coordinatises the plane p# w obtained from the spread semifield S # w is isotopic to a pre-semifield #Sw via the Andre´ construction. Then the semifield S with multiplication given by ðx; yÞ ðu; vÞ ¼ ðxv þ yu þ gðuxÞ; yv þ f ðuxÞÞ:
Proof. The proof of this theorem is similar to that of Theorem 6.1. For all u; vAGF ðqÞ ˆ þ ugðxÞ ˆ fðx; y; ufðyÞ þ xv; xu þ yvÞ j x; yAGF ðqÞg
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
85
# : Again we dualise with respect to the inner-product is an element of the spread S ððx; y; z; wÞ; ða; b; c; dÞÞ ¼ Trðxa þ yb þ zc þ wdÞ: So we want to find ða; b; c; dÞ such that ˆ þ ugðxÞ ˆ Trðxa þ yb þ cðufðyÞ þ xvÞ þ dðxu þ yvÞÞ ¼ 0 for all x; yAGF ðqÞ: If we put x ¼ 0 then Trðyðb þ f ðucÞ þ dvÞÞ ¼ 0 and hence b ¼ ð f ðucÞ þ dvÞ: If we put y ¼ 0 then Trðxða þ gðucÞ þ cv þ duÞÞ ¼ 0 # w is and hence a ¼ ðgðucÞ þ cv þ duÞ: The corresponding spread element of S therefore fððgðucÞ þ cv þ duÞ; ð f ðucÞ þ dvÞ; c; dÞ j c; dAGF ðqÞg: # w is as By a straightforward change of coordinates the multiplication for S claimed. & # w is the commutative semifield of rank 2 over its middle Note that the semifield S nucleus that we saw in Section 3. In this section, we have constructed potentially six non-isotopic semifields from a Cohen–Ganley pair of functions ð f ; gÞ: The following table we list the left, right and middle nuclei of these six semifields. We have determined the multiplication of a presemifield isotopic to each these six semifields. However, to determine the nuclei one must first calculate the multiplication in the semifield itself using the method described in Section 1. In Table 1 we list the multiplication in a pre-semifield isotopic to the relevant semifield. This table does not apply to the linear example in which all nuclei are S itself since associativity holds. In this case all the six semifields are GF ðq2 Þ: # S # ; S # w are pairwise non-isotopic Theorem 6.3. The six semifields S; S ; Sw ; S; # S CS # ; unless ð f ; gÞ; is of Dickson–Kantor–Knuth type in which case SCS; # w are three pairwise non-isotopic, or linear in which case they are all isotopic Sw CS to GF ðq2 Þ: Proof. As mentioned before in the case of the linear example all the six semifields are GF ðq2 Þ so we consider only the so-called proper semifields. If two planes p and p0 coordinatised by semifields S and S0 respectively are isomorphic then the ranks of S and S0 over their left/right nucleus are equal. Hence it is only possible that # S CS # and Sw ¼ S # w : By Theorem 5.3 SCS # if and only if S is a linear SCS;
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
86
Table 1 The nuclei of the six semifields associated with a semifield flock Semifield
ðx; yÞ3ðu; vÞ
L
M
R
S S Sw # S # S # Sw
ðyu þ xf ðvÞ þ ygðvÞ; xu þ yvÞ ðxv þ uf ðyÞ þ vgðyÞ; xu þ yvÞ ˆ ˆ ðxv þ fðyuÞ þ gðyvÞ; xu þ yvÞ ˆ þ xgðuÞ; ˆ ðyu þ xfðvÞ xu þ yvÞ ˆ þ ugðxÞ; ˆ ðxv þ ufðyÞ xu þ yvÞ ðxv þ yu þ gðxuÞ; yv þ f ðuxÞÞ
GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ
GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ
GF ðq0 Þ GF ðqÞ GF ðq0 Þ GF ðq0 Þ GF ðqÞ GF ðq0 Þ
or Dickson–Kantor–Knuth example and hence in these cases we also have that # w : In the Dickson–Kantor–Knuth examples GF ðq0 ÞaGF ðqÞ S CS and Sw CS and so by the preceding argument SI / S ; SI / Sw and S I / Sw : &
7. Literature The literature concerning this subject is somewhat confusing. As proved in [3] the so-called sporadic examples of Cohen–Ganley pairs of functions in [7] and the examples in [21] are equivalent to Dickson–Kantor–Knuth examples. The fact that the commutative semifield of rank 2 over its middle nucleus in these examples is isotopic to a Dickson–Kantor–Knuth example now follows from Theorems 4.4, 6.1, 5.2 and 6.3. Finally for q odd the only non-existence theorem concerning Cohen–Ganley pairs of functions is the following result which is a consequence of the main theorem in [4]. Theorem 7.1. Let ð f ; gÞ be a Cohen–Ganley pair of functions in GF ðqÞ½X ; q odd, and suppose that GF ðq0 Þ; q ¼ qn0 ; is the largest subfield of GF ðqÞ such that both f and g are linear over GF ðq0 Þ: If ð f ; gÞ is not linear over GF ðqÞ and not of Dickson–Kantor– Knuth type then q0 o4n2 8n þ 2:
Acknowledgments We thank W.M. Kantor for his comments and suggestions.
References [1] A.A. Albert, Finite division algebras and finite planes, Proc. Sympos. Appl. Math. 10 (1960) 53–70. [2] J. Andre´, U¨ber nicht-Desarguessche Ebenen mit transitiver Translationgruppe, Math. Z. 60 (1954) 156–186.
ARTICLE IN PRESS S. Ball, M.R. Brown / Advances in Mathematics 189 (2004) 68–87
87
[3] S. Ball, M. Lavrauw, Commutative semifields of rank 2 over their middle nucleus, in: G.L. Mullen, et al., (Eds.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Springer, Berlin, 2002, pp. 1–21. [4] S. Ball, A. Blokhuis, M. Lavrauw, On the classification of semifield flocks, Adv. Math. 180 (2003) 104–111. [5] I. Bloemen, Substructures and characterizations of finite generalized polygons, Ph.D. Thesis, University of Ghent, 1995. [6] I. Bloemen, J.A. Thas, H. Van Maldeghem, Translation ovoids of generalized quadrangles and hexagons, Geom. Dedicata 72 (1998) 19–62. [7] S.D. Cohen, M.J. Ganley, Commutative semifields, two dimensional over their middle nuclei, J. Algebra 75 (1982) 373–385. [8] P. Dembowski, Finite Geometries, Springer, Berlin, 1968. [9] L.E. Dickson, Linear algebra in which division is always uniquely possible, Trans. Amer. Math. Soc. 7 (1906) 514–527. [10] H. Gevaert, N.L. Johnson, Flocks of quadratic cones, generalized quadrangles and translation planes, Geom. Dedicata 27 (1988) 301–317. [11] H. Gevaert, N.L. Johnson, J.A. Thas, Spreads covered by reguli, Simon Stevin 62 (1) (1988) 51–62. [12] J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985. [13] J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991. [14] D.R. Hughes, F.C. Piper, Projective Planes, Springer, Berlin, 1973. [15] W.M. Kantor, Ovoids and translation planes, Canad. J. Math. 34 (1982) 1195–1207. [16] W.M. Kantor, Commutative semifields and symplectic spreads, preprint. [17] D.E. Knuth, Finite semifields and projective planes, J. Algebra 2 (1965) 182–217. [18] M. Lavrauw, Scattered subspaces with respect to spreads and eggs in finite projective spaces, Ph.D. Thesis, Technical University of Eindhoven, The Netherlands, 2001. [19] G. Lunardon, Flocks, ovoids of Qð4; qÞ and designs, Geom. Dedicata 66 (1997) 163–173. [20] T. Penttila, B. Williams, Ovoids of parabolic spaces, Geom. Dedicata 82 (2000) 1–19. [21] A.R. Prince, Two new families of commutative semifields, Bull. London Math. Soc. 32 (2000) 547–550. [22] J.A. Thas, Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987) 441–452. [23] J.A. Thas, Generalized quadrangles of order ðs; s2 Þ: II, J. Combin. Theory Ser. A 79 (1997) 223–254. [24] J.A. Thas, S.E. Payne, Spreads and ovoids in finite generalized quadrangles, Geom. Dedicata 52 (1994) 227–253. [25] M. Walker, A class of translation planes, Geom. Dedicata 5 (1976) 135–146.