Non-classical polar unitals in ﬁnite Dickson semiﬁeld planes

. We give three proofs, two intrinsic and one extrinsic, that every Dickson–Ganley unital U ( σ ), parametrized by a ﬁeld automorphism σ , is non-classical if σ is not the identity, extending a result of Ganley’s (Math Z 128:34–42, 1972); we prove that U ( σ 1 ) is isomorphic to U ( σ 2 ) if and only if σ 1 = σ 2 or σ 1 = σ − 1 2 ; and we determine the (design) automorphism group of U ( σ ) as the collineation subgroup of the ambient Dickson semiﬁeld plane stabilizing the unital. This contains as a special case the corresponding result of O’Nan’s (J Algebra 20:495–511, 1965) on the classical unital.


Introduction
A unital U is a t − (v, k, λ) design, where t = 2, v = m 3 + 1, k = m + 1, and λ = 1: there is exactly one block on two points, there are m + 1 points on each block, and there is a total of m 3 + 1 points. Let Π be a projective plane, i.e. a 2 − (n 2 + n + 1, n + 1, 1) design. A unitary polarity ρ of Π is an involutory correlation with ( √ n) 3 + 1 absolute points. The absolute points and non-absolute lines of ρ constitute an example of U . In this case U is called a polar unital. If Π is the classical plane P G(2, q 2 ) coordinatized by the finite field F q 2 , then U is called a classical unital of order q. As a subvariety of P G(2, q 2 ) a classical unital is a hermitian curve [7,15,17].
The problem of characterizing polar unitals seems difficult, as it involves the questions of embedding as well as the existence of unitary polarities in a projective plane. As a starting point one considers the characterization of the classical unital. A fundamental result is the determination of the automorphism group of a classical unital as a design. In [22], this is shown to be P ΓU (3, q 2 ). In the same paper, it is also shown that in the design there is never a configuration of four blocks in general position intersecting in six points, which is now referred to as an O'Nan configuration. In [25], it is conjectured that if a unital does not admit any O'Nan configurations then it is classical. The conjecture remains open. A weaker yet remarkable result in this direction is Wilbrink's characterization of a classical unital [30]. There are three characterization conditions. In this paper we shall be concerned with the first two. The first condition is the absence of an O'Nan configuration, and the second condition is a notion of parallelism in unitals which will be recalled later in the paper. It is shown in [30] that if the order of the unital is even, then these two conditions imply the third.
Since all unitary polarities in P G(2, q 2 ) are projectively equivalent, a polar unital in the classical plane is the classical unital. Consider now a polar unital U embedded in a non-classical plane Π. As remarked in [20], the question as to whether U can be classical is not yet answered. It is shown in that paper that a class of polar unitals embedded in the finite Figueroa planes are not classical. Here we turn our attention to a class of polar unitals embedded in the Dickson semifield planes. In [12], Ganley showed that the projective plane Π(K) defined over a Dickson semifield K admits a unitary polarity which thus defines a polar unital U. The unital U = U(σ) is parametrized by a non-identity field automorphism σ. In [13] (see also [1]) Ganley studied the collineation subgroup Col(U) of Π(K) stabilizing U, and proved that in some special cases these polar unitals are not classical by demonstrating the existence of O'Nan configurations.
In this paper, we begin by proving the existence of O'Nan configurations in all cases. We also provide an alternative proof by demonstrating the invalidity of the second condition of Wilbrink's in these unitals.
Next we note that a Dickson-Ganley unital meets the line at infinity at one point, (∞), and Col(U) acts transitively on the affine points of the unital [13]. Such a unital is also called a transitive parabolic unital (see for example [1]). For our purpose the point (∞) is special for a further reason. It satisfies Wilbrink's second condition (called condition (II) in [30]) in strong form (see Corollary 4.2 below). This makes it possible to construct from U a design S which is isomorphic to the residual of a classical inversive plane.
The classical circle geometry allows us to work out the algebra to show that an isomorphism between two Dickson-Ganley polar unitals determines an isotopism between their corresponding Dickson semifields. Since two semifields define isomorphic projective planes if and only if the semifields are isotopic, this has the consequence that U(σ 1 ) is isomorphic to U(σ 2 ) if and only if σ 1 = σ 2 or Vol. 104 (2013) Non-classical polar unitals 471 6,26]. As our method of proof extends to the case of the identity automorphism, this provides a third and extrinsic proof that U(σ) is not classical.
Finally, we prove that the isotopism above gives rise to an isomorphism between the ambient Dickson semifield planes. As a consequence we conclude that the automorphism group Aut(U) of the design U equals Col(U). When σ is the identity automorphism, this gives O'Nan's result [22] for the classical unital.
In the next section, we recall the construction of the Dickson semifield plane and the Dickson-Ganley unital as well as their basic properties. In Sect. 3 we extend Ganley's result and show that O'Nan configurations exist in all Dickson-Ganley unitals, with the consequence that these unitals are not classical. In Sect. 4 we prove that Wilbrink's condition (II) does not hold at any affine absolute point, thus giving an alternative proof of the same conclusion. We also prove that Wilbrink's condition (II) holds at the absolute point (∞) in a strong form. In Sect. 5 we recall the relevant geometry of finite inversive planes and prove some results on automorphisms of associated designs essential for our purpose. In Sect. 6 the geometry and algebra of Sect. 5 will be applied to a Dickson-Ganley unital to obtain crucial information on Aut(U). In Sect. 7 we solve the isomorphism problem for the Dickson-Ganley unitals, prove the main extension theorem, and determine their (design) automorphism groups.

Dickson semifield planes and Dickson-Ganley unitals
A finite semifield is an algebraic system that satisfies all axioms of a division ring except the associativity of multiplication. Such a system has various names in the literature, but the term semifield is due to Knuth [21]. Some examples of finite semifields relevant to the study of unitals are the Dickson semifields [9,10], Albert's twisted fields [2], and Albert's generalized twisted fields [3,4].
In more details, a finite semifield K(+, ·) is a finite algebraic system with two binary operations, addition and multiplication, which satisfy the following axioms: 3. There is an element 1 ∈ S such that for any a ∈ S, 1a = a1 = a. 4. For any a, b, c ∈ S, a(b + c) = ab + ac and (a + b)c = ac + bc.
The additive group K(+) of K is an elementary Abelian group and so K(+) is isomorphic to (Z/pZ) N for some prime p (called the characteristic of the semifield) and positive integer N . Thus K can also be considered as a vector space over the finite field F p , and |K| = p N (see [21]).
Since a semifield is a particular kind of ternary ring, it can be used to coordinatize a projective plane Π(K) [14]. Here we follow the notations of Ganley [13]. The set of points of Π Incidence is set-theoretic inclusion. We call such a projective plane a semifield plane. Any semifield plane has the property that [∞] and (∞) are respectively a translation line and a translation point [24]. Two semifields K 1 (+ 1 , · 1 ) and K 2 (+ 2 , · 2 ) are said to be isotopic if there exists an ordered triple (A, B, C) of additive bijections from is called an isotopism from K 1 to K 2 , and an autotopism if K 1 = K 2 . In case A = B = C, K 1 and K 2 are isomorphic. The geometric significance of isotopy is the following: two semifields coordinatize isomorphic planes if and only if they are isotopic [3,Theorem 6].
We now turn our attention to the Dickson (commutative) semifields [9,10]. Consider the finite field F q , where q = p e for an odd prime p, and e > 1. Then there is a non-square element δ in F q and a non-identity automorphism σ of F q . Let K be a two-dimensional vector space over F q with basis elements 1 and λ. Define multiplication of K by (x + λy)(u + λv) = xu + δy σ v σ + λ(xv + yu). Then K(+, ·) is a commutative semifield where + is the vector space addition of K over F q . K is called a Dickson semifield of order q 2 .
Consider the projective plane Π(K) coordinatized by a Dickson semifield K. By a result of Ganley [12,Theorem 5], any projective plane coordinatized by a finite commutative semifield which has a non-trivial involutory automorphism admits a unitary polarity. In particular, since the Dickson semifield K admits an involutory automorphism α : x + λy → x − λy, Π(K) admits a unitary polarity ρ given by It is readily verified that the absolute points of ρ are (∞) and all (affine) points of the form (x+λy, − 1 2 (x 2 −δy 2σ )+λv) where x, y, v ∈ F q (see [13]). Note that the non-absolute lines not incident with (∞) are those of the form [m 1 + λm 2 , k 1 + λk 2 ] where k 1 = 1 2 (m 1 2 − δm 2 2σ ). As for the lines on (∞), the absolute line is the line at infinity [∞], the remaining q 2 affine lines being non-absolute lines. Denote by U = U(δ, σ) the polar unital defined by ρ, i.e. the unitary block design (of order q) whose points are the absolute points of ρ and whose blocks are the non-absolute lines of ρ. We call U a Dickson-Ganley unital.

The existence of O'Nan configurations in Dickson-Ganley unitals
In a unital, an O'Nan configuration is a set of four unital lines in general position, intersecting in six unital points. It is known that a classical unital does not contain any O'Nan configuration [22]. In this section, we extend Ganley's result to the existence of an O'Nan configuration in any Dickson-Ganley unital. Indeed, we prove the existence of an O'Nan configuration on any affine (non-absolute) line through (∞) (Theorem 3.1). Our strategy is to set up three possibly repeated nonabsolute lines not incident with (∞) and meeting the given affine line in three distinct affine absolute points, so that these three non-absolute lines are distinct, non-concurrent, and meet each other in absolute points, provided that it is possible to choose a non-square δ not satisfying certain polynomials. This we prove to be always possible, which means that the four lines and six points under investigation do constitute an O'Nan configuration.
Recall from Sect. 2 that Col(U) is transitive on the affine absolute points. Since Col(U) fixes the absolute point (∞), it is transitive on the affine (non-absolute) lines through (∞). We have the following theorem:  Proof. By the remark preceding the theorem, we may assume without loss of generality that the given affine line is [0].
Then the P i 's and [0] ∩ l i 's are all absolute points. For all i = j, P i ∈ l j if and only if Hence we have, for i = j, the twelve equations (A ij ) and (B ij ): It is easy to check that rank(M) = 5. We want to show that b is in col(M), which is equivalent to having We introduce parameters α, β, γ, m and specialize as follows: We shall solve for x i , y i , k i , l i 's in terms of α, β, γ and m, excluding any solutions leading to P i ∈ [0], for i = 1, 2, 3.
Similarly, using (A 21 ) and (A 23 ), we have By our specialization, this becomes y 2 Vol. 104 (2013) Non-classical polar unitals 475 We now choose m = 1 and let γ = 1 m−1 . Then y 2 σ = − x2 δ . Substituting y 2 σ into (A 21 ), we obtain and so x 2 = 0, it follows that By (A 31 ), (A 32 ), and so x 3 = 0, it follows that Then In terms of α and β, we obtain: Since m = 1, this system of linear equations in k 2 , l σ 2 has a unique solution. Indeed, and

Then (3.1) becomes
Now choose m further so that it is not fixed by σ, and α such that α σ −α+1 = 0. Then (3.1) further becomes Since σ does not fix m, (3.1) is satisfied by a unique nonzero β which is independent of δ. Then we compute l σ 2 and k 2 , and then a set of

2) does not vanish implies that
. Together with the fact that y 3 − y 2 = 2β τ = 0, we conclude that P 1 , P 2 and P 3 are three distinct absolute points not on [0]. Hence l 1 = P 2 · P 3 , l 2 = P 3 · P 1 , l 3 = P 1 · P 2 and any of these lines is distinct from  Proof. If not, take an O'Nan configuration on the affine absolute point (∞) Φ for some Φ ∈ Aut(U). Then Φ −1 carries the O'Nan configuration to (∞), contradicting that fact that there exists no O'Nan configuration on (∞).
The following result is immediate from Theorem 3.1 and the nonexistence of O'Nan configuration in the classical unital:

Wilbrink's condition (II) and Dickson-Ganley unitals
Throughout this section, let q = p e , where p is an odd prime and e > 1, δ be a non-square in F q and σ : x → x p s a non-identity automorphism of F q , where 1 ≤ s < e. Moreover, let U = U(σ) denote the polar unital as defined in Sect. 2. In this section, we show that U is not classical by proving that Wilbrink's second condition does not hold at any absolute point except the point (∞).
) if and only if the line is non-absolute. Proof. Suppose (x + λy, u + λv) is a unital point on l, so that Note that the q lines [m 1 + λm 2 , k 1 + λk 2 ], where k 2 ∈ F q , intersect N at distinct absolute points. If there is another line intersecting the lines of N at absolute points, it will give rise to an O'Nan configuration containing (∞), contradicting Ganley's result [13,Theorem 1]. This proves the converse.
The following Corollary is immediate from Lemma 4.1. Note that by , where k 2 ∈ F q , are parallel in the sense that they intersect the same lines through (∞) at distinct absolute points. In this sense, we say that (∞) is a vertex of Wilbrink's condition (II) in strong form.

Corollary 4.2. (∞) is a vertex for Wilbrink's condition (II) in strong form.
As the λ-free part and the λ-part of the coordinates of unital points on a given non-absolute line missing (∞) are not related as simply as those on a line through (∞), we suspect that any other choice of an absolute point might fail to be a vertex for Wilbrink's condition (II). This is indeed the case: because the absolute points on [0] are (∞) and (0, λw) where w ∈ F q . We aim to show that each of these lines meets a line in N at a non-absolute point.

Now for each nonzero
Suppose further this is an absolute point. Then we have the system of equations: We aim to show that for each w = 0, there exists t = 1 such that there is no solution to this system (4.4)-(4.8).

Inversive plane geometry
The point (∞) of a Dickson-Ganley unital U is special. In particular, it satisfies Wilbrink's condition (II) in strong form (Corollary 4.2). This makes it possible to construct from U a design S which turns out to be isomorphic to the residual of a classical inversive plane. The structure of Aut(S), the automorphism group of S, plays a key role in our study on the structure of Aut(U), the automorphism group of U. In this section, we recall the geometry of inversive planes needed for our purpose and establish some lemmas which allow us to study S and Aut(S) in some details.
An inversive plane is a set of points with distinguished subsets of the points, called circles satisfying the following axioms: (IP1) Any three distinct points are contained in exactly one common circle. (IP2) If P and Q are points and if c is a circle containing P but not Q, then there is a unique circle d such that P, Q ∈ d and c ∩ d = {P }.
(IP3) There are four points not on a common circle.
We are concerned only with the finite inversive planes. These are exactly the class of 3-designs with parameters (n 2 + 1, n + 1, 1) with n ≥ 2. (We refer to Dembowski [7] for standard results on finite inversive planes.) We call n the order of a finite inversive plane.
Let I be a finite inversive plane of order n. A bundle of circles of I, denoted by [P, Q], is the set of all circles through the points P, Q of I, with P = Q. The points P, Q are called the carriers of the bundle. The following lemma studies how a circle not belonging to a bundle meets the circles of the bundle. Note that by axiom (IP1), two distinct circles intersect in 0, 1, or 2 points. Accordingly we say that the two circles are disjoint, tangent, or intersecting, respectively.

Lemma 5.1. Let P, Q be two distinct points not on a circle c in an inversive
plane I of order n. Then one of the following statements is true: 1. n is odd, and c is respectively disjoint from (n + 1)/2, tangent to 0, and intersecting (n + 1)/2, circles of [P, Q]. 2. n is odd, and c is respectively disjoint from (n − 1)/2, tangent to 2, and intersecting (n − 1)/2, circles of [P, Q]. 3. n is even, and c is respectively disjoint from n/2, tangent to 1, and intersecting n/2, circles of [P, Q].

n is even, and c is tangent to all n + 1 circles of [P, Q].
Proof. This is (2.9) and (2.10) of [8] and simple counting from axioms.
We shall also need to consider another type of circle set. A flock is a set F of mutually disjoint circles in I which partitions the points of I except two distinct points P and Q. These points are again called the carriers of the flock. The existence of flocks is not as clear as that of bundles, and a flock need not be determined by its carriers. However, in an egglike inversive plane, it is known that every flock is uniquely determined by its carriers, as we recall below.
Let O be an ovoid in P G(3, q), q > 2, i.e. a set of q 2 +1 points in general position (no three of which are collinear) in P G (3, q). An example is the elliptic quadric which has canonical form f (x 0 , x 1 ) + x 2 x 3 = 0, where f is an irreducible quadratic form. As in the case of an elliptic quadric, at each point of an ovoid there is a unique tangent plane, and all other (secant) planes meet the ovoid in an oval, i.e. a set of q + 1 points in general position (see [15]). Given an ovoid  Such a flock is called linear (see for example [28]). Concerning linear flocks, there is the following result ( [23,27]; see also [11]):

Theorem 5.2. Every flock of an egglike inversive plane is linear.
Since in P G (3, q), any two tangent planes of an ovoid O meet in an external line of O, every flock of an egglike inversive plane is uniquely determined by its carriers. We denote such a flock by F(P, Q) with carriers P and Q.
We are almost ready to prove the main result of this section. Let I be an egglike inversive plane. Let X be a point of I, and consider the residual I X of I at X. Recall that given any design S and a point X of S, the residual (design) S X at X is obtained by deleting the point X and all blocks containing X. We are going to show that the automorphism group Aut(I X ) is isomorphic to the subgroup Aut(I) X of the automorphism group Aut(I) fixing X.
We shall need the result that any egglike inversive plane satisfies the Bundle Theorem [7,29]. The latter theorem is as follows: Bundle Theorem. Let {P i , Q i |i = 0, 1, 2, 3} be a set of points in an inversive plane I, where P i , i = 0, 1, 2, 3, are distinct. Let c 0 , c 1 , c 2 , c 3 be circles in I such that c i ∩ c i+1 = {P i , Q i }, with subscripts taken mod 4. Then P 0 , Q 0 , P 2 , Q 2 are on a common circle if and only if P 1 , Q 1 , P 3 , Q 3 are on a common circle.
We call the configuration (see Fig. 1) described in the Bundle Theorem a B-configuration.
The following lemma proves that in an egglike inversive plane, any four points on a circle can be completed to a B-configuration.
Proof  4 and missing X such that c 2 meets c 3 at one or two new distinct points, say P 2 , Q 2 , with P 2 and Q 2 not necessarily distinct. By the Bundle Theorem, P 0 , Q 0 , P 2 , Q 2 are on a circle, say c 5 .
We are now ready to prove the main result of this section. Proof. Any element in Aut(I) X induces an element in Aut(I X ) by restriction. On the other hand, given f ∈ Aut(I X ), letf be the extension of f to I by setting Xf = X. To show thatf defines an element in Aut(I) X , it suffices to show thatf maps any circle on X to a circle on X.
Since the case when q = 3 is elementary and is readily verified, we consider the case when q ≥ 4. Let c be a circle on X, and let P 0 , Q 0 , P 3 , Q 3 be any four distinct points on c\{X}. By Lemma 5.3, there is a B-configuration containing P 0 , Q 0 , P 3 , Q 3 , and none of its circles, except c, are on X. Thus the five circles which are not on X are mapped byf to circles not on X. By the Bundle Theorem, P 0f , Q 0f , P 3f , Q 3f are on a common circle, say d. Note that d is a circle on X. For, if d does not pass through X, then df −1 is a circle on P 0 , Q 0 , P 3 , Q 3 but not X, contradicting axiom(IP1). Now d is determined by P 0f , Q 0f , P 3f , and since Q 3 is arbitrary,f maps c to d.
It is clear that the above defines an isomorphism between the two groups.

Classical Inversive Plane and Dickson-Ganley unitals
We return to the Dickson-Ganley unitals. Consider a Dickson unital U = U(σ) defined by the polarity ρ in a Dickson semifield plane Π(K(δ, σ)) as given in Sect. 2. We shall construct from U a design S with respect to the special point (∞). Since (∞) satisfies Wilbrink's condition (II) in strong form, we can follow the construction of S from U as in Wilbrink [30] (see also [5] for further development).  (q 2 , q + 1, q) design. We call the blocks of S circles. Note that the parameters of S are exactly those of a residual design of an inversive plane of order q. Indeed, we shall prove below the main result of this section, namely, that S is the residual of a classical inversive plane.
To this end, consider the inversive plane I = (X , C) whose points are points of the projective line P G(1, q 2 ) and whose circles are all sublines P G(1, q) in P G(1, q 2 ). I is Miquelian (see [16]) and hence classical.
By [16], the automorphism group of a Miquelian inversive plane of order q is given by P ΓL(2, q 2 ). Thus,
Both proofs are intrinsic in nature, that is, they concern incidence patterns in U(σ). But how does one distinguish among these non-classical unitals? In this section, we shall answer this question by constructing from an isomorphism between two Dickson unitals an isotopism between their corresponding Dickson semifields. In particular this also provides a third and extrinsic proof that U(σ) is not classical. Furthermore, we prove that the isotopism gives rise to an isomorphism between the ambient planes, resulting in our main extension theorem. As a consequence we conclude that the automorphism group of U(σ) equals the collineation subgroup of the ambient plane stabilizing the unital.
The maps A, B and C are defined as follows.
Then m B z C = (xu + δy σ1 v σ1 ) (a 1 2 − δa 2 2 ) Suppose there is an additive bijection A : K 1 −→ K 2 completing B, C to an isotopism. Then A is uniquely determined by B and C. Consider the following definition of A: (7.14) We show that this is sufficient to yield our first main result.
Theorem 7.4. If U 1 and U 2 are isomorphic, then K 1 and K 2 are isotopic.
Hence (A, B, C) is an isotopism.
Second, suppose one of them, which we may assume to be σ 2 , is non-identity. We derive necessary conditions on a 1 and a 2 .
Let ω be a primitive element of