Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 4 (2011) 291–311
Wada Dessins associated with Finite Projective
Spaces and Frobenius Compatibility
Cristina Sarti
Mathematisches Seminar, Goethe Universität
Postfach 111932, D-60054 Frankfurt a. M., Germany
Received 20 October 2009, accepted 16 July 2011, published online 19 August 2011
Abstract
Dessins d’enfants (hypermaps) are useful to describe algebraic properties of the Rie-
mann surfaces they are embedded in. In general, it is not easy to describe algebraic prop-
erties of the surface of the embedding starting from the combinatorial properties of an em-
bedded dessin. However, this task becomes easier if the dessin has a large automorphism
group.
In this paper we consider a special type of dessins, so-called Wada dessins. Their under-
lying graph illustrates the incidence structure of finite projective spaces Pm(Fn). Usually,
the automorphism group of these dessins is a cyclic Singer group Σ` permuting transitively
the vertices. However, in some cases, a second group of automorphisms Φf exists. It is a
cyclic group generated by the Frobenius automorphism. We show under what conditions
Φf is a group of automorphisms acting freely on the edges of the considered dessins.
Keywords: Dessins d’enfants, Wada dessins, bipartite graphs, graph embeddings, difference sets,
finite geometries, Riemann surfaces, Frobenius automorphism, Singer groups.
Math. Subj. Class.: 05C10, 30F10, 05B10, 05B25, 51E20, 51D20
1 Introducing dessins d’enfants
The term dessins d’enfants was first used by Grothendieck (1984) to refer to objects which
are very simple but important to describe Riemann surfaces. Dessins d’enfants D can be
defined as hypermaps on compact orientable surfaces. A hypermap in its Walsh representa-
tion [15] is a bipartite graph drawn without crossings on a surfaceX and cutting the surface
into simply connected cells (faces).
For a vertex of the graph we call valency the number of incident edges. For a cell, the
valency is the number of edges on its boundary. This number is always even for hypermaps.
E-mail address: sarti@math.uni-frankfurt.de (Cristina Sarti)
Copyright c© 2011 DMFA Slovenije
292 Ars Math. Contemp. 4 (2011) 291–311
Edges bordering a cell from both sides have to be counted twice (see [14] and [11]). A
characteristic property of a dessin is its signature (p, q, r), where
1. p is the least common multiple of all valencies of the white vertices,
2. q is the least common multiple of all valencies of the black vertices,
3. 2r is the least common multiple of all face valencies.
A dessin is called uniform if all white vertices have the same valency p, all black vertices
have the same valency q and all cells have the same valency 2r. Riemann surfaces with
embedded dessins are algebraic curves of special type. According to Belyı̆’s theorem (see
[2]) a surface X admits a model over the field Q if and only if there exists a meromorphic
function β
β : X → P1(C)
ramified above at most three points. Without loss of generality, these can be identified with
{0, 1,∞}. The preimages of 0 and 1 are, respectively, the set of white and black vertices of
the bipartite graph on X . The preimages of∞ correspond to points within the faces. Each
face contains exactly one point which is commonly called face center (see e.g. [11]). This
means that we have a dessin on a surface if and only if we may describe it with equations
whose coefficients are in the field Q.
In general, it is not easy to relate the combinatorial properties of a dessin to algebraic
properties of the surface, such as defining equations or the moduli field, but the task be-
comes easier if the embedded dessin has a large automorphism group. In the best case, this
group acts transitively on the edges of the dessin (for recent results see, for instance, [10])
and we say that the dessin is not only uniform but even regular.
In this paper, we consider dessins whose underlying graph describes the incidence
structure of points and hyperplanes of projective spaces Pm(Fn) (see Sections 2 and 2.1).
The possibility to construct this kind of dessins was first studied by Streit and Wolfart [14]
for projective planes P2(Fn). Starting from some of their results, we examine here special
kinds of uniform dessins called Wada dessins (see Section 2.2). The cells of these dessins
have all the same valency 2` and there always exists a cyclic group Σ` transitively permut-
ing the edges of type ◦—• and of type •—◦ on their boundaries (see [14] and [12]). We
are interested in determining when the full automorphism group contains other groups of
automorphisms beside Σ`.
We consider the Frobenius automorphism acting on points and on hyperplanes of pro-
jective spaces Pm(Fn) (see Section 3) and we establish under what conditions it induces
an automorphism of the associated Wada dessins (see Sections 3.1 and 3.2). In general, it
is not easy to predict the necessary restrictions on the parameters m and n such that these
conditions are satisfied. In Section 4 we prove that if n and m+ 1 are primes the problem
can be solved.
Since in the literature automorphisms of dessins are defined in slightly different ways,
we point out that here we consider only orientation-preserving automorphisms.
2 Finite projective spaces
Let Fm+1n be the vector space over the finite field Fn, n = pe a prime power. We de-
fine the finite projective space Pm(Fn) as the vector space Fm+1n \{0} factorized by the
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 293
multiplicative group F∗n of order n− 1:
Pm(Fn) := (Fm+1n \{0})/F∗n . (2.1)
The integer n is called order of Pm(Fn) ([3], [5]).
The number of points of Pm(Fn) is given by
` := |Pm(Fn)| =
|Fm+1n \{0}|
|F∗n|
=
nm+1 − 1
n− 1
. (2.2)
By duality, the integer ` is also the number of hyperplanes of Pm(Fn).
Hyperplanes are subspaces Um−1(Fn) of dimension m − 1 in Pm(Fn), thus they cor-
respond to subspaces of dimension m in Fm+1n . Similarly to equation (2.2), we compute
the number of points on each of them as:
q := |Um−1(Fn)| =
|Fmn \{0}|
|F∗n|
=
nm − 1
n− 1
. (2.3)
By duality, the integer q is also the number of hyperplanes through each point.
The finite field Fnm+1 may be considered as a vector space over the field Fn. A well
known fact is that the multiplicative group of a finite field is cyclic. Therefore the m-
dimensional projective space over Fn may be identified with the quotient
Pm(Fn) ∼= F∗nm+1/F
∗
n , (2.4)
which, being considered as a group, is also cyclic of order `. Let g be a generator of this
group.
Due to the identification above we make correspond points Pb and, by duality, hyper-
planes hw of Pm(Fn) to powers of the generator g:
gb ↔ Pb, gw ↔ hw, b, w ∈ Z/`Z . (2.5)
In this way, we obtain a numbering of points and of hyperplanes by the integers b, w ∈
{0, . . . , `− 1}. Point and hyperplane numberings are closely related to each other and are
not arbitrary as we will explain more in detail in Section 2.1.
Among the projective linear transformations of Pm(Fn) we find cyclic projectivities
permuting in a single cycle, i.e. transitively, the set of points. By duality they also tran-
sitively permutes the set of hyperplanes. These projectivities generate so-called Singer
groups (see e.g. [7]). Due to (2.4), it is easy to prove that the group Σ` ∼= F∗nm+1/F
∗
n –with
` defined as in (2.2)– is cyclic and it acts transitively permuting the set of points and the set
of hyperplanes of Pm(Fn). Hence, we formulate the following
Proposition-Definition 2.1. Every Pm(Fn) admits a point (hyperplane) transitive cyclic
group of automorphisms. This group is a Singer group.
This proposition was first proved by Singer ([13]) for projective planes P2(Fn). Now
it is widely known ([1], [5], [7]) that it also holds for spaces of higher dimension. For
an exhaustive proof see [9, Chapter II, Section 7]. The existence of a Singer group cycli-
cally permuting points and hyperplanes justifies the fact that projective spaces Pm(Fn) are
commonly called cyclic projective spaces.
294 Ars Math. Contemp. 4 (2011) 291–311
Let γ be a generating element of Σ`. Thus, due to (2.5) and to (2.4), the action of
each element γa ∈ Σ`, a ∈ Z/`Z, on the points Pb and on the hyperplanes hw is naturally
expressed by:
γa :Pb 7−→ Pb+a ,
ha 7−→ hw+a . (2.6)
2.1 Constructing dessins
In order to construct dessins associated with a projective space Pm(Fn), we need to con-
struct the corresponding bipartite graph. We first introduce the conventions in Table 1.
point black vertex •
hyperplane white vertex ◦
incidence joining edge —
Table 1: Conventions
Incidence between a point Pb and an hyperplane hw is illustrated by the bipartite graph
through a joining edge between a black vertex b and a white vertex w. Recall that points
Pb and hyperplanes hw are numbered with integers b, w ∈ {0, . . . , ` − 1} given by the
exponents of a generator g of F∗nm+1/F
∗
n (see Section 2, Relation (2.5) ). This numbering
is not arbitrary. We consider the q points on a line. Thanks to the existence of the Singer
group defined in Proposition 2.1, Singer could show (see [13]) that for projective planes
the integers resulting from differences of a line index with the indices of the incident points
form a difference set. Difference sets are defined in the following way:
Definition 2.2 ([1]). A (v, k, λ)-difference set D = {d1, . . . , dk} is a collection of k
residues modulo v, such that for any residue α 6≡ 0 mod v the congruence
di − dj ≡ α mod v (2.7)
has exactly λ solution pairs (di, dj) with di and dj in D.
In particular, sets (D + s) mod v with s ∈ Z/vZ are also difference sets and we call
them shifts of D. If D̂ ≡ (t ·D + s) mod v with s ∈ Z/vZ, t ∈ (Z/vZ)∗ then D̂ and D
are said to be equivalent.
For projective planes v is equal to `, which is the total number of points (lines), and k is
equal to q, which is the number of points on a line and, by duality, of lines through a point.
Singer’s construction tells us that a point Pb and a line hw are incident if and only if
b− w ≡ di mod ` , i ∈ Z/qZ , (2.8)
where the q elements di are the elements of a difference set D. It is a well known fact
that Singer’s construction can be extended to projective spaces of higher dimension (see,
for instance, [7]), i.e. differences of point indices with the index of the common incident
hyperplane build an (`, q, λ) - difference set. This difference set determines a numbering
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 295
of points depending on hyperplane numbering and viceversa (see Relation (2.8)). Nev-
ertheless, the difference set is not unique since we may have several difference sets with
parameters (`, q, λ) (see [1]) which are equivalent to D or not. We thus fix one difference
set D and one ordering of its elements, unique up to cyclic permutations.
According to (2.8) and identifying each point and each hyperplane with its index b or
w, we choose the local incidence pattern given in Figure 1 1.
Figure 1: Local incidence pattern.
We construct the embedding of the bipartite graph starting with a white vertex w and
with an incident edge (w,w + di). Thus according to the local incidence pattern of the
black and of the white vertices we have chosen, the next incident edge going clockwise
around a cell is (w + di, w + di − di+1) followed by (w + di − di+1, w + 2di − di+1).
Repeating this procedure, we obtain a cell boundary with the sequence of edges given in
Figure 2.
The construction is applied to successive white vertices until q ·` different edges of type
◦—• are constructed. Different means here that every two edges differ at least in one of the
two indices b orw. The construction terminates since all q ·` incidences between points and
hyperplanes are represented by the bipartite graph 2. For each cell, depending on the value
of differences di−di+1 we reach the starting edge after 2r steps, with r := `gcd((di−di+1),`) .
Thus we obtain cells with valencies 2r (see Figure 2), where r does not necessarily have
the same value for all cells.
2.2 Wada dessins
We now consider a special case of the above construction.
Suppose that we may find at least one cyclic ordering of the elements of D such that all
differences di − di+1 are prime to `. In this case, all cells of the dessin we construct have
the same valency 2` as it is easy to check. The dessin is therefore uniform with signature
1We remark here that this is not the only possible incidence pattern we may choose. According to the fixed
ordering of the elements of the difference set, the white vertices incident with a black vertex are ordered anticlock-
wise and the black vertices incident with a white one are ordered clockwise. We have chosen these orderings since
we are interested in special uniform dessins called Wada dessins wich we will introduce later on in this section
and which are the main topic of study of this paper.
The choice of different orderings is also possible and may give rise to dessins which are not of Wada type (see
[12, Chapter 6]).
2For reasons of duality between hyperplanes (white vertices) and points (black vertices) we may also carry
out the construction starting with edges incident with black vertices. In this case we come to an end when q · `
different edges of type •—◦ are constructed.
296 Ars Math. Contemp. 4 (2011) 291–311
Figure 2: Construction of a cell incident with a white vertex w. After 2r steps we reach the
starting edge (w,w + di).
(q, q, `) and with q cells. Such dessins have the following nice property. On the boundary
of each cell each white and each black vertex with a given index occurs precisely once. If
two white vertices had the same index, then according to the incidence pattern in Figure 1
and to the construction described above we should have
w + α · (di − di+1) ≡ w + β · (di − di+1) , α, β ∈ Z/`Z ,
but this is possible only for α ≡ β mod ` since differences (di − di+1) are prime to `. In
a similar way we may prove that it is not possible to have two black vertices with the same
index on the same cell boundary.
This property was first described by Streit and Wolfart [14] for bipartite graphs of pro-
jective planes P2(Fn) and is called Wada property. The choice of the name Wada goes
back to the theory of dynamical system and of the Lakes of Wada. It is, in fact, possible to
divide the euclidean plane in three regions such that all the points on the boundary of one
region are also on the boundary of the other two (see e.g. [4, Chapter 4]). To emphasize
the analogy between this phenomenon and the one observed for the cells of their dessins,
Streit and Wolfart called them Wada dessins.
More in general, the Wada property may also be observed for embeddings of bipartite
graphs associated with projective spaces of higher dimension. By construction, the Wada
dessins we obtain are always uniform. Moreover, due to the action of the cyclic group Σ`
on the elements of projective spaces (see Section 2), these dessins have the property that at
least the group Σ` is a group of automorphisms. This group acts transitively on the black
and on the white vertices permuting them cyclically on the cell boundaries. This action
induces a transitive action on the set of edges of type ◦—• and on the set of edges of type
•—◦ belonging to the boundary of each cell (see [12, Chapter 5] for more details).
We conclude by giving the following definition of Wada compatibility for orderings of
elements of difference sets D associated with projective spaces:
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 297
Definition 2.3. We call orderings of the q elements of a difference set D associated with a
projective space Pm(Fn) Wada compatible if differences mod ` of consecutive elements
di, di+1 of D are prime to `, i.e.:
gcd ((di − di+1), `) = 1 ∀i ∈ Z/qZ . (2.9)
Remark 2.4. As we have remarked above the difference set D with the chosen element
ordering is not unique. Multiplying D with integers t ∈ (Z/`Z)∗ and shifting it with
integers s ∈ Z/`Z we obtain new difference sets equivalent to D. If we do not change the
element ordering of the original difference set or if we only permute it cyclically, the Wada
dessins we construct are isomorphic to each other. However, for non-cyclic permutations
and for difference sets with the same parameters (`, q, λ) but non-equivalent to D, even
if the chosen ordering is Wada compatible, the dessins we construct are in general not
isomorphic to each other. We may, in fact, obtain a different embedding of the graph into
an orientable surface and a different Belyı̆ function.
For more general and open questions about if and how many non-equivalent difference
sets with the same parameter set exist, see e.g. [1].
3 The Frobenius Automorphism
We consider again the identification of the projective space Pm(Fn), n = pe, with the
quotient of multiplicative cyclic groups:
Pm(Fn) ∼= F∗nm+1/F
∗
n .
Since Fnm+1 is a finite field, we know that there exists an automorphism σ, the Frobenius
automorphism, acting on the elements a ∈ Fnm+1 in the following way:
σ : Fnm+1 −→ Fnm+1 ,
a 7−→ ap .
The Frobenius automorphism generates the Galois group Gal(Fnm+1/Fp) ∼= Φf as a cyclic
group of order f = e · (m+ 1).
As we have seen in Section 2, we may identify pointsPb and hyperplanes hw of Pm(Fn)
with powers of a generator of F∗nm+1/F
∗
n (see Relation (2.5)). Thus each σ
u ∈ Φf , where
u ∈ Z/fZ, acts on Pb and hw as:
σu : Pb 7−→ Pbpu ,
hw 7−→ hwpu .
(3.1)
The action of Φf subdivides the set of points P and the set of hyperplanes H into orbits
with lengths ϕ ∈ N, ϕ | f . Each set always contains at least one orbit of length one, i.e.
the orbit of the point P0 ∈ P and the orbit of the hyperplane h0 ∈ H. In fact,
σu : P0 7−→ P0 ,
h0 7−→ h0 ∀u ∈ Z/fZ .
3.1 Frobenius difference sets
Not every automorphism of projective spaces Pm(Fn) leads to an automorphism of asso-
ciated dessins d’enfants. Whether the Frobenius automorphism induces an automorphism
298 Ars Math. Contemp. 4 (2011) 291–311
of associated dessins or not, depends, basically, on the difference set we choose for the
construction. First of all, let us look at an example.
Example 3.1. We consider the projective space P4(F2) with parameters ` = 31 and q =
15. For the construction of associated dessins we use the difference set D and the shift
D′ ≡ D − 1 mod 31 with their elements ordered in the following way:
D = {1, 3, 15, 2, 6, 30, 4, 12, 29, 8, 24, 27, 16, 17, 23} mod 31 ,
D′ ≡ D − 1 mod 31
= {0, 2, 14, 1, 5, 29, 3, 11, 28, 7, 23, 26, 15, 16, 22} mod 31 .
Recalling the incidence pattern given in Figure 1, we construct two dessins D and D′. We
are sure that they have the Wada property, since ` = 31 is prime. The dessins have signature
(15, 15, 31) and 15 cells. Due to the action on points and on hyperplanes of P4(F2) (see
Figure 3: Sketch of dessins D and D′ associated with P4(F2). In red we emphasize the
action of the Frobenius automorphism σ on the cells. The action of σ turns out to be an
automorphism of the dessin D but not of the dessin D′ (see text for details).
Relation (3.1)), the Frobenius automorphism σ acts on the vertices of each dessin by a
multiplication with the prime two:
σ : b 7−→ b · 2 ,
w 7−→ w · 2 . (3.2)
Lookig at Figure 3 and with an easy check of the action on the vertices, we observe that σ
is an automorphism of D but not of D′. On D it rotates the cells around the fixed vertex
w = 0 –and by duality also around b = 0– by an angle ω = 2π5 .
The reason for the different action is due to the different behaviour of the sets D and
D′ under the action of σ. According to Singer’s construction, elements of difference sets
associated with projective spaces Pm(Fn) correspond to differences of indices of points
and of incident hyperplanes. Thus, recalling the action of σ on points and hyperplanes (see
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 299
Relation (3.1)), its action on the elements of D and D′ is a multiplication with the integer
two. Under such multiplication the setD is fixed up to a cyclic permutation of its elements.
On the contrary, the set D′ is not fixed, as it is easy to check:
σ(D) = {2, 6, 30, 4, 12, 29, 8, 24, 27, 16, 17, 23, 1, 3, 15} mod 31 ,
σ(D′) = {0, 4, 28, 2, 10, 27, 6, 22, 25, 14, 15, 21, 30, 1, 13} mod 31 .
We will analyse more closely the ordering of the elements of D in the next section.
Here we formulate the following
Proposition-Definition 3.2. Let Pm(Fn) ∼= F∗nm+1/F
∗
n be a projective space, n = p
e, p
prime, containing ` points and ` hyperplanes whose incidence can be described using dif-
ference sets. The Frobenius automorphism determines at least one difference set Df fixed
under the action of Φf ∼= Gal(Fnm+1/Fp). Multiplying Df with integers t ∈ (Z/`Z)∗, we
obtain further difference sets fixed under the action of Φf .
We call Df and all difference sets t ·Df Frobenius difference sets.
Proof. As we have seen in Section 3, Φf divides the set of points P and the set of hy-
perplanes H of Pm(Fn) into orbits with possibly different lengths. In particular, there is
always at least one orbit of length one in each set. In fact, Φf always fixes the elements
P0 ∈ P and h0 ∈ H. Let us consider the hyperplane h0. Since Φf is a group of auto-
morphisms of Pm(Fn), each σu ∈ Φf preserves incidence. If we now consider the set
P0 = {Pi0 , Pi1 , . . . , Piq−1} of points on h0, since
σu : h0 7−→ h0 ∀σu ∈ Φf ,
by incidence preservation we have as well:
σu : P0 7−→ P0 ∀σu ∈ Φf .
According to Singer’s construction, differences of point indices with the index of the com-
mon incident hyperplane describe a difference set (see Section 2.1), thus the indices of the
points belonging to P0 form the Frobenius difference set Df we are looking for. It is easy
to see that multiplying Df with integers t ∈ (Z/`Z)∗ we still obtain Frobenius difference
sets. In fact, Φf acts on the elements of Df and of each t · Df by multiplication with
powers pu, u ∈ Z/fZ, so we have:
pu · t ·Df = t · pu ·Df ≡ t ·Df mod ` ∀t ∈ (Z/`Z)∗ ,
i.e. each t ·Df is fixed under the action of Φf .
Remark 3.3. The difference set Df and the sets t · Df determined by Φf may not be
unique. In fact, as we have already remarked in Sections 2.1 and 2.2, other difference sets
Df with parameters (`, q, λ) may exist which are equivalent or non-equivalent to Df and
are fixed by Φf . The case of difference sets resulting from shifts of Df will be analysed
more closely later on in this section.
The above proof implies
300 Ars Math. Contemp. 4 (2011) 291–311
Corollary 3.4. The cyclic group Φf acts on the elements of a Frobenius difference set Df
by a multiplication with powers pu, u ∈ Z/fZ and we have
puDf ≡ Df mod ` ∀u ∈ (Z/fZ) .
Due to the action of Φf on points and hyperplanes, from the above proof it also follows
that Φf divides the points on h0 into orbits with lengths ϕ, ϕ ∈ N, ϕ | f . AsDf consists of
the indices of these points, this means that Φf also divides the elements of Df into orbits
with lengths ϕ. Hence, we formulate the following
Corollary 3.5. Under the action of Φf , the elements of a Frobenius difference set Df are
subdivided into orbits with lengths ϕ, ϕ ∈ N, ϕ | f . These orbits correspond to orbits of
points in Pm(Fn).
Example 3.6. In the above example for P4(F2) the Frobenius automorphism generates the
Galois group Φ5 ∼= Gal(F52/F2). This group acts on the points Pi and on the hyperplanes
hi of P4(F2) by a multiplication with powers 2u, u ∈ Z/5Z:
σu ∈ Φ5 : Pi 7−→Pi2u ,
hi 7−→ hi2u , u ∈ (Z/5Z) .
The elements of the difference setD5 may be identified with the indices of the 15 points on
the fixed hyperplane h0. The difference set D5 is fixed under the action of σ and therefore
of Φ5. Its elements are subdivided by Φ5 into three orbits of length five:
{1, 2, 4, 8, 16} , {3, 6, 12, 24, 17} , {15, 30, 29, 27, 23} .
Example 3.7. Let us consider the projective space P3(F3) ∼= F∗34/F∗3 with 40 points and,
by duality, with 40 hyperplanes. The Frobenius automorphism σ generates the Galois group
Φ4 ∼= Gal(F34/F3) which acts on the points Pi and on the hyperplanes hi as
σu ∈ Φ4 : Pi 7−→Pi3u ,
hi 7−→ hi3u , u ∈ (Z/4Z) .
The elements of the following (40, 13, 4)-difference set ([1]) may be identified with the
indices of the 13 points on the hyperplane h0 fixed by Φ4:
D4 = {21, 22, 23, 25, 26, 29, 34, 35, 38, 0, 5, 7, 15} mod 40 .
The set D4 is a Frobenius difference set and it is easy to prove that multiplying it with
powers 3u we only have a permutation of its elements, i.e. we have
3uD4 ≡ D4 mod 40 ∀u ∈ (Z/4Z) .
The cyclic group Φ4 subdivides the elements of D4 into the following five orbits:
{21, 23, 29, 7}, {22, 26, 34, 38}, {25, 35}, {5, 15}, {0} .
We have seen that difference sets which result from multiplications of Frobenius dif-
ference sets Df by elements t ∈ (Z/`Z)∗ are still Frobenius difference sets fixed by Φf .
Therefore, it is reasonable to ask about shifts. Indeed, depending on the number of fixed
points and of fixed hyperplanes of Φf we can have more than one Frobenius difference set:
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 301
Proposition 3.8. Let Pm(Fn) ∼= F∗nm+1/F
∗
n be a projective space, n = p
e, p prime.
The projective space Pm(Fn) contains ` points and ` hyperplanes whose incidence can be
described using a Frobenius difference set Df . No shifts of Df are allowed if and only if
only h0 and equivalently only P0 are fixed by Φf ∼= Gal(Fnm+1/Fp). If there are more
elements fixed, we will have shifts of Df which are still Frobenius difference sets and:
#Frobenius difference sets as shifts of Df = # fixed hyperplanes (fixed points) of Φf .
Proof. Let us assume that no shifts of Df are allowed but there is at least another hy-
perplane hs fixed by Φf . This would mean that h0 and hs share the same point set P0.
Nevertheless, sharing the same point set means h0 = hs.
Now we assume that only the hyperplane h0 is fixed by Φf , but there are two difference
sets fixed: the Frobenius difference set Df and a shift of it D′f ≡ (Df + s) mod `. Both
Df and D′f are defined up to multiplication with elements t ∈ (Z/`Z)∗. According to
Singer’s construction if we identify the points on h0 with the elements of Df , then we may
identify the points on hs with the elements of D′f ≡ (Df + s) mod `. As D′f is fixed
under the action of Φf so should also hs be, but this would be a contradiction to the fact
that h0 is unique. It thus follows that no shifts of Df are allowed.
As we have already seen, the indices of the points on every hyperplane hs, s ∈ Z/`Z
can be identified with the elements of a shift Df + s. If Φf fixes some of the hyperplanes
hs, it directly follows that the indices of the points on each fixed hs describe Frobenius
difference sets Df + s and we have:
#Frobenius difference sets as shifts of Df = # fixed hyperplanes (fixed points) of Φf .
Remark 3.9. From the proof of Proposition 3.8 follows that, once we know a Frobenius
difference set Df , it is easy to determine other equivalent difference sets fixed by Φf .
We only need to know which of the hyperplanes hs is fixed by Φf . The corresponding
difference set is then Df + s.
Example 3.10. For the projective space P4(F2) there are only one hyperplane h0 and one
point P0 fixed under the action of Φ5. It follows that the Frobenius difference set
D5 = {1, 3, 15, 2, 6, 30, 4, 12, 29, 8, 24, 27, 16, 17, 23} mod 31
is unique up to multiplication with elements t ∈ (Z/31Z)∗.
Example 3.11. For the projective space P3(F3) there are two hyperplanes and, by duality,
two points fixed by Φ4. The fixed hyperplanes are h0 and h20 and the fixed points are P0
and P20. As we have seen in Example 3.7, we may choose
D4 = {21, 22, 23, 25, 26, 29, 34, 35, 38, 0, 5, 7, 15} mod 40
as the Frobenius difference set corresponding to the set of points on h0. We, therefore, have
that
D′4 ≡ D4 + 20 mod 40 = {1, 2, 3, 5, 6, 9, 14, 15, 18, 20, 25, 27, 35} mod 40
is the Frobenius difference set corresponding to the set of points on h20 and it is easy to
prove that 3u(D4 + 20) ≡ (D4 + 20) mod 40 for all u ∈ (Z/4Z).
302 Ars Math. Contemp. 4 (2011) 291–311
Figure 4: Sketch of dessinsD andD′′ associated with P4(F2). As in Figure 3 we emphasize
in red the action of the Frobenius automorphism on the cells (see text for more details).
3.2 Frobenius compatibility
The existence of Frobenius difference sets is a necessary but still not a sufficient condition
for the construction of dessins having the Frobenius automorphism as a dessin automor-
phism. We consider the following example:
Example 3.12. In addition to the dessins D and D′ we have considered in Example 3.1
we construct a third dessin D′′ associated with the projective space P4(F2). For the con-
struction we use the difference set D′′ whose elements are the elements of D but with a
different ordering3 :
D = {1, 3, 15, 2, 6, 30, 4, 12, 29, 8, 24, 27, 16, 17, 23} mod 31 ,
D′′ = {1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 23, 24, 27, 29, 30} mod 31 .
According to the incidence pattern given in Figure 1 we may construct the Wada dessinD′′
sketched in Figure 4. In Example 3.1 we have seen that σ is an automorphism of D acting
with a rotation of the cells around the white vertex w = 0. However, for D′′ we observe
that it is not an automorphism. The different behaviour of the Frobenius automorphism on
the dessins D and D′′ depends on the ordering of the elements of the difference sets. We
consider orbits of elements under the action of the cyclic group Φ5 generated by σ. As we
have seen in Example 3.6 the elements of D are subdivided into three orbits of length five:
{1, 2, 4, 8, 16} , {3, 6, 12, 24, 17} , {15, 30, 29, 27, 23} .
Thus, we are able to construct five blocks: the first block only contains the first elements
of each orbit, the second block the second elements of each orbit, etc. The Frobenius
automorphism maps the elements of each block onto the elements of the next block. The
3We use here again the notation D for the difference set D5 in order to be consistent with the notation used in
Example 3.1.
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 303
element ordering of D corresponds to this subdivision into blocks, the element ordering of
D′′ does not.
We call the ordering of the elements of D given in the above example Frobenius com-
patible.
According to the action of Φf on the elements of a Frobenius difference set Df (see
Corollary 3.4), we give the following definition:
Definition 3.13. Consider the following cyclic orderings of the q elements of a Frobenius
difference set Df :
Df ={d1, . . . , dk, pjd1, . . . , pjdk, . . . . . . . . . , p(f−1)jd1, . . . , p(f−1)jdk} ,
for some j ∈ (Z/fZ)∗ , q
f
= k . (3.3)
We call such cyclic orderings Frobenius compatible orderings.
Since Wada compatibility is also necessary for the construction of Wada dessins (see
Definition 2.3) we formulate moreover
Proposition 3.14. The elements of a Frobenius difference set Df ordered in a Frobenius
compatible way are also ordered in a Wada compatible way iff differences of consecutive
elements belonging to the subset {d1, . . . , dk} are prime to `
gcd (di − di+1, `) = 1 ∀i ∈ {1, . . . , (k − 1)} (3.4)
and iff
gcd
(
dk − pjd1, `
)
= 1 . (3.5)
Proof. If the elements of Df ordered in a Frobenius compatible way also satisfy the Wada
condition, then conditions (3.4) and (3.5) necessarily hold.
If condition (3.4) holds then we also have
gcd
(
pj(di − di+1), `
)
= 1 ∀j ∈ (Z/fZ)∗ . (3.6)
In fact, since ` = n
m+1−1
n−1 = n
m+nm−1+· · ·+1 with n = pe we have gcd(`, p) = 1. This
means that all differences of consecutive elements belonging to the subset {pjd1, . . . , pjdk}
are prime to `. Of course, this may be extended to each other subset {ph·jd1, . . . , ph·jdk},
h ∈ Z/fZ.
Condition (3.5) is necessary to make sure that when passing from one subset to the next
the Wada condition is also satisfied. If (3.5) holds then for the same reasons as in (3.6) we
also have:
gcd((ph·jdk − p(h+1)·jd1), `) = 1 ∀h ∈ Z/fZ .
We proceed considering Wada dessins constructed with Frobenius difference sets whose
elements are ordered in a Frobenius compatible way. For these dessins the cyclic group Φf
is a group of automorphisms. We examine this fact more in detail in the next sections.
304 Ars Math. Contemp. 4 (2011) 291–311
4 A ‘nice’ case
In Section 3.1 we have seen that for each projective space Pm(Fn) there always exists at
least one Frobenius difference set Df . Unfortunately, we cannot always order its elements
in a Frobenius compatible way. In fact, it may happen that the cyclic group Φf divides the
elements of Df into orbits with different lenghts. In Example 3.7 we have seen that the
cyclic group Φ4 divides the elements of the difference set D4 associated with P3(F3) into
orbits with lenghts four, two and one. In general, it is not easy to determine the necessary
conditions for the parameters m and n of Pm(Fn) so that we can predict the existence of
Frobenius compatible orderings. Here we consider a ’nice’ case and we see that if the order
of the cyclic group Φf is prime, under some conditions, Frobenius compatible orderings
exist.
We first formulate two lemmas:
Lemma 4.1. Let f be the order of the cyclic group Φf ∼= Gal(Fnm+1/Fp) generated by the
Frobenius automorphism acting on a projective space Pm(Fn) with n = pe, e ∈ N\{0}.
Let f be prime, thus we have:
n = p , f = m+ 1 , Φf ∼= Gal(Fpm+1/Fp) . (4.1)
If p 6= m + 1 and p 6≡ 1 mod (m + 1), then f divides the valency q of the white and of
the black vertices.
Proof. The proof follows from Fermat’s little theorem.
For n = p the integer q is given by (recall Relation (2.3)):
q =
pm − 1
p− 1
.
The integer f = m+ 1 divides q if
pm ≡ 1 mod (m+ 1) ,
and this is true due to Fermat’s little theorem since we have chosen p 6= m + 1. Now we
need the denominator not to ’destroy’ the divisibility property. In fact, we have to choose
p 6≡ 1 mod (m+ 1) ,
since for p ≡ 1 mod (m+ 1) we obtain:
q = pm−1 + · · ·+ 1 ≡ m mod (m+ 1)
and gcd(m, (m+ 1)) = 1.
Lemma 4.2. We consider the Frobenius difference set Df fixed by the cyclic group Φf .
Under the conditions of Lemma 4.1 on f and n, no shifts of Df are fixed by Φf and the
following properties hold:
1. 0 6∈ Df and
2. all Φf -orbits of elements di ∈ Df have length f .
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 305
Proof. As we have seen in Corollary 3.4, the group Φf acts on the elements of Df by
multiplication with powers of p and it divides the elements into orbits whose lengths are
divisors of f . Since f is prime, we will only have orbits of length 1 or of length f . Suppose
we have at least one orbit of length one. Since f divides q (see Lemma 4.1 above), we
necessarily have at least f − 1 other orbits of length 1. Let 0 ∈ Df and let {0} be an orbit
of length one.
Recalling some ideas of Baumert [1] about projective planes, we first show that gcd(p−
1, `) = 1.
Dividing ` by p− 1 we obtain:
` = (p− 1)(pm−1 + · · ·+ (m− 1)p+m) + (m+ 1) .
We know
gcd(p− 1,m+ 1) = 1
due to p 6≡ 1 mod (m + 1) and to m + 1 being prime. From the Euclidean algorithm it
therefore follows that
gcd(p− 1, `) = 1 .
Let us suppose that there exists another Frobenius difference set D′f ≡ (Df + s) mod `
fixed by Φf . For the integer s ∈ Z/`Z we have:
p · s ≡ s mod ` i.e. (p− 1) · s ≡ 0 mod ` . (4.2)
Since gcd(p − 1, `) = 1, equation (4.2) can be satisfied only for s ≡ 0 mod `. Thus no
shifts of Df are allowed.
Now, if (f − 1) elements di ∈ Df with di 6≡ 0 mod ` are fixed by p, then they should
satisfy the congruence relation:
p · di ≡ di mod ` =⇒ (p− 1) · di ≡ 0 mod ` .
Again, since gcd(p − 1, `) = 1, this congruence can be satisfied only for di ≡ 0 mod `.
This means that {0} is the only possible orbit of length one under the action of p. Never-
theless, we must exclude it since in this case f would not divide q.
It follows that 0 6∈ Df and that Df is decomposed only into Φf -orbits of length f .
Thanks to the above lemma we may choose cyclic orderings of the elements of Df as
we have defined them in 3.13. We have
Corollary 4.3. Under the conditions of Lemma 4.1, the elements of a Frobenius difference
set Df fixed by the cyclic group Φf may be ordered in a Frobenius compatible way. This
ordering is fixed under the action of Φf up to cyclic permutations.
Using the Frobenius difference setDf with Frobenius compatible element orderings we
construct dessins associated with projective spaces. If the dessins have the Wada property,
then the cyclic group Φf is a group of automorphisms.
Proposition 4.4. Let Φf be the cyclic group generated by the Frobenius automorphism
acting on a finite projective space Pm(Fn), n = pe. Let f be prime and p 6= m + 1,
p 6≡ 1 mod (m+1). LetDf be a Frobenius difference set fixed by Φf whose elements are
ordered in a Frobenius compatible way. If the elements of Df are also ordered in a Wada
compatible way so that we can construct a (q, q, `)-Wada dessin D, then Φf is a group
of automorphisms of D acting freely on the edges and rotating the set of cells around the
vertices b = w = 0 fixed by Φf .
306 Ars Math. Contemp. 4 (2011) 291–311
Figure 5: Local incidence patterns with Frobenius compatible ordering of the elements of
Df .
Proof. We suppose that at least one Frobenius compatible ordering of the elements of Df
considered up to cyclic permutations is also Wada compatible and we construct a (q, q, `)-
Wada dessin D. Since f |q (see Lemma 4.1 above), the group Φf has a suitable size to
be a group of automorphisms of D acting on the q cells of D. The cyclic group Φf is
generated by the Frobenius automorphism σ. We consider the action of σ on the points and
hyperplanes of Pm(Fn) and therefore on the vertices of D. As we have seen in Section 3 it
acts by multiplication of the indices with the prime p:
σ :Pb 7−→ Pbp ,
hw 7−→ hwp .
According to the notation introduced in Definition 3.13, we write the edges of D as
ewν,i = (w,w + p
νdi) (◦ — •) and ebν,i = (b, b− pνdi) (• — ◦) with ν ∈ Z/fZ,
i ∈ {1, . . . , k}. The action of σ on the edges is given by:
σ : ewν,i = (w,w + p
νdi) 7−→ p · ewν,i = (p · w, p(w + pνdi)) ,
ebν,i = (b, b− pνdi) 7−→ p · ebν,i = (p · b, p(b− pνdi)) .
If w and b are not fixed under the action of Φf , then none of the edges ewν,i and e
b
ν,i is fixed
by σ. If, on the contrary, w and b are fixed, then σ(ewν,i) 6= ewν,i and σ(ebν,i) 6= ebν,i only
if di is not fixed by p. Indeed, this is true, otherwise we could not have chosen Df with a
Frobenius compatible ordering of its elements. Thus Φf does not fix any of the edges and
we say that its action is free on them.
We now consider the action of Φf on the cells around the vertices w and b of D which
are fixed by Φf . Since Φf only fixes the difference set Df (see Lemma 4.2), this means
that it only fixes the vertices b = w = 0 of the dessin. In fact, according to Singer’s con-
struction (see Section 2.1), elements of difference sets D associated with projective spaces
correspond to indices of points on hyperplanes (and by duality of hyperplanes through
points). As Df is the only difference set fixed by Φf , only h0 and P0 are fixed by Φf , from
which it follows that w = b = 0 are the only vertices fixed by Φf .
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 307
Recall the incidence pattern of each vertex given in Figure 5 where the elements of
Df are ordered in a Frobenius compatible way. The incidence pattern is fixed up to cyclic
permutations of the elements of Df . The action of σ on the cells Cc, c ∈ Z/qZ around
w = 0 results in a mapping of every cell to a following cell (see Figure 6) such that:
σ : Cc 7−→ Cc+mk mod q,
c ∈ Z/qZ ,m ∈ Z/fZ , k = q
f
. (4.3)
We obtain the following relation between powers of p and the cells we run through:
p-powers cells
p0 Cc
p1 Cc+mk mod q
p2 Cc+2mk mod q
p3 Cc+3mk mod q
...
...
pf−1 Cc+(f−1)mk mod q
Evidently, σ describes a rotation of the set of cells around 0. The order of the rotation
is f .
Figure 6: Local ordering of the cells around w = 0 with a Frobenius difference set Df .
The elements of Df are ordered in a Frobenius compatible way.
308 Ars Math. Contemp. 4 (2011) 291–311
Remark 4.5. 1. A consequence of the above proof is that Φf acts freely not only on
the edges but also on the cells and it divides them into k orbits of length f .
2. We consider the vertices fixed by Φf as fixed points x on the target surface X of the
embedding. Then, introducing local coordinates z we may suppose that z(x) = 0
and that Φf acts on a neighbourhood of z = 0 by multiplication with powers of a
root of unity ζf :
z 7−→ ζaf z , a ∈ Z/fZ .
The root of unity ζf is called multiplier of the automorphism σ ∈ Φf on X (see
[14]).
Example 4.6. For the projective space P4(F2), we have considered in several of the above
examples, we have f = m + 1 = 5 and p 6= 5, p 6≡ 1 mod 5, thus the conditions of
Proposition 4.4 are satisfied. We construct a (15, 15, 31)-Wada dessin with the Frobenius
difference set:
D5 = {1, 3, 15, 2, 6, 30, 4, 12, 29, 8, 24, 27, 16, 17, 23} mod 31
whose elements are ordered in a Frobenius compatible way (see Example 3.1 and 3.12).
The cyclic group Φ5 ∼= Gal(F25/F2), generated by the Frobenius automorphism, acts
freely on the edges and on the cells. On the cells it acts with a rotation by the angle ω = 2π5
around the fixed vertices b = w = 0. The cells are subdivided into three orbits of length
five.
Other dessins, whose parameters satisfy the conditions of Proposition 4.4, are given
in Table 2. For most of the projective spaces listed there ` is prime, so we are sure we
may construct Wada dessins regardless of the chosen element orderings of the associated
Frobenius difference set.
For the spaces P4(F3) and P10(F2) the integer ` is not prime, thus we have to check
whether Frobenius compatible orderings of elements are also Wada compatible. According
to Proposition 3.14, we only need to check differences of the first block of elements and the
one difference at the ’transition’ between the first and the second block. Other differences
are multiplications with powers of the prime p for which we have gcd(`, p) = 1.
For both spaces the prime factors of the integer ` are quite big. In fact, we have ` =
121 = 11 · 11 and ` = 2047 = 89 · 23, so it is very likely to find Frobenius and Wada
compatible orderings. For instance, for P4(F3) it is easy to check that the ordering of the
elements of the following Frobenius difference set is Frobenius and Wada compatible.
D5 = {1, 4, 7, 11, 13, 34, 25, 67, 3, 12, 21, 33, 39, 102, 75, 80, 9, 36, 63, 99, 117, 64, 104,
119, 27, 108, 68, 55, 109, 71, 70, 115, 81, 82, 83, 44, 85, 92, 89, 103} mod 121 .
In this case, the group generated by the Frobenius automorphism is cyclic of order five and
it divides the elements of D5 into eight orbits of length five.
5 Concluding remarks
We consider projective spaces Pm(Fn) with general m and n, which do not necessarily
satisfy the conditions of Proposition 4.4. In this case, it is more difficult to predict whether
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 309
q ` f Wada Frobenius
P2(F5) 6 31 3 (6, 6, 31) Φ3 ∼=Gal(F53/F5)
P4(F2) 15 31 5 (15, 15, 31) Φ5 ∼=Gal(F25/F2)
P4(F3) 40 121 = 112 5 (40, 40, 121) Φ5 ∼=Gal(F35/F3)
P4(F7) 400 2801 5 (400, 400, 2801) Φ5 ∼=Gal(F75/F7)
P6(F2) 63 127 7 (63, 63, 127) Φ7 ∼=Gal(F27/F2)
P6(F3) 364 1093 7 (364, 364, 1093) Φ7 ∼=Gal(F37/F3)
P6(F5) 3906 19531 7 (3906, 3906, 19531) Φ7 ∼=Gal(F57/F5)
P10(F2) 1023 2047 = 23 · 89 11 (1023, 1023, 2047)(?) Φ11 ∼=Gal(F211/F2)
(?) = only if there exists a Frobenius compatible ordering which is also Wada compatible.
Table 2: Some projective spaces whose parameters satisfy the conditions of Proposition
4.4.
the cyclic group Φf or a subgroup Φg ⊂ Φf is a group of automorphisms of associated
Wada dessins.
Let Φg ⊂ Φf be generated by a power σs of the Frobenius automorphism, with
s ∈ (Z/fZ) and gcd(s, f) 6= 1. The action of Φg on the elements of the Frobenius differ-
ence set we use to construct the associated dessin is a multiplication by the integer t = ps.
As we have seen in Sections 3.1 and 3.2, for Φf to be a group of automorphisms of
the constructed dessin we need not only a Frobenius difference set Df , but we also need
the ordering of the elements of Df to be compatible with the action of Φf . Equivalent
conditions are necessary if we need a subgroup Φg ⊂ Φf to be a group of automorphisms
of the dessin. This means that under the action of Φg the elements of a fixed difference set
Dg have to be subdivided into orbits of equal length. This can be achieved if the following
conditions hold:
1. The order g of Φg divides q, the number of elements of Dg;
2. The integer t satisfies gcd(t− 1, `) = 1.
We need the first condition since it implies that Φg has a suitable size to subdivide the
elements of Dg into orbits of the same length g.
The second condition makes sure that all orbits have the same length. In fact, having
gcd(t− 1, `) = 1 means that the relation
t · di ≡ di mod ` ∀di ∈ Dg
is satisfied only for di ≡ 0 mod `. But we exclude this possibility, otherwise Dg would
contain the only orbit {0} of length one and all other orbits would have length g. This
would be a contradiction to the first condition g | q, so we have di 6≡ 0 mod ` for all
di ∈ Dg and all orbits have length g.
Under these conditions, it is possible to order the elements of a difference set fixed
by Φg in a way compatible with its action, i.e. such that Φg acts permuting the elements
310 Ars Math. Contemp. 4 (2011) 291–311
cyclically. If the ordering is also Wada compatible, then Φg is a group of automorphisms
of the constructed Wada dessin. Similarly to the action of Φf (see Section 4) also Φg acts
on the cells by rotating them around the fixed vertices w = b = 0 (for a more detailed
description see [12]). We conclude with some examples:
Example 5.1. For the projective space P4(F4) we have q = 85 and ` = 341. The group
generated by the Frobenius automorphism is the cyclic group Φ10. The order of Φ10 does
not divide q, so it cannot be a group of automorphisms of any of the dessins associated
with P4(F4). Nevertheless, the subgroup Φ5 ⊂ Φ10 with 5 = m + 1 can be. The power
σ2 of the Frobenius automorphism can be chosen as a generator of Φ5. The action of
Φ5 on the vertices of associated dessins is expressed by multiplication with 22. Since
gcd(22 − 1, 341) = 1, also the second of the two above conditions is satisfied. This means
that the elements of the difference setD5 fixed by Φ5 are subdivided into Φ5-orbits of equal
length: there are 17 orbits of legth five. The elements may, therefore, be ordered in a way
compatible with the action of Φ5. If this ordering is such that all differences of consecutive
elements (di − di+1), di, di+1 ∈ D5 are prime to ` = 341 = 31 · 11, i.e. if it is Wada
compatible, we may construct an (85, 85, 341)-Wada dessin for which Φ5 is a group of
automorphisms acting freely on the 85 cells.
Example 5.2. For P6(F4) we have q = 1365, ` = 5461. The Frobenius automorphism
generates the cyclic group Φ14 with gcd(14, 1365) = 7. This means that the subgroup
Φ7 ⊂ Φ14 has a suitable size to be a group of automorphisms of possible Wada dessins
associated with P6(F4). As in the above example, we may choose the power σ2 of the
Frobenius automorphism as a generator of Φ7. The action of Φ7 on the vertices of associ-
ated dessins is expressed by multiplication with powers 22. Since gcd(22 − 1, 5461) = 1,
also the second of the two above conditions is satisfied and Φ7 divides the elements of
the associated difference set D7 into 195 orbits of length 7. We order the elements of the
difference set D7 in a way compatible with the action of Φ7. If this ordering is also Wada
compatible we may construct a (1365, 1365, 5461)-Wada dessin for which Φ7 is a group
of automorphisms acting freely on the cells.
In general, it is not an easy task to construct difference sets with parameters (v, k, λ)
or, more specifically, with parameters (`, q, λ) if they are associated with projective spaces
Pm(Fn). The difference sets used in our examples are known difference sets we took from a
difference set list in [1]. In [1] as well as in [8] some construction techniques are described.
Of course, knowing that a difference set is a Frobenius difference set associated with a
projective space Pm(Fn) and that its elements may be ordered in a Frobenius compatible
way may help in the construction (see e.g. the sections about multipliers of difference sets
in [1, Section III.c] and in [8, Section 2.5]). In this case the structure of the difference set
is, in fact, completely determined by the action of the cyclic group Φf or of a subgroup
Φg of it, as we have seen in Section 3.2 and in the present section. Nevertheless, for
projective spaces Pm(Fn) with arbitrary parameters m and n it is difficult to predict when
such orderings occur, as we have remarked above. More research is needed in this direction.
The question about the existence of Wada compatible orderings of elements has been
answerd by [14] and more recently by [6] for difference sets associated with projective
planes P2(Fn). It is still an open and challenging question for projective spaces of higher
dimension.
C. Sarti: Wada Dessins associated with Finite Projective Spaces. . . 311
Acknowledgements
This is a part of the PhD thesis of the author at the University of Frankfurt. She would like
to warmly thank her advisors Jürgen Wolfart and Gareth Jones for introducing her to the
beautiful topic of dessins d’enfants and for many useful hints and discussions.
She would also like to thank Benjamin Mühlbauer, Alessandra Sarti and Ayberk Zeytin
for carefully reading the first version of this paper. Many thanks also to the anonymous
reviewer of this paper for very useful comments.
References
[1] L. Baumert, Cyclic Difference Sets, Springer - Verlag, Berlin/Heidelberg/New York, 1971.
[2] G.V. Belyı̆, On Galois extensions of a maximal cyclotomic field, Math. USSR Isvestija 14
(1980), 247–256.
[3] A. Beutelspacher and U. Rosenbaum, Projektive Geometrie, Vieweg, Braun-
schweig/Wiesbaden, 2nd edition, 2004.
[4] Gerard Buskes and Arnoud C. M. van Rooij, Topological Spaces: From Distance to Neighbor-
hood. Springer, New York/Berlin/Heidelberg etc., Reprint, 1997.
[5] P. Dembowski, Finite Geometries, Springer, Berlin/Heidelberg/New York, 2nd edition, 1997.
[6] R. Goertz, Coprime ordering of cyclic planar difference sets, Discrete Math. 309 (2009),
5248–5252.
[7] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, 2nd
edition, 2005.
[8] D. R. Hughes and F. C. Piper, Design Theory, Cambridge University Press, Cambridge, New
York, etc., 1985.
[9] B. Huppert, Endliche Gruppen I, Springer, Berlin/Heidelberg/New York, 1979.
[10] G. Jones, M. Streit and J. Wolfart, Wilson’s graph operations on regular dessins and cyclotomic
filelds of definition, Proc. London Math. Soc 100 (2010), 510–532.
[11] S.K. Lando and A.K. Zvonkin, Graphs on Surfaces and Their Applications, Springer,
Berlin/Heidelberg/New York, 2004.
[12] C. Sarti, Automorphism Groups of Wada Dessins and Wilson Operations, Dissertation, Uni-
versity of Frankfurt, Germany, 2010.
[13] J. Singer, A theorem in finite projective geometry and some applications in number theory,
Trans. Amer. Math. Soc., 43 (1938), 377–385.
[14] M. Streit and J. Wolfart, Cyclic projective planes and Wada dessins, Documenta Math. 6,
39–68.
[15] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory Ser. B 18 (1975), 155–163.