Quasiplatonic Surfaces, and Automorphisms

Abstract

Quasiplatonic Riemann surfaces or algebraic curves, sometimes also called curves with many automorphisms or triangle curves, can be characterised in many equivalent ways, for example as those curves having a regular dessin, one with the greatest possible degree of symmetry. The sphere and the torus each support infinitely many regular dessins, easily described in both cases. For each genus g > 1 there are, up to isomorphism, only finitely many regular dessins; this chapter gives complete lists for genera 2, 3 and 4, and discusses methods for counting and classifying them. These methods often involve counting generating triples for finite groups, in some cases with the aid of character theory (which we briefly summarise) and Möbius inversion. We present several important infinite families of quasiplatonic curves, such as Hurwitz and Macbeath-Hurwitz curves, Lefschetz and Accola-Maclachlan curves. We prove that like their counterparts, the curves with trivial automorphism group, quasiplatonic curves can be defined over their field of moduli. Many of the automorphism groups appearing in this chapter are 2-dimensional linear or projective groups over finite fields, so we summarise their most relevant properties in the final section.

5.5 Appendix: Linear and Projective Groups

Many of the groups appearing in this book are linear or projective groups, so we summarise their definitions and properties here. See [13], [27, Sect. II.6–II.8], [55, Sect. 4.5], or [57] for details.

If R is a commutative ring with identity, then for each integer n ≥ 1 the general linear group $\mathop{\mathrm{GL}}\nolimits _{n}(R)$ consists of the n × n matrices M over R which are invertible, that is, those for which the determinant $\det M$ is a unit in R. The function $\det$ is an epimorphism from $\mathop{\mathrm{GL}}\nolimits _{n}(R)$ to the multiplicative group U(R) of units in R, so its kernel, the special linear group $\mathop{\mathrm{SL}}\nolimits _{n}(R)$ consisting of those M with $\det M = 1$, is a normal subgroup of $\mathop{\mathrm{GL}}\nolimits _{n}(R)$ with $\mathop{\mathrm{GL}}\nolimits _{n}(R)/\mathop{\mathrm{SL}}\nolimits _{n}(R)\mathop{\cong}U(R)$. In particular, if R is a field K then $U(R) = K^{{\ast}}:= K\setminus \{0\}$.

For any field K and n ≥ 2, the natural action of $\mathop{\mathrm{GL}}\nolimits _{n}(K)$ on the vector space V = K ⁿ induces an action of this group on the (n − 1)-dimensional projective space $\mathbb{P}^{n-1}(K)$ formed by the 1-dimensional subspaces of V. The kernel of this action is a normal subgroup $Z\mathop{\cong}K^{{\ast}}$ consisting of the scalar matrices $\lambda I_{n}\;(\lambda \in K^{{\ast}})$, and the group of transformations induced on $\mathbb{P}^{n-1}(K)$ is the projective general linear group $\mathop{\mathrm{PGL}}\nolimits _{n}(K)\mathop{\cong}\mathop{\mathrm{GL}}\nolimits _{n}(K)/Z$. In this action, the subgroup $\mathop{\mathrm{SL}}\nolimits _{n}(K)$ induces the projective special linear group $\mathop{\mathrm{PSL}}\nolimits _{n}(K)\mathop{\cong}\mathop{\mathrm{SL}}\nolimits _{n}(K)/(\mathop{\mathrm{SL}}\nolimits _{n}(K) \cap Z)\mathop{\cong}\mathop{\mathrm{SL}}\nolimits _{n}(K)Z/Z$. This is a simple group for all n ≥ 2 and all fields K, except when n = 2 and | K |  = 2 or 3, in which case the group is isomorphic to S ₃ or A ₄. The kernel $\mathop{\mathrm{SL}}\nolimits _{n}(K) \cap Z$ of the induced epimorphism $\mathop{\mathrm{SL}}\nolimits _{n}(K) \rightarrow \mathop{\mathrm{PSL}}\nolimits _{n}(K)$ is isomorphic to the group of nth roots of 1 in K, a cyclic group of order dividing n, while $\mathop{\mathrm{PGL}}\nolimits _{n}(K)/\mathop{\mathrm{PSL}}\nolimits _{n}(K)$ is isomorphic to the quotient of K ^∗ by its subgroup of nth powers (so $\mathop{\mathrm{PSL}}\nolimits _{n}(K) =\mathop{ \mathrm{PGL}}\nolimits _{n}(K)$ whenever K is algebraically closed).

The affine group $\mathop{\mathrm{AGL}}\nolimits _{n}(K)$ consists of the affine transformations of V, those of the form v ↦ Av + b where $A \in \mathop{\mathrm{GL}}\nolimits _{n}(K)$ and b ∈ V. This is a semidirect product of a normal subgroup, isomorphic to the additive group of V, consisting of the translations v ↦ v + b, and a complement $\mathop{\mathrm{GL}}\nolimits _{n}(K)$, consisting of the affine transformations with b = 0, that is, the linear transformations of V.

These groups $\mathop{\mathrm{GL}}\nolimits _{n}(K)$, $\mathop{\mathrm{SL}}\nolimits _{n}(K)$, $\mathop{\mathrm{PGL}}\nolimits _{n}(K)$, $\mathop{\mathrm{PSL}}\nolimits _{n}(K)$ and $\mathop{\mathrm{AGL}}\nolimits _{n}(K)$ can all be extended by the Galois group $\mathop{\mathrm{Gal}}\nolimits K$ of the field K, acting on coordinates of vectors and points, to form groups

$$\displaystyle{\Gamma \mathrm{L}_{n}(K)\;,\;\Sigma \mathrm{L}_{n}(K)\;,\;\mathrm{P}\Gamma \mathrm{L}_{n}(K)\;,\;\mathrm{P}\Sigma \mathrm{L}_{n}(K)\quad \text{and}\quad \mathrm{A}\Gamma \mathrm{L}_{n}(K)}$$

containing them as normal subgroups with quotients isomorphic to $\mathop{\mathrm{Gal}}\nolimits K$.

If K is the finite field $\mathbb{F}_{q}$ of order q, then these groups such as $\mathop{\mathrm{GL}}\nolimits _{n}(K)$ are often denoted by $\mathop{\mathrm{GL}}\nolimits _{n}(q)$, etc., with the finite simple group $\mathop{\mathrm{PSL}}\nolimits _{n}(q)$ written as L _n (q) in ATLAS notation [9]. In this case, $\mathop{\mathrm{SL}}\nolimits _{n}(q) \cap Z\mathop{\cong}\mathop{\mathrm{PGL}}\nolimits _{n}(q)/\mathop{\mathrm{PSL}}\nolimits _{n}(q)\mathop{\cong}C_{d}$, where $d =\gcd (n,q - 1)$. We have q = p ^e for some prime p, so that $\mathop{\mathrm{Gal}}\nolimits K$ is a cyclic group of order e, generated by the Frobenius automorphism x ↦ x ^p. Counting linearly independent row-vectors gives $\vert \mathop{\mathrm{GL}}\nolimits _{n}(q)\vert =\prod _{ i=0}^{n-1}(q^{n} - q^{i})$, and the orders of the other related groups are easily deduced from this.

The case n = 2 is particularly important. The projective line $\mathbb{P}^{1}(K)$ can be identified with $K \cup \{\infty \}$ by identifying the point with homogeneous coordinates [x, y], that is the subspace of V = K ² spanned by the non-zero vector (x, y), with the element $x/y\;(= \infty$ if y = 0). If we let matrices act on the left of column vectors, so that each $A \in \mathop{\mathrm{GL}}\nolimits _{2}(K)$ acts as

$$\displaystyle{A =\Big (\,\begin{array}{*{10}c} a& &b\\ c & &d \end{array} \,\Big)\;:\;\Big (\,\begin{array}{*{10}c} x\\ y \end{array} \,\Big)\mapsto \Big(\,\begin{array}{*{10}c} ax + by\\ cx + dy \end{array} \,\Big),}$$

then the projective transformation of $\mathbb{P}^{1}(K)$ corresponding to A is given by

$$\displaystyle{z = \frac{x} {y}\;\mapsto \;\frac{ax + by} {cx + dy} = \frac{az + b} {cz + d}\,.}$$

This action of $\mathop{\mathrm{PGL}}\nolimits _{2}(K)$ on $\mathbb{P}^{1}(K)$ is sharply 3-transitive, meaning that the group acts regularly on ordered triples of distinct points, so a non-identity element has at most two fixed points. The subgroup fixing any one point is conjugate to the subgroup fixing $\infty$; this is induced by the matrices A with c = 0, and it is a semidirect product of an abelian normal subgroup isomorphic to the additive group of K, given by also taking $a = d = 1$, and a complement isomorphic to the multiplicative group K ^∗, given by taking $b = c = 0$. This complement is the subgroup of $\mathop{\mathrm{PGL}}\nolimits _{2}(K)$ fixing 0 and $\infty$, and the subgroup fixing any two points is conjugate to it. This describes all the elements with fixed points. There are also elements with no fixed points in $\mathbb{P}^{1}(K)$. When $K = \mathbb{F}_{q}$, so that $\vert \mathop{\mathrm{PGL}}\nolimits _{2}(K)\vert = q(q^{2} - 1)$, these elements have orders dividing q + 1; they have two fixed points when regarded as elements of $\mathop{\mathrm{PGL}}\nolimits _{2}(q^{2})$ acting on $\mathbb{P}^{1}(\mathbb{F}_{q^{2}})$ via the inclusion of $\mathbb{F}_{q}$ in $\mathbb{F}_{q^{2}}$. The elements of $\mathop{\mathrm{PGL}}\nolimits _{2}(q)$ fixing two points of $\mathbb{P}^{1}(\mathbb{F}_{q})$ have orders dividing q − 1, and those with one fixed point have order p, where q = p ^e for a prime p.

We are particularly interested in the groups $\mathop{\mathrm{PSL}}\nolimits _{2}(q) =\mathop{ \mathrm{SL}}\nolimits _{2}(q)/\{ \pm I\}$, often denoted by L ₂(q) in ATLAS notation [9]; see [13, Chap. XII] or [27, Sect. II.8] for full details and proofs of their following properties.

The group $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ is a subgroup of $\mathop{\mathrm{PGL}}\nolimits _{2}(q)$, of order $q(q^{2} - 1)/d$ and of index $d =\gcd (2,q - 1)$ equal to 1 or 2 as p = 2 or p > 2. If p = 2 then $\mathop{\mathrm{GL}}\nolimits _{2}(q) =\mathop{ \mathrm{SL}}\nolimits _{2}(q) \times Z$ with $\mathop{\mathrm{SL}}\nolimits _{2}(q) =\mathop{ \mathrm{PSL}}\nolimits _{2}(q) =\mathop{ \mathrm{PGL}}\nolimits _{2}(q)$, so in this case the above description of $\mathop{\mathrm{PGL}}\nolimits _{2}(q)$ applies to $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$. For any q, a non-identity element of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ has order dividing $(q - 1)/d$, equal to p, or dividing $(q + 1)/d$, as it fixes two, one or no points in $\mathbb{P}^{1}(\mathbb{F}_{q})$. Equivalently, if t is its trace (defined only up to multiplication by − 1), then t ² − 4 is respectively a non-zero square, equal to 0, or a non-square in $\mathbb{F}_{q}$. The automorphism group of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ can be identified with $\mathrm{P}\Gamma \mathrm{L}_{2}(q)$, acting by conjugation on its normal subgroup $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$.

Dickson described the subgroups of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ in [13, Chap. XII], see also [27, Sect. II.8], and from this one can describe the maximal subgroups:

Proposition 5.3

Any maximal subgroup of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ has one of the following forms, where $d = (2,q - 1)$ :

• 1. the stabiliser of a point in $\mathbb{P}^{1}(\mathbb{F}_{q})$ , isomorphic to the unique subgroup of order $q(q - 1)/d$ in $\mathop{\mathrm{AGL}}\nolimits _{1}(q)$ ;

• 2. a dihedral group of order 2(q ± 1)∕d;

• 3. a group isomorphic to $\mathop{\mathrm{PSL}}\nolimits _{2}(r)$ where $\mathbb{F}_{r}$ is a maximal subfield of $\mathbb{F}_{q}$ (that is r = p ^f with e∕f prime);

• 4. a group isomorphic to $\mathop{\mathrm{PGL}}\nolimits _{2}(r)$ where q = r ² is a perfect square;

• 5. a group isomorphic to A ₄ , S ₄ or A ₅ . $\square$

Maximal subgroups of types (1) and (2) exist for all q, and those of types (3) and (4) exist whenever r satisfies the stated conditions. Subgroups isomorphic to A ₄ exist if and only if q is odd or q = 2^e with e even; subgroups isomorphic to S ₄ exist if and only if $\,q \equiv \pm 1\bmod 8\,$; subgroups isomorphic to A ₅ exist if and only if p = 5 or $\,q \equiv \pm 1\bmod 5\,$; when they exist, these subgroups of type (5) are not always maximal.

We also need the character table for $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$. In fact we give three tables, for $\,q \equiv 1\bmod 4\,$, $\,q \equiv -1\bmod 4\,$ and q = 2^e.

In each table, the first row indicates the orders of the elements in the corresponding column: 1 for the identity element in the first column, p (where q = p ^e) for the parabolic elements (two conjugacy classes when q is odd, one when q = 2^e), and then in the last two columns, representing the hyperbolic and elliptic classes, the divisors n > 1 of $(q - 1)/2$ and $(q + 1)/2$ (or of q − 1 and q + 1 when q = 2^e). The hyperbolic and elliptic classes are each represented by an element a ⁱ or b ^j, where a and b are elements of orders (q ∓ 1)∕2 respectively (or q ∓ 1 when q = 2^e). In Table 5.5, where $\,q \equiv 1\bmod 4\,$, we have $i,j = 1,\ldots,r:= (q - 1)/4$; for instance, the involutions are represented by the hyperbolic element a ^r. In Table 5.6, where $\,q \equiv -1\bmod 4\,$, we have $i = 1,\ldots,r - 1$ and $j = 1,\ldots,r:= (q + 1)/4$, with the involutions represented by the elliptic element b ^r. In Table 5.7, where q = 2^e, we have $i = 1,\ldots,(q - 2)/2$ and $j = 1,\ldots,q/2$; in this case the involutions are parabolic.

Table 5.5 The character table of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ for $\,q \equiv 1\bmod 4\,$

Table 5.6 The character table of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ for $\,q \equiv -1\bmod 4\,$

Table 5.7 The character table of $\mathop{\mathrm{PSL}}\nolimits _{2}(q)$ for q = 2^e

In each table, the entries in the first column are the degrees χ(1) of the corresponding irreducible characters χ. Each row represents a single character, except in the cases χ(1) = q ± 1. In Table 5.5, the last two rows represent r − 1 and r characters, with $k = 1,\ldots,r - 1$ and l = 1, …, r respectively. In Table 5.6, the last two rows each represent $r - 1 = (q - 3)/4$ characters, with $k,l = 1,\ldots,r - 1$. In both tables, the preceding four rows each represent a single character, as do the first two rows of Table 5.7, where the last two rows represent $(q - 2)/2$ and q∕2 characters respectively. In Tables 5.5 and 5.6 we have $\zeta =\zeta _{(q-1)/2}$ and $\xi =\zeta _{(q+1)/2}$, but in Table 5.7 we have $\zeta =\zeta _{q-1}$ and $\xi =\zeta _{q+1}$, where $\zeta _{n}:=\exp (2\pi i/n)$.