Regular dessins with abelian automorphism groups

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1 Jürgen Wolfart and Benjamin Mühlbauer 1 Institut für Mathematik der Goethe Universität, D Frankfurt a.m., Germany wolfart@math.uni-frankfurt.de Regular dessins with abelian automorphism groups Dedicated to the memory of Wolfgang Schwarz Abstract. A quasiplatonic curve can be defined over the rationals if it has a regular dessin with abelian automorphism group. 1 Basic facts and definitions The following facts and definitions are fundamental for the theory of dessins d enfants on Riemann surfaces. Proofs and more details may be found in [11] and the references given there. 1.) Compact Riemann surfaces Y are smooth projective algebraic curves, holomorphic mappings between such Riemann surfaces are morphisms between the algebraic curves. 2.) As a curve, Y can be defined over a number field if and only if there is a Belyĭ function β : Y Ĉ = P1 (C) on Y, that is a meromorphic function, non constant and ramified at most above 0, 1, ([1, 7, 9, 12]). In this case we call Y a Belyĭ surface or a Belyĭ curve, (Y, β) a Belyĭ pair. 3.) The β preimage of the real interval [0, 1] defines a bipartite graph on Y with β 1 (0) and β 1 (1) as the sets of white and black vertices, respectively. The graph subdivides Y into simply connected cells, and we can consider the poles of β as their midpoints. This embedded graph D is called the dessin d enfant [8] corresponding to β. 4.) On the other hand, if we embed a bipartite graph D in a compact oriented 2 manifold Y such that Y \ D is the disjoint union of simply connected cells, then there is a unique conformal structure on Y and a unique Belyĭ function β corresponding to D ([8, 14] or [11], Thm. 3.15). 5.) For every Belyĭ pair (Y, β), that is for every dessin, there are cocompact Fuchsian triangle groups and finite index subgroups Φ such that Y = Φ\H where H denotes the hyperbolic plane and such that the Belyĭ function can be written ([17], Thm. 1 part of the DFG project Wo 199/4 1 1

2 3, [11], Thm. 3.10) as the canonical mapping Φ\H \H = Ĉ : Φz z. The entries in the signature (l, m, n) of are common multiples of the zero orders of β, 1 β and 1/β, respectively. If l, m, n are chosen minimally, we call (l, m, n) the type of the dessin. For some triples of small numbers will be an euclidean or a spherical triangle group and H has to be replaced with C or the Riemann sphere Ĉ. 6.) The dessin is called regular if there is a group G Aut Y acting transitively on the set of edges of D and preserving the colours of the vertices. If Y has a regular dessin, it is called quasiplatonic (or curve with many automorphisms in older papers, triangle curve in the literature about Beauville surfaces). There are many other characteristic properties ([17] or [11], Thm. 5.1), for example the Belyĭ function β defines a normal covering, its monodromy group (the so called cartographic group of the dessin) is isomorphic to its covering group, Φ is a normal subgroup of, and one has /Φ = G. Every Belyĭ surface (or every dessin) has a quasiplatonic (respectively regular) covering of finite degree. 7.) The field L is called a field of definition for Y if all coefficients of the polynomials defining Y are contained in L. More generally, we call L a field of definition for Y if there is a curve Y isomorphic (over C) to Y defined over L. For σ Gal Q/Q we denote by Y σ the curve obtained by the action of σ on all coefficients of the defining equations of Y. In the context of the present paper, we may suppose that L is a number field; it is not at all uniquely determined, however the moduli field M(Y ) of Y, that is the fixed field of the subgroup := {σ Gal Q/Q Y σ = Y } Gal Q/Q V Y of the absolute Galois group depends only on the isomorphism class of Y. It is contained in all fields of definition of Y. For any number field K we can define similarly a field of definition K and a moduli field M K (Y, G) for the pair consisting of Y and a group G Aut Y of automorphisms as the fixed field of the group V (Y,G)/K := {σ Gal Q/K f σ : Y Y σ isomorphism with f σ α = α σ f σ α G}. 8.) It is well known that the moduli field is not always a field of definition of Y ([5, 13]). According to Weil [16] it is a field of definition if there is a choice of isomorphisms f σ : Y Y σ for all σ Gal Q/M(Y ) satisfying the cocycle condition f στ = f τ σ f τ for all σ, τ Gal Q/M(Y ). 2

3 9.) By consequence, Y can be defined over M(Y ) if the identity is the only automorphism of Y because then all f σ are uniquely determined. By variations of this argument, also regular dessins, that is Belyĭ pairs (Y, β) with normal β can be defined over their moduli field M(Y, β) ([3], Proposition 2.5), obviously defined as the fixed field of V (Y,β) := {σ Gal Q/Q f σ : Y Y σ isomorphism with β σ f σ = β}. From this result one can derive that also quasiplatonic curves can be defined over their moduli field ([4], [17], Thm. 5, [11], Thm. 5.4). Similarly, (Y, Aut Y ) can be defined over its moduli field if Aut Y contains its centre as a direct factor ([6], Thm. 1). 2 Abelian automorphism groups Theorem 1 Let D be a regular dessin on a quasiplatonic curve X with abelian automorphism group Aut D, and let β denote the corresponding Belyĭ function. Then, (X, β) is defined over Q. Corollary. Quasiplatonic curves having a regular dessin with abelian automorphism group can be defined over Q. Theorem 1 is already given as Proposition 3 in [2], together with a sketch of a proof. For the corollary, and under slightly different hypotheses, there are two more recent proofs in [10] and [15], Cor. 1.2, based on quite different ideas. We will use instead the following lemma which seems to be well known; unfortunately, we do not know a quotable source or where it originally comes from (the first author learned it from David Singerman and Gareth Jones many years ago). Lemma. Every regular dessin D with abelian automorphism group is a quotient of some regular dessin D r of type (r, r, r) on the Fermat curve F r of exponent r, with automorphism group Aut D r = Cr C r. Proof. (See also [11], Example 5.5 and Exercise 5.8) If we describe F r by its affine model x r + y r = 1, it is easy to see that β(x, y) := x r defines a normal Belyĭ function corresponding to the normal inclusion of the commutator subgroup in the triangle group = (r, r, r). This commutator subgroup is the surface group of F r, and for the corresponding dessin D r we have Aut D r = / = Cr C r (for r > 3 it is an index 6 normal subgroup of the full automorphism group Aut F r ). Now suppose that X is quasiplatonic with a regular dessin of type (l, m, n) and abelian automorphism group A. Let r be the lcm of l, m, n. The surface group N of X is then the (torsion free) kernel of a group epimorphism of the triangle group (l, m, n) onto A, inducing as well an epimorphism = (r, r, r) A 3

4 with some kernel Γ which is in general not torsion free. We have however X = Γ\H, and because A = /Γ is abelian, we get Γ with a quotient Γ/ =: U C r C r, whence X = U\F r and D = U\D r. Proof of Theorem 1. With the same notations and the same affine model for F r as above we have now X = U\F r. Every automorphism α U acts on F r by α : (x, y) (ζ k r x, ζ h r y) with a primitive r th root of unity ζ r and exponents k, h Z/rZ. So, F r and its Belyĭ function β are defined over Q and Aut F r is defined over Q(ζ r ). Moreover, the Belyĭ function B on X can be considered as induced by the Belyĭ function β on the U orbits in F r (well defined because β has the same value for all arguments in every U orbit). Clearly, X σ = U σ \F r for all σ Gal Q/Q where σ acts in the obvious way on the automorphisms α U, hence B σ is induced by β on the U σ orbits. However, since every Galois automorphism σ Gal Q/Q acts on ζ r as ζ r ζr s for some s Z/rZ only depending on σ, we have α σ = α s and α σ U for all α U, hence U σ = U for all σ. Because D r, that is the pair (F r, β) is defined over Q, also the quotient D = U\D r is invariant under Gal Q/Q. Therefore its moduli field is Q, and by Proposition 2.5 of [3] (see point 9. of Section 1) D can be defined over Q. 3 Examples and generalisations Quasiplatonic surfaces having dessins with abelian automorphism groups exist in all genera g > 1, compare the Accola Maclachlan curves described in [11], Exercise 5.20: for n = 2g + 1 the automorphism group A = C 2n is an image of the triangle group (2, n, 2n) under an epimorphism with torsion free kernel Γ such that the quotient curve X = Γ\H has an affine model y 2 = x n 1. In fact, there are much more examples. Among the 22 isomorphism classes of quasiplatonic curves in the genera g = 2, 3, 4 there are only one curve in genus 3 and three curves in genus 4 not having a regular dessin with abelian automorphism group (they are also defined over Q, but by other reasons, compare [2]), see the Tables 5.2 to 5.4 in [11]. Recall that a curve Y of genus > 1 has only finitely many automorphisms. If Y is defined over a number field K, its automorphisms can therefore also be defined over a fixed number field L K : by the action of field automorphisms of C on Aut Y fixing K elementwise we would otherwise get an infinity of automorphisms. A similar argument shows that a minimally chosen L is a normal extension of K. Recall also that Y = Γ\H for a torsion free normal subgroup Γ of the triangle group which is the normaliser of Γ in PSL 2 R. 4

5 Theorem 2 Let Y be a quasiplatonic curve, β its canonical Belyĭ function Y Aut Y \Y, let K be the moduli field ( = minimal field of definition) of (Y, β), and let the Galois extension L K be a field of definition for (Y, Aut Y ). 1) For any subgroup U Aut Y, the curve X := U\Y and the Belyĭ function B induced by β on X can be defined over L. 2) Its relative moduli field M K := M K (X, B), defined as the fixed field of V (X,B)/K := {σ Gal Q/K f σ : X X σ isomorphism with B σ f σ = B}, is the same as the fixed field of the group G U := {σ Gal L/K U σ is conjugate to U in Aut Y }. Proof. 1) can be proved similarly to Cor. 1 of [6] considering the intermediate fields in the function field extension L(Y )/L(β) whose Galois group 2 is antiisomorphic to Aut Y, compare [11], Thm ) First we have to prove that if there is an a Aut Y with the property aua 1 = U σ, we get an isomorphism f : U\Y = X X σ = U σ \Y with B = B σ f which is in fact induced by Y Y : y a(y). Second we have to show that for any σ Gal L/K with the property (X σ, B σ ) = (X, B) there is such an automorphism a Aut Y inducing this isomorphism via the conjugation of U, hence σ G U. Since σ fixes Y and β, both subgroups U and U σ lift to Fuchsian groups Φ, Φ σ between Γ and. The isomorphism f : X X σ lifts to some α PSL 2 R acting on H and conjugating Φ into Φ σ since B = B σ f. Moreover, α normalises Γ, hence belongs to and induces an automorphism a Aut Y. If U Aut Y is a normal subgroup, X is itself quasiplatonic. If moreover K = Q, then M is the minimal field of definition of (X, B). References [1] Belyĭ, G.V.: On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43, , 479 (1979) [2] Conder, M.D.E., Jones, G.A., Streit, M., Wolfart, J.: Galois actions on regular dessins of small genera. Rev. Mat. Iberoam. 29, (2013) 2 [6] contains a little mistake: since β is used in the base field of the Galois extension of function fields, we have to be sure that β is also defined over the field of constants; this is guaranteed here by the construction of K L. In Cor. 1 of [6] this may fail if the hypotheses of Lemma 1 or Lemma 2 of [6] are not satisfied. There one can fill the gap in the hypothesis by replacing the moduli field M(S, G) with a sometimes slightly larger moduli field M(S, β, G) in whose definition all f σ satisfy in addition β σ f σ = β. 5

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