ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS

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1 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS PETE L. CLARK Abstract. Here we give the first serious consideration of a family of algebraic curves which can be characterized (essentially) either as the compact Riemann surfaces uniformized by congruence subgroups of the Fuchsian triangle groups (a, b, c) or as the algebraic curves X/H, where X admits a G = P SL 2 (F q)- Galois Belyi map and H is a subgroup of G. This family contains in particular the classical modular curves and XX commensurability classes of quaternionic Shimura curves. Our perspective is that all of these curves ought to be viewed, in some ways, as generalized Shimura curves, notwithstanding the fact that the socalled non-arithmeticity of most of their uniformizing Fuchsian groups means that the Hecke algebra of modular correspondences will be too small to allow us to employ some of the usual automorphic techniques. Nevertheless the theory of Cohen and Wolfart provides a modular embedding of each of our curves as a cycle in a higher-dimensional quaternionic Shimura variety, and this embedding allows the theory of these curves to retain at least an automorphic flavor. Our work here is in two parts. The first part is foundational: our curves can be defined and studied using their canonical Belyi maps (using the arithmetic theory of branched coverings), in terms of Takeuchi s theory of Fuchsian groups with trace ring an order in a totally real number field, or as subvarieties of quaternionic Shimura varieties via the Cohen-Wolfart embedding. These three perspectives give quite complementary information on the properties of the curves, but it requires some care to check that the various constructions do in fact yield the same algebraic curves (or compact Riemann surfaces), and our first order of business is to establish this compatibility. Secondly, we are interested in determining both the minimal field of definition of our curves as well as the field of rationality of the automorphism group G = P SL 2 (F q). This latter field F is of obvious interest, both intrinsically and because by Hilbert irreducibility one gets P SL 2 (F q) as a Galois group over F. Alas one almost never has F = Q and in fact our analysis shows what researchers on the Inverse Galois Problem must have already suspected or known: apart from one or two exceptions, it is impossible to attain P SL 2 (F q) as the Galois group of a Galois cover of the projective line which is ramified at only three points! Nevertheless we are able to show in many cases that F is an abelian number field and conjecture that this is always the case, a strange sort of converse to Belyi s original work which attains (in particular) P SL 2 (F q) as a Galois group over Q ab. We have no really satisfactory explanation for this abelian phenomenon as we explain, it is easy to construct, for any number field K, a curve X acted upon by a finite group G, such that X X/G is a Belyi map, and such that every field of definition of X and G contains K but it confirms that our restriction to congruence subgroups of possibly non-arithmetic triangle groups gives us a family with distinguished arithmetic-geometric properties. We hope to persuade the reader that these curves are unusually worthy of further study. 1

2 2 PETE L. CLARK 1. Introduction The aim of this paper is to introduce a certain class of complex algebraic curves (of genus g 2) which include the classical elliptic modular curves and certain Shimura curves. There are many indications that the rich geometric, arithmetic and automorphic theories of modular curves should have analogues for our class of curves, despite the fact that their uniformizing Fuchsian groups are in general (socalled!) non-arithmetic, i.e., not commensurable with a Fuchsian group derived from a quaternion algebra. Let us enunciate the property of modular curves that we want to generalize. Let p 7 be a prime, and let X(p) be the (compactified) modular curve with full level p-structure. The group G = P SL 2 (F p ) acts effectively on X(p), and the natural map X(p) X(p)/G has the following property: there exists an isomorphism X(p)/G = P 1 such that the composite map X(p) P 1 is ramified only above the points 0, 1,. Otherwise put, there exists a subgroup G = P SL 2 (F p ) Aut(X(p)) such that quotienting by G gives a Galois Belyi map. Here we will be interested in the class of algebraic curves X /C of genus g 2 with the property that there exists a subgroup G = P SL 2 (F q ) Aut(X) such that the map X X/G is a Belyi map (again, this means that X/G has genus zero and there are at most three ramification points). Why is this an appealing class of algebraic curves? First of all, such curves can be defined over Q. Indeed, if k K is an inclusion of algebraically closed fields, Y /k is an algebraic curve, and ϕ : X Y is a branched covering defined over K but with k-rational branch points, then both X and ϕ can be defined over k. Conversely, Belyi proved that every algebraic curve defined over Q admits a map to P 1 with at most 3 ramification points. In other words, the class of algebraic curves admitting Belyi maps is enormously vast. On the other hand, the class of algebraic curves X admitting Galois Belyi maps X P 1 is much more specialized: there are only finitely many such curves of any given genus g 2 or with any given automorphism group. 1 We shall call such curves Wolfart curves, after J. Wolfart, who has studied them extensively over the complex numbers. For N 4, the Fermat curve X N +Y N = Z N is also a Wolfart curve. Wolfart curves admit many characterizations: for instance, the locus on the moduli space M g of curves of genus g at which the function C # Aut(C) attains a strict local maximum consists precisely of the genus g Wolfart curves. Thus in particular they include the class of curves attaining the Hurwitz bound # Aut(C) 84(g 1) (Hurwitz curves), which have been much studied in the literature. 2 Or, all-importantly for us, a Riemann surface C is a Wolfart curve if and only if its uniformizing Fuchsian group Γ is a (necessarily torsionfree) finite-index normal 1 If G Aut(X) is such that X X/G is a Belyi map, then so too is X X/ Aut(X); thus the distinction between subgroups of automorphisms and the entire automorphism group is not a critical one. Moreover, that X X/G is a Belyi map means that G is already very large; in practice, this usually forces G = Aut(X). 2 In fact we find the Wolfart condition to be so much more natural than the Hurwitz condition as to find it slightly odd that Hurwitz curves in particular have been so extensively studied.

3 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS3 subgroup of a Fuchsian triangle group (a, b, c), i.e., the group generated by the elements τ b τ a, τ c τ b, τ a τ c, where τ a, τ b, τ c are reflections in the sides of a hyperbolic triangle with angles 2π a, 2π b, 2π c (with a, b, c Z + ). Then gives the Galois Belyi map. Γ\H (a, b, c)\h Viewed as groups of automorphisms of H (i.e., as subgroups of P SL 2 (R)), for any positive integers a, b, c such that 1 a + 1 b + 1 c < 1, there exists a triangle group (a, b, c), unique up to conjugacy. One often allows any of a, b, c to take the value, which corresponds to allowing the vertices of the triangle to lie on the boundary RP 1 of H. E.g. (2, 3, ) = P SL 2 (Z) the standard modular group. As Wolfart notes in his work, it would be very interesting to undertake an arithmetic study of this distinguished class of curves The aim of this paper is to begin the explicit arithmetic study of a certain class of complex algebraic curves. Namely, we consider (always smooth, projective geometrically connected) algebraic curves X/C of genus g 2 such that the natural map q : X X/ Aut(X) is a Belyi map: that is, X/ Aut(X) has genus zero and q is ramified over three points, so that with an appropriate choice of coordinate function, we may identify X/ Aut(X) with P 1 and the ramification points with {0, 1, }. We dub such curves Wolfart curves, after J. Wolfart, who has studied them extensively. Wolfart curves can also be characterized as the points X on the moduli space M g at which the function X # Aut(X) attains a strict local maximum; for this reason they are often called Riemann surfaces with many automorphisms. (In particular, they include the curves X of genus g for which Hurwitz s bound # Aut(X) 84(g 1) is attained (Hurwitz curves), which are themselves much studied in the literature.) On the other hand, these special curves (and their quotients by subgroups H of Aut(X)) include many if not most of the examples of curves which have been most intensively studied in the literature, e.g. the Fermat curves, Klein s quartic curve, the classical modular curves, and certain Shimura curves. As for any curve admitting a Belyi map, a Wolfart curve can be defined over Q, or equivalently over some number field. This suggests that Wolfart curves should be studied from an arithmetic perspective (e.g., rational points, places of good and bad reduction), and of course the above examples also furnish the list of curves most familiar to arithmetic algebraic geometers. Of course, given a Wolfart curve X /C, before embarking upon such an arithmetic study we need a model of X over some number field K. In this sort of business, there are several different ways of construing the statement X can be defined over K and it is critically important to distinguish carefully between them. The weakest possible notion is that of the field of moduli M(X) of X, which is the fixed field of the largest subgroup A of Aut(C) such that σ A = X σ C = X. Clearly, if there exists some field K and a curve defined over K whose basechange to C is isomorphic to X /C, then K contains the field of moduli M(X). In general, the converse is false. However, it can be repaired as follows: any Galois covering t : X P 1 of compact Riemann surfaces can be defined over its field of moduli M(X, t) [Kock, Thm. 2.2]. Now for any X /C be any curve (of genus g 2) such

4 4 PETE L. CLARK that X/ Aut(X) = P 1, it is easy to see that the field of moduli of X as a curve is equal to the field of moduli of the covering t : X X/ Aut(X), so it follows that any Wolfart curve can be defined over its field of moduli. On the other hand, even if X can be defined over M(X), there will in general not be a unique model defined over M(X): indeed, after fixing a single model X /M(X), the set of M(X)-models is given by the Galois cohomology set H 1 (M(X), Aut(X)). Note that here Aut(X /Q ) is being viewed as a module over the absolute Galois group of M(X). This action cuts out a finite Galois extension F (X)/M(X) which is the minimal field such that the covering X P 1 is Galois over M(X). Definition: An aut-canonical model for an algebraic curve X /Q is given by a number field K and a K-model X for X /Q such that: (i) All the automorphisms of X are defined over K. (ii) For any number field L for which there exists an L-model X for X Q such that all automorphisms of X are defined over L, there exists a field homomorphism ι : K L such that X ι L = X. In particular, there exists at most one autcanonical model up to isomorphism. Example: For a prime number p 7, consider the modular curve X(p) = Γ(p)\H. Then G = Aut(X(p)) = P SL 2 (F p ) and the quotient map X X/G is the natural map corresponding to the inclusion of Fuchsian groups Γ(p) Γ(1) = P SL 2 (Z). Thus X/G = P 1 = C(j), and the map is ramified over j = 1728, 0, with indices 2, 3 and p. (So in particular, X(p) is a Wolfart curve.) Then the field of moduli of X(p) is Q, and indeed X(p) admits models over Q. 3 It turns out that X(p) does not admit a Q-model with all of its automorphisms defined. Letting p = ( 1) p2 1 2 p, K.-y. Shih showed that X(p) has an autcanonical model over Q( p ). Remark 1: The standard moduli interpretation of X(p) gives a model over Q(ζ p ) with all automorphisms defined. This model is derived from the autcanonical model by basechange (and indeed Q( p ) embeds in Q(ζ p )). Remark 2: We assume chosen on the rational curve X/ Aut(X) a Q-model so that each of the three ramification points are Q-rational, or equivalently so that the Galois action on the ramification divisor D is trivial. (In work on the Inverse Galois Problem, it is often convenient to equip D with a nontrivial Galois module structure in such a way as to cancel out certain other nontrivial Galois actions. We shall not do this here.) As we shall recall later, this class contains many of the families of curves which have received the most intense arithmetic study: e.g. certain Shimura curves including the classical modular curves and Klein s quartic curve, and the Fermat curves. On the other hand, there are only finitely many Wolfart curves of any given genus (equivalently, with any fixed automorphism group). They can be characterized 3 Every elliptic curve E/Q gives rise to such a model, two such models being isomorphic iff E[p] = E [p] as Galois modules. This exhibits infinitely many distinct Q-models, and there are others besides.

5 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS5 as the points on the moduli space M g of genus g curves at which the function X # Aut(X) has a strict local maximum, so are in a sense the maximally special points on M g. In particular they include the family of curves attaining the Hurwitz bound Aut(X) = 84(g 1) which have been intensely studied. 4 Having said all this, in this paper our focus of attention is the Wolfart curves with automorphism group isomorphic to P SL 2 (F q ), a class which has decidedly nicer arithmetic properties. Especially, we are interested in the following Question 1. Let X/C be a Wolfart curve with Galois group G = P SL 2 (F q ). Is it the case that X, together with all its automorphisms, can be defined over an abelian number field? We shall answer this question affirmatively in many cases. Note well that this is not a general property of Wolfart curves: Proposition 2. For any number field K, there exists a Wolfart curve X/C such that any field of definition of X and Aut(X) contains K. Proof: Let E be an elliptic curve with j-invariant j(e) K \ O K. Since E can be defined over K, by Belyi s theorem there exists a Belyi map β : E P 1 (possibly defined over a larger field). We claim that β is not a Galois cover, and indeed that the only elliptic curves admitting Galois Belyi maps are those with j-invariant 0 or Indeed, suppose G Aut(E) is a (finite!) subgroup such that β : E E/G is a Belyi map. Then G has a maximal normal subgroup G τ of translations (by torsion points); thus G = G/G τ is a group of automorphisms fixing the origin of the isogenous elliptic curve E := E/G τ and such that E E /G is a Galois Belyi map. But unless j(e) {0, 1728}, we have G {±1}, and the quotient by 1 is certainly not a Belyi map, being ramified over the four 2-torsion points. Let X P 1 be the Galois Belyi map obtained by taking the Galois closure of β : E P 1. If X were an elliptic curve it would again have j-invariant 0 or 1728, and then the natural map X E would be an isogeny, so that E would have complex multiplication, contradicting the non-integrality of j(e). Thus X has genus at least 2 so is a Wolfart curve in our sense. If X is defined together with all its automorphisms over some field L, then the subgroup H G such that X/H = E acts L-rationally, so that E is defined over L. Thus K L, completing the proof. So an affirmative answer to Question 1 would exhibit a curious property of the groups P SL 2 (F q ) calling out not just for verification but for some sort of explanation. The answer seems to be that every P SL 2 (F q )-Wolfart curve is a Shimura curve. Taken literally this claim asserts the arithmeticity of infinitely many Fuchsian triangle groups, so is known to be false by work of Takeuchi. However, work of Beasley-Cohen and Wolfart nevertheless allows us to obtain our Galois Belyi map X X/P SL 2 (F q ) by embedding P 1 as a cycle in a higher-dimensional quaternionic Shimura variety and pulling back to the P SL 2 (F q )-covering associated to an appropriate congruence subgroup. In this way the abelian phenomenon can be related to the Shimura-Deligne theory of canonical models. But in fact things are not so simple, as the work of Shimura and Deligne furnishes 4 In fact Wolfart curves are so similar in their arithmetic-geometric behavior to Hurwitz curves that it seems rather artificial to restrict attention to the latter class.

6 6 PETE L. CLARK us with a canonical model over an abelian extension not of Q but over a certain totally real number field. This field of definition is well-known to be the minimal one consistent with a certain moduli interpretation, but it is also known that it can be a proper extension of the minimal field of definition of the automorphisms of X: in the special case of the P SL 2 (F p )-covering X(p) X(1) of classical modular curves, this discrepancy goes back (at least) to work of Shih. Nowadays Shih s work can also be understood in the context of the rigidityrationality theory developed by workers on the Inverse Galois Problem. In the general case rigidity does not hold, and using related criteria of Volklein leaves us within a quadratic extension of fully determining the minimal field of definition of Aut(X). In the special case where X can as a curve be defined over Q, we are able to put together all of these considerations to find the precise minimal field of definition, an abelian number field. The general case seems to require a more complete theory of canonical models than we currently possess. 2. Statements of the Main Results Let G be a finite group. By a G-Wolfart curve we mean an algebraic curve with Aut(X) = G and such that q : X X/G is a Belyi map. Theorem 3. Let X be a complex algebraic curve admitting a P SL 2 (F q )-Galois Belyi map map β : X P 1, unramified outside {0, 1, }. Then: a) The field of moduli of X is an abelian number field. b) There exists an abelian number field K and an extension L/K with [L : K] 2 such that X and all of its automorphisms can be defined over L. More precise results are possible: we can compute the minimal field of definition and the Galois orbit of the corresponding point on the moduli space. The following result is representative. For more detailed investigations, we shall find it useful to restrict our attention to a particular family of curves. Namely, suppose p is a prime number and N 7 is a positive integer which is either equal to p or coprime to p. We define a positive integer a = a(n, p) as 1 if N = p and as the order of p in (Z/2NZ) /(±1) when (N, p) = 1. We also make the following auxiliary hypothesis: a(n, p) is an odd number. Theorem 4. There exists a curve X(N, p)/q with the following properties: i) G = Aut(X(N, p)) = P SL 2 (F p a(n,p)). ii) X X/G is a Belyi map with ramification indices (2, 3, N). iii) The field of moduli F w of X(N, p) is the fixed field of Q(ζ 2N ) under the action of H = 1, p (Z/2NZ), and X(N, p) can be defined over F w. iv) Every Wolfart curve with Galois group P SL 2 (F p a(n,p)) and ramification indices (2, 3, N) lies in the same Galois orbit as X(N, p), and the Galois orbit has size [F w : Q]. Remarks: It would be more accurate to say, There is a Galois orbit of curves... since our construction does not single out a unique curve. This could be remedied for instance by defining X(N, p) to be the disjoint union of all [F w : Q] distinct curves in the Galois orbit. But the abuse of notation simplifies matters so we will

7 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS7 stick with it. This theorem generalizes the main result of [Streit], which considered (by different methods) the special cases N = 7 (Hurwitz curves) and a = 1. The curves X(p, p) are the classical modular curves X(p), and the covering X(p, p) P 1 is the one associated to the congruence subgroup Γ(p) GL 2 (Z), i.e., to the kernel of the reduction map GL 2 (Z) GL 2 (F p ). When (1) N {7, 9, 11, 14, 18}, let F N = Q(ζ 2N + ζ 1 2N ), and let X N /F N be the Shimura curve associated to a maximal order O in the quaternion algebra over F N which is split at every finite place and ramified at all but one infinite place. Since O is split at every prime above p, O F p = M2 (O FN ) F p = M 2 (F p a(n,p)), so the analogous congruence subgroup Γ(p) O gives rise to the P SL 2 (F a(n,p) p )-covering X(N, p). Note that (1) is not the complete list of all Shimura curves uniformized by Fuchsian triangle groups (see Takeuchi); we have only taken those with ϕ [FIGURE OUT WHETHER THIS IS CONSISTENT WITH TAKEUCHI]. The proposal of the introduction can now be stated in a more concrete form: we suggest that all the curves X(N, p) be viewed as Shimura curves. Of course, Theorem 3 applies to the curves X(N, p) to give that their full automorphism group can be defined over an at-most-quadratic extension of an abelian number field. In certain cases we can show that the minimal field of definition is itself abelian: Theorem 5. Under any of the following assumptions, the minimal field of definition of X(N, p) and all its automorphisms is F 2N ( p ): a) N = p (classical modular case). b) (6, N) > 1 (rigid cases). c) [F 2N : Q] = ϕ(2n) 2 is odd and F w = Q. [AFTER THIS POINT, WE REVISIT AN EARLIER DRAFT OF THE PAPER] To be more precise, we can show the following. Theorem 6. Let X/Q be a Belyi-Wolfart curve with Galois group G = P SL 2 (F q ) with q > 3. Then the field of moduli of X is an abelian number field, and there exists an abelian number field L, and extension K/L with [K : L] 2 such that K is a field of definition for all the automorphisms of X. We have some reason to believe that the field extension K/L, if nontrivial, is obtained by adjoining D for some rational number D. If this is true, then the conjecture follows immediately. At least the conjecture holds for infinitely many cases among the Belyi-Wolfart curves X(N, p) to be introduced presently, namely whenever p generates (Z/2NZ) /(±1). In this case the field of moduli is Q, so that K/Q is a Galois number field, whose Galois group is an extension of a cyclic group by a group of order at most two. By elementary group theory, all such groups are abelian. Thus one perspective on the present work is to understand fields of definition of automorphism groups of Belyi-Wolfart curves. But there are other ways to look at what is being done here:

8 8 PETE L. CLARK Group theory: we use essentially the rigidity - rationality results developed by researchers on the Inverse Galois Problem. Indeed, around the same time as the proof of Belyi s theorem a fundamental advance on the inverse Galois problem was made by Belyi, Fried and Thomson, who found a purely algebraic condition on a finite group sufficient for its regular Galois realizability over an explicit cyclotomic field, namely rigidity. This is a certain simple transitivity condition on triples of elements in G (recalled explicitly, for what it s worth, in Section 2.3). Many finite groups have been realized as Galois groups over Q (and many more over Q ab ) using rigid triples, but at a certain point one has to deal with the fact that the existence of a rigid triple is rather restrictive. Much less restrictive (but still too much to hope for in general) is a condition of weak rigidity (also Section 2.3), and (despite my very limited knowledge in this area) I think it is fair to characterize the greatest part of the work of the last ten years on the Inverse Galois Problem as studying when the condition of weak rigidity is enough to ensure Galois realizations over Q or over Q ab (see [Malle-Matzat] and especially [Volklein]). In this context we can mention the key fact, which is that the groups P SL 2 (F q ) possess weakly rigid triples which, while not rigid in general, are very close. Indeed, results of [Macbeath 1969] imply that (as we shall make more precise later!) we are at most a single outer automorphism away from rigidity, so in what I call a semirigid situation. This is the same factor of 2 giving rise to the mysterious quadratic extension in Theorem 1. I have not seen this semirigidity property of P SL 2 (F q ) in the literature, 5 but at the least it seems to be a useful way to understand some known Galois realizations of groups P SL 2 (F q ). Indeed, while in the standard references [Malle-Matzat] and [Volklein] one finds Galois realizations of P SL 2 (F q ) over Q ab using only reduction to the rigid case, the semi-rigidity implies that we are within a factor of 2 of understanding all possible regular realizations of groups P SL 2 (F q ) as Galois groups via triples. 6 There is an infinite family of groups P SL 2 (F p 3) in which the field L of Theorem 2 is Q. In case p = 2 we have not only semirigidity but rigidity i.e., P SL 2 (F 8 ) occurs regularly as a Galois group over Q, a result due to G. Malle. This underscores the importance of determining the mysterious at-most quadratic extension K: can it be trivial?!? Generalized Shimura curves: A final perspective to which we hew the most closely in the present draft is obtained by specializing to the case of ramification degrees (2, 3, N). In this case we get an explicit two parameter family of generalized Shimura curves X(N, p)/q where p is a prime and N 7 is a positive integer either equal to p or prime to p. We impose an additional constraint on N and p, as follows: define a positive integer a = a(n, p) as 1, if N = p, 5 One should note the carefully phrased weakness of this statement. 6 The last two words are important: conjecturally, every finite group is the Galois group of a finite branched covering of P 1 /Q, but certainly more than three branch points are required in general: e.g. (Z/2Z) n requires n + 1 branch points even over C.

9 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS9 the order of p in (Z/2NZ) /(±1), if gcd(p, N) = 1. Note that a(n, p) will be odd for all primes p if N is itself prime and congruent to 3 modulo 4; to fix ideas, the reader might like to keep this case in mind. (Indeed, keep in mind the case N = 7 no matter what.) We illustrate Theorem 1 in this case: Theorem 7. Let N 7 be a positive integer, p a prime number such that either N = p or (N, p) = 1, and such that the integer a = a(n, p) defined above is odd. Then there is an algebraic curve X(N, p)/q which is not necessarily connected. It has the following properties (we assume here that gcd(6n, p) = 1; the other cases, which are easier, are treated in Section 3.5): a) Each connected component of X(N, p) has full automorphism group G = P SL 2 (F p a), and the canonical map X(N, p) X(N, p)/g = P 1 is ramified only over (0, 1, ) with indices (2, 3, N). b) The field of moduli F w of each connected component is the unique subfield of Q(ζ 2N ) which is of degree ϕ(2n) 2a over Q. The curve X(N, p) itself is the disjoint union of the distinct Galois conjugates of any of its connected components, so admits Q-rational models. c) Let K be the splitting field of the G-action on any component of X(N, p). Then F 2N K and [K : F 2N ] 2. Thus P SL 2 (F p a) occurs regularly as a Galois group over the number field K, which has degree equal to either ϕ(2n) 2 or ϕ(2n). This underscores the last point that we want to make: the general theory of Shimura varieties as exposed in [Deligne 1971] gives disconnected canonical models for Shimura varieties X (p) with level p structure, such that the degree of the field of definition of each component over the field of definition F of the base Shimura variety X goes to infinity with p. But the semirigidity of P SL 2 (F q ) points to the existence of models of X (p) over at-most quadratic extensions K of F which still have a canonicity property, not in the sense of moduli interpretations or fields generated by special points, but in the more elementary sense of being characterized by the property that K(X (p)) K(X ) is a Galois extension. We might tentatively call these models subcanonical models to emphasize their smaller field of definition. 3. Background on Wolfart curves Such Riemann surfaces have been studied by J. Wolfart [Wolfart 1997], [Wolfart 2000]. In particular, he gives the following intriguing characterization. Theorem 8. (Wolfart) For a compact Riemann surface X of genus at least two, the following are equivalent: a) X admits a Galois Belyi map. b) The canonical map q : X X/ Aut(X) is a Belyi map. c) X is uniformized by a Fuchsian group Γ which is a finite index normal subgroup of a hyperbolic triangle group (a, b, c). d) There exists an open neighborhood U of X in the complex manifold M g (C) such that # Aut(X) > # Aut(Y ) for all Y U \ {X}.

10 10 PETE L. CLARK In order of Wolfart s work, we shall call a curve satisfying the equivalent conditions of Theorem 8 a Belyi-Wolfart curve. Theorem 9. (Properties of Belyi-Wolfart Curves) a) If X is a G-Belyi-Wolfart curve with ramification degrees (a, b, c), then g(x) = 1 + #G 2 (1 1 a 1 b 1 c ). b) There exist at most finitely many Belyi-Wolfart curves of a given genus g or with full automorphism group any given finite group G. c) A Belyi-Wolfart curve can be defined over its field of moduli. d) Suppose Γ is a finite index normal subgroup of (a, b, c), where (a, b, c) is not of one of the following forms, said to be non-maximal: Then (2, b, 2b), (3, b, 3b), (a, a, c). Aut(Γ\H) = (a, b, c)/γ. Remark: By part a), #G as a function of g is maximized when (a, b, c) = (2, 3, 7). Combining with Theorem 8 we recover the Hurwitz bound, # Aut(X) 84(g(X) 1). A curve attaining the Hurwitz bound is called a Hurwitz curve, and the automorphism group of a Hurwitz curve is called a Hurwitz group. The problem of determining all Hurwitz groups is purely algebraic: we are asking which finite groups are generated by an element x of order 2 and an element y of order 3 such that xy has order 7. Here we shall be concerned with the following questions on a Belyi-Wolfart curve X with Aut(X) = G. First, we wish to compute its field of moduli F w, which is the unique minimal field of definition for X. Second, we wish to compute the field of definition of G: that is, Aut(X/Q) is naturally a g = Gal Q/Fw -module, so cuts out a Galois extension K/F w, the unique minimal field such that the automorphisms are defined over K. This is also the minimal field K such that the corresponding extension of function fields K(X) K(X/G) = K(P 1 ) = K(t) is a Galois extension, with Galois group G. Thus there is an application to the regular Inverse Galois Problem: applying Hilbert s irreducibility theorem, we get that the finite group G occurs as a Galois group over K. Thus one is interested in conditions to ensure that K is as small as possible: ideally, such that K = Q, or more realistically, such that K is an abelian number field. Even this latter condition certainly is not satisfied for every Belyi-Wolfart curve. In the Appendix we give a simple argument showing that for any number field L, there exists a Belyi-Wolfart curve X such that the minimal field of definition of (X, Aut(X)) contains L. The construction, however, tells us absolutely nothing about G = Aut(X). Here we study the special case of G = P SL 2 (F q ) (actually, we restrict attention to the case of q = p a for a odd in order to better analyze a particular subclass of curves, but this restriction is not necessary for the present paragraph) and get results which seem to indicate that the case of P SL 2 (F q ) is especially well behaved.

11 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS The proof of Theorem P SL 2 (F q ): elementary results. In this section we recall the basic facts on conjugacy classes and automorphisms of the groups P SL 2 (F q ), which will be needed in our subsequent analysis. As is traditional, we represent elements of P SL 2 (F q ) by matrices in SL 2 (F q ) and must keep in mind the twofold ambiguity arising from the map SL 2 (F q ) P SL 2 (F q ) = SL 2 (F q )/(±1) (note that there is no such ambiguity when q is even: ±1 = 1). First we recall the classification of conjugacy classes, first in SL 2 (F q ) and then in P SL 2 (F q ). Let g SL 2 (F q ), with characteristic polynomial X 2 tr(g)x + 1. Suppose first that g 1 is semisimple, i.e., its minimal polynomial has distinct roots, so accordingly here coincides with the characteristic polynomial. (Equivalently, g can be diagonalized over F q, although not necessarily over F q. This occurs if and only if tr(g) ±2. Then the rational canonical form of g is [ tr(g) Notice that R(g) has minimal polynomial X 2 + tr(g) + 1, so to any α k \ 2 we associate a unique conjugacy class C(α) in SL 2 (F q ) of matrices of trace α, such that C(α), C( α) become identified in P SL 2 (F q ). Any element g C(α) is diagonalizable over F q if and only if its eigenvalues λ, λ 1 are in F q. Otherwise g is diagonalizable over F q 2. It follows that the order of any semisimple element of SL 2 (F q ) divides q 2 1. This bound can be improved: suppose g SL 2 (F q ) has order N prime to p. Then tr(g) = ζ N + ζ 1 N F q, and conversely if F q contains ζ N + ζ 1 N we will get an element of order N. That is, the inertial degree of F N = Q(ζ N + ζ 1 N ) at p must divide a = log p(q), i.e., a must be a multiple of the order of p in (Z/NZ) /(±1). It follows (by looking at some congruences) that the maximal order of a semisimple element in SL 2 (F q ) is q + 1. When q is even, SL 2 (F q ) = P SL 2 (F q ), and when q is odd, quotienting out by 1 halves the order of any even order conjugacy class, and we get that the maximal order of a semisimple element in P SL 2 (F q ) is q+1 2. ]. Consider now the conjugacy classes of nontrivial elements g of SL 2 (F q ) of trace ±2; by multiplying by ±1, we may look only at the trace 2 case, in which we are getting unipotent matrices, every one of which is conjugate to a matrix [ ] 1 u U(u) =, 0 1 for some u F q. (By convention and with apologies to Lincoln 7, we do not call the identity matrix unipotent.) Moreover, one can check that U(u) is conjugate to U(v) if and only if u = v (F q 2 ). That is, if q is odd there are exactly two conjugacy classes of nontrivial unipotent elements, and if q is even there is exactly one. These elements always have order p, and hence still have order p after descent to P SL 2 (F q ). Note then that in P SL 2 (F p ) there is an element of order N if and only 7 Q: How many legs does a dog have if you count its tail as a leg? A: Four. Calling its tail a leg doesn t make it one. Abraham Lincoln.

12 12 PETE L. CLARK if N = p or N is prime to p. Finally, we will need to know the outer automorphism group of P SL 2 (F q ) recall that the outer automorphism group of any group G is the quotient of the full automorphism group by the subgroup of inner automorphisms G/Z(G). One way to get an automorphism of P SL 2 (F q ) is to restrict an inner automorphism of P GL 2 (F q ) to P SL 2 (F q ) via the normal embedding P SL 2 (F q ) P GL 2 (F q ). Since the quotient is isomorphic to F q /F 2 q (induced by the determinant map), the subgroup has index 2 if q is odd and index 1 if q is even, so we get at most one nontrivial outer automorphism this way and only then if q is odd. Conversely, if q is odd, conjugation by the nontrivial coset induces a nontrivial outer automorphism of order two on P SL 2 (F q ), denoted τ. On the other hand we can produce outer automorphisms using the Galois theory of F p a/f p : i.e., the Frobenius f p : x x p acts entrywise on SL 2 (F p a) has order a, and descends to the projectivization with the same order. And we re done: indeed Out(P SL 2 (F p a)) = τ f p, e.g. [Suzuki]. In particular it is abelian, and of order 2a if q is odd and a if p is even P SL 2 (F q ): Macbeath s theory. We summarize here some of the results of [Macbeath 1969], which lead to a complete classification of subgroups of P SL 2 (F q ) generated by two elements. Consider triples of elements (g 1, g 2, g 3 ) SL 2 (F q ) satisfying g 1 g 2 g 3 = 1. Let (a, b, c) be, respectively, the orders of g 1, g 2 and g 3. Note that, contrary to elsewhere in this paper, we do not make any assumption on the subgroup generated by g 1, g 2 and g 3 : indeed the point is to find conditions to ensure that the projective image of g 1, g 2, g 3 is all of P SL 2 (F q ). We define tr(g 1, g 2, g 3 ) := (tr(g 1 ), tr(g 2 ), tr(g 3 )) F 3 q. (Notice that this is very close to just recording the conjugacy classes of g 1, g 2 and g 3.) Also, if (α, β, γ) F 3 q, put T (α, β, γ) to the set of all triples (g 1, g 2, g 3 ) such that tr(g 1, g 2, g 3 ) = (α, β, γ). We say that an F q -triple (α, β, γ) is commutative if there exists some triple (g 1, g 2, g 3 ) T (α, β, γ) which is commutative: such that g 1 g 2 = g 2 g 1. Note that this is only possible if the order of g 3 divides the lcm of the orders of g 1 and g 2, so in particular no F q -triple consisting of elements of orders 2, 3, N with N 7 (or N = 5) can be commutative. We say a triple is exceptional if its sequence (a, b, c) of orders is in the following list: (2) (2, 2, n), (2, 3, 3), (3, 3, 3), (3, 4, 4), (2, 3, 4), (2, 5, 5), (5, 5, 5), (3, 3, 5), (3, 5, 5), (2, 3, 5) The exceptional triples are precisely the orders of triples of elements in the finite spherical triangle groups. No (2, 3, N) for N 7 is exceptional. Finally, let k F q be a subfield. Then clearly P SL 2 (k) is a subgroup of P SL 2 (F q ). This is called a regular projective subgroup. If moreover the unique quadratic extension k 2 of k is contained in F q, then P GL 2 (k) is also a subgroup of P SL 2 (F q )

13 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS 13 since we can rescale by the squareroot of the determinant an irregular projective subgroup. Suppose (α, β, γ) is an F q -triple with F q = F p (α, β, γ). If (g 1, g 2, g 3 ) T (α, β, γ) generate an irregular projective subgroup, then that subgroup is P GL 2 (F q). Especially, if q is an odd power of p, then irregular subgroups do not exist. (This is indeed our reason for restricting to odd a.) Here now is Macbeath s classification of subgroups of P SL 2 (F q ) generated by two elements. Theorem 10. (Macbeath) a) Every triple (g 1, g 2, g 3 ) SL 2 (F q ) is either exceptional, commutative or generates a projective subgroup of P SL 2 (F q ). In particular, if q = p a with a odd, then any triple of elements with orders (2, 3, N) with N 7 generates a subgroup P SL q (k) of P SL 2 (F q ). b) For any k-triple (α, β, γ), the set T (α, β, γ) of triples of elements of SL 2 (F q ) with traces α, β, γ is nonempty. c) P SL 2 (F q ) acts on noncommutative triples by simultaneous conjugacy and preserves each T (α, β, γ) (clearly). The number of orbits of this action on T (α, β, γ) is 1 if p = 2 and 2 if p > 2. In all cases, the inner automorphism group of SL 2 (F p ) acts transitively on T (α, β, γ), and it follows that in the generating case, Aut(P SL 2 (F q )) acts transitively on PT (α, β, γ). For the proof see [Macbeath 1969, Theorem 5]. We shall explain the it follows that in part c), since this is a key point for us that is not stated explicitly in his paper. Indeed, if g and h are two triples with common trace triple (α, β, γ) and g (hence also h) generates P SL 2 (F q ), then the matrix m P SL 2 (F p ) which conjugates g to h carries a generating set for P SL 2 (F q ) into a generating set for P SL 2 (F q ), hence m must stabilize P SL 2 (F q ) and thus induces an automorphism of P SL 2 (F q ). Here are some immedidate consequences of this remarkable theorem: applying parts b) and c) together, we can just choose three elements (α, β, γ) F 3 q, and there exists a triple (g 1, g 2, g 3 ) SL 2 (F q ) with the given elements as traces. Moreover, since Aut(P SL 2 (F q )) acts transitively on the projectivized triples PT (α, β, γ), any two such triples generate isomorphic subgroups which are, except in very special degenerate cases, isomorphic either to P GL 2 (F q ) or to P SL 2(F q ) for some F q F q. In particular, if log p (q) is odd, the first possibility does not occur. Suppose we have (g 1, g 2, g 3 ) T (α, β, γ) such that g 1, g 2, g 3 = P SL 2 (F q ) P SL 2 (F q ). Let k := F p (α, β, γ) be the subfield of F q generated by the traces of the elements of the triple. The theorem implies that the triples, unique up to automorphisms, could also be constructed in k, and it follows that F q k. (This is really not obvious: each element g i can certainly be conjugated into a matrix with k-rational entries, but it was not clear that all three elements could be simultaneously conjugated into k, which is what has just been shown.) The converse statement k F q is obvious, so we conclude that the projective group generated by any (nonexceptional, noncommutative, regular) (g 1, g 2, g 3 ) T (α, β, γ) is P SL 2 (F p (α, β, γ)). Since P SL 2 (F p ) always has elements of orders 2 and 3 (use either the above description of conjugacy classes or even the fact that 6 (p2 1)(p 2 p) = #P SL 2 2(F gcd(2,p) (p 1) p )) the field generated is the trace field of the element of order N, which recall, exists

14 14 PETE L. CLARK in P SL 2 (F p ) when N = p or gcd(n, p) = 1. The following is an easy consequence of Macbeath s theorem. Proposition 11. Fix N and p as above, and assume that a = a(n, p) is odd [e.g. this holds for all p when N 1(4) is prime]. Then q = p a is the unique power of p for which there exists a nondegenerate epimorphism ϕ : (2, 3, N) P SL 2 (F q ). Proof: It follows immediately that there is at most one power p a of p such that a triple of elements of orders (2, 3, N) can generate P SL 2 (F p a). If N = p then the element of order N is unipotent of order p, so its trace, 2, generates F p. If p = 2 then P SL 2 (F q ) = SL 2 (F q ) and we want the field generated by the trace of an element of order N in SL 2 (F q ), i.e., F p (ζ N + ζ 1 N ). If p > 2 is prime to N, then our element of order N comes from an element of order 2N in SL 2 (F p ), so has trace ζ 2N + ζ 1 2N.8 This completes the proof. Remark: I do not claim that it is necessary to consider only odd-degree field extensions of F p to get groups P SL 2 (F q ), only that it is sufficient. Indeed, Macbeath remarks that by taking a triple of elements with orders (2, 3, q + 1) in F q (such an element is shown to exist in Section 3.1), then because GL 2 (F q) does not have an element of order q + 1, it must be that the associated triples generate projective subgroups isomorphic to P SL 2 (F q ). In this way Macbeath showed that P SL 2 (F q ) is generated by elements of order 2 and 3 for every q 13. The same result can be shown for the smaller values of q, except q = 9, in which case the result is false. Rather than try to write down more complicated necessary and sufficient conditions for the triple to generate P SL 2 (k) rather than P GL 2 (k), we will be content to look at odd-degree extensions of the prime subfield The Weak Rigidity - Weak Rationality Lemma. Let now G be a finite group with trivial center, and let g = (g 1,..., g n ) be a tuple of elements in G: from now on this means that g 1 g n = 1 and G = g 1,..., g n. To g we associate the tuple of conjugacy classes C = (C 1,..., C n ). C is rigid if G = Inn(G) acts simply transitively on the set of tuples with conjugacy class tuple C. In fact the simple is redundant here, under the requirement that G has trivial center, since any inner automorphism which fixes each g i in a tuple corresponds to an element h of g which commutes with every g i, hence is central in G since the g i form a generating set, hence is trivial. C is weakly rigid if Aut(G) acts transitively on the set of tuples of C. Also, for any n, g Q, the absolute Galois group of Q, acts on the set of n-tuples of conjugacy classes in a finite group G: if G has order N, then the action by definition factors through G Q(ζN )/Q = (Z/NZ) and then C σ := (C σ 1,..., C σ n) is just 8 When N is odd, the passage to 2N is not necessary: we could also find an element of order N in SL 2 whose associated cyclic group maps injectively into the projectivization. On the other hand it is harmless, since when N is odd, (Z/2NZ) = (Z/NZ). When N is even one must indeed use ζ 2N instead of ζ N (unless p = 2).

15 ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS 15 interpreted as raising each conjugacy class to the appropriate power modulo N. 9 This allows us to define the field of rationality F r (C) of C as the field extension cut out by this g Q -action: note that F r Q(ζ N ). Similarly we have the field of weak rationality F w (C), which is the subfield of Q(ζ N ) fixed by the subgroup of (Z/NZ) consisting of all exponents i modulo N with the property that there exists an automorphism ϕ of G with C i = ϕ(c) (note that the automorphism ϕ can depend on the exponent i but must be uniform for all n conjugacy classes in C). Clearly we have F w F r. Now all of our results about fields of definition and fields of moduli for curves X(N, p) and for their automorphism groups will use the following lemma. Lemma 12. ( WRWR Lemma ) Let G be a finite group of order N with trivial center, let g = (g 1,..., g n ) be a (generating!) tuple with associated conjugacy class tuple C = (C 1,..., C n ). Assume that C is weakly rigid. Then: a) There exists a unique pair (X, β) with X = X(G, C)/Q and subgroup G Aut(X) such that β : X(G, C) X/G = P 1 is a branched covering with ramification type C. b) X(G, C) can be defined over its field of moduli, which is the field of weak rationality F w. c) There is a canonical bijection between the g Q -orbits {X σ } and {C σ }, the latter being merely the set of C i as i ranges over elements of (Z/NZ). d) There is a unique minimal field of definition K for the group of automorphisms G, and we have: F r K a natural embedding ρ : G K/Fr Stab Out(G) (C), where the latter term is the subgroup of outer automorphisms of G which stabilize each of the conjugacy classes in C. In particular, K = F r if G is rigid, and in general we have [K : F r ] #Out(G). e) Finally, X(G, C) admits a model over F r which is canonical in the sense of being uniquely determined by the condition that the automorphisms of G are F r -rational. 9 It is reasonable to ask what right gq really has to act on the conjugacy classes of our poor finite group G. This has a good answer: let X = P 1 Q \D be the projectiv line over Q minus any Q-rational divisor D of degree n (the nth roots of unity are a popular choice, although {, 0, 1,..., n 2} would work just as well). Then the étale fundamental group of X fits in a short exact sequence 1 π 1 (X) π 1 (X) g Q 1, with the first term being the profinite completion of the geometric fundamental group, i.e., the profinite completion of the discrete group with the canonical presentation g 1,..., g n g 1 g n = 1. So a tuple g in G corresponds to an epimorphism π 1 (X) G, whence, by the Riemann existence theorem, an algebraic curve X covering the projective line and branched only over the points of D. Now the short exact sequence gives rise to a continuous homomorphism ρ : g Q Out(π 1 (X)) i.e., there is a compatible inverse system of actions of g Q on each finite quotient, so in particular we get ρ : g Q Out(G). To try to say exactly what ρ is (for each G compatibly) is an entire field of mathematics see e.g. [Deligne 1989] but the first step is to determine the induced action on the conjugacy classes of G, and this action is indeed by the cyclotomic character. Note by the way, that the geometric meaning of weak rigidity is that the resulting branched covering depends, up to isomorphism, only on C and not on the choice of tuple g.

16 16 PETE L. CLARK This lemma follows almost immediately from results found in [Volklein], although it is not stated therein explicitly. One needs to combine Remark 3.9, parts a) - d) with Proposition 9.2b). It would be nice to have a complete self-contained proof of the WRWR Lemma in which all terms are expressed both group-theoretically and geometrically. Probably I will write this up some day, but not today. Important Remark: In view of our intended application to the (2, 3, N)-case, we have in our account of the definitions of weakly rigid and weakly rational n- tuple made a simplifying assumption: we have implicitly assumed that the branch divisor in P 1 consists of n points P 1,..., P n with each P i P 1 (Q). But in general for a branched covering X P 1 /Q we can say only that the ramification divisor D is defined over Q as a divisor. To such a choice of D = P P n Div n (P 1 )(Q) and an n-tuple C = (C 1,..., C n ) we assign the ramification type (G, C, D), and include the g Q -action on the points of D in the definition of (weak) rationality: e.g. the field of rationality of the type is the subfield of Q(ζ N ) fixed by elements σ G Q(ζN )/Q such that for all 1 j n, C σ P j = C σ(pj), with an analogous definition for weak rationality. But observe that the g Q -action can only permute ramification points with equal ramification indices, so when the ramification indices are pairwise distinct e.g. (2, 3, N) for N 7 the ramification divisor is forced to be pointwise Q-rational. In practice, this means that one will usually get more economical realizations of Galois groups by using tuples with at least some coincident orders. In particular, Galois realizations of nonexceptional groups P SL 2 (F q ) over Q using triples with distinct orders (a, b, c) do not exist (as can be shown easily using the results of Section 3.2 and Lemma 5). On the other hand, since a favorable choice of the divisor D can only cut down the degree of the field of rationality of the type by a factor of n!, one can deduce an absolute bound on the exponent a such that any group P SL 2 (F p a) can be the Galois group of a covering X P 1 /Q ramified at three branch points. Indeed, Macbeath s theorem shows that the traces of the triple must generate F p a, and therefore the Galois orbit of C = (C 1, C 2, C 3 ) must have size at least a (with inequality occuring only if there are unipotent conjugacy classes). If the orders of two of the conjugacy classes coincide, then perhaps we can choose the divisor D such that the field of rationality of the type has degree a/2. If all three orders coincide, then the degree of the field of rationality is still at least a/6. In fact, since the g Q action on conjugacy classes factors through Q ab, it is clearly impossible for g Q to permute the three conjugacy classes via S 3, and the degree of the field of rationality is at least a/3. It follows that only P SL 2 (F p ), P SL 2 (F p 2) or P SL 2 (F p 3) can be Galois groups of Belyi-Wolfart curves with all automorphisms defined over Q. The first two cases occur for sets of primes p of positive density: e.g. P SL 2 (F p ) can be realized if any of 2, 3 or 5 is a quadratic nonresidue mod p, and P SL 2 (F p 2) can be realized if 5 is a quadratic nonresidue mod p. For exponents a 3, exactly one group P SL 2 (F p a) is known to occur as a Galois group over Q (by any means): p a = 2 3 = 8, via a rigid (9, 9, 9) triple [Malle-Matzat] Semirigidity in P SL 2 (F p a). Let g be any generating triple in P SL 2 (F q ) which is not exceptional, i.e., the triple of orders of the elements (a, b, c) is not one