Seminaire de Theorie des Nombres , Paris 1987 - 88

The method of choice nowadays for achieving a group G as a Galois group of a regular extension of ((x) goes under the heading of rigidity. It works essentially, only, to produce Galois extensions of Q(x) ramified over 3 points. The three rigidity conditions ((0.1) below) imply that G is generated in a very special way by two elements. Generalization of rigidity that considers extensions with any number r of branch points has been around even longer than rigidity { 5.1). Of the three conditions, the generalization of the transitivity condition, 0.1 c), requires only the addition of an action of the Hurwitz monodromy group H (a quotient of the Artin braid group). But it also adds a 4th condition that in many situations amounts to asking for a Q-point on the Hurwitz space associated the data for the generators of G. Theorem 1 below -our main theoremis that in the case r = 4 this is equivalent to finding a Q-point on a curve derived from a quotient of the upper half plane by a subgroup of PSLy(l). Although the description of this curve is quite explicit, there is one big problem : while it is sometimes a modular curve (5 4), more often it is not. For this exposition we apply the theory to a simple example that illustrates the main points that arise in the arithmetic of 4 branch point covers (5 5.2 and 5.3). The group is just A. in this case, but this allows us to compare the generalizations of m d @ with the historical progenitor of this, Hilbert's method for realizing alternating groups as Galois groups ( 5 5.3). Description of the main results. The theory of the arithmetic of covers of the sphere arises in many diophantine investigations. The most well known, of course, is a version of the inverse problem of Galois theory : does every finite group G arise as the group of a Galois extension L/Wx) with fl n L = ( (i.e. L/ x ) is a regular extension) ? For this lecture we use the dual theory of finite covers rf : X+ IP1 of projective nonsingular curves. We shall consistently assume in this notation that P1 is identified with C U co = IP; , a copy of the complex plane uniformized by x, together with a point at CO. Such a rover corresponding to the field extension L/Q(x) would have the property that it is defined over (( (5 1.1) and the induced map on the function field level recovers the field extension L/Q(x). It is valuable, as we shall see in the key example of the paper, to consider covers rf : X + Pi that may not be Galois. Branch points and monodromy groups : Denote the degree of such an extension by n. The branch points of the cover are the values of x for which the cardinality of the fiber p { x ) is inferior to n. We will consistently denote the branch points of the cover by xl, ..., x (almost always assuming that each is a genuine branch point). The key parameter in all investigations is r. From Riemann's existence theorem, degree n extensions L of C(x) ramified over r places x,,..., x are in one-one correspondence up to a natural equivalence with the degree n equivalence classes of connected covers of IPa{xl, ..., zr). These are in turn in one-one correspondence with equivalence classes of transitive permutation representations T : r , Ñ 5% on the set {1,2, ..., n} where T, denotes the fundamental group of W X ~ , . . . ~ ~ ~ ) . The Galois group of the normal closure of the extension L/t(x) is identified with the monodromy group of the cover, the group G = T(ri). Rigidity when r = 3 : Excluding solvable groups, most of the success in achieving groups as Galois groups has come through the arithmetic theory of covers in the case r = 3. The apparatus that reduces this to a computation related to a description of the cover through Riemann's existence theorem has been named rigidgig following Thompson's usage in [TI (the name has the unfortunate aspect of potential confusion with the concept of rigidifying data, a tool that does appear in the proofs of results that generalize rigidity. We do only an exposition on rigidftg so no problems are likely to occur). The rigiditg test starts with an r-tuple C = {C,, ..., CJ of conjugacy classes of G (3 1.2). Our version (3 5.1) includes the faithful permutation representation T : G Ñ S of the monodromy group of the cover as part of the data of the statement. For the moment we use a stronger set of conditions on (C, T ) than. is necessary it is still more general than used by most practitioners in order to simplify our exposition on the distinction between r = 3 and r > 3. A little more notation will help to keep the key statements relatively memorable. Denote the normalizer of G in Sn by Ns (G). We denote the subgroup n of this that maps {C,, ..., C } into itself (by conjugation) by % (C). The n group generated by the entries of vl, ..., or is G(g). Recall that a conjugacy class of a group is said to be rational if it is closed under putting elements to powers relatively prime to the orders of elements in the class. If the following hold, then G is the Galois group of a regular extension of ((x) ramified at any r points xl, ..., x ? Q : b) each of the classes C,, ..., C is rational; and c) G acts transitively by conjugation on the following set of r-tuples { ( T .., TJI G(r) = G, r i6 C, and T, ... T = 1). Condition (0.1 a) is not necessary, but weakening it is no triviality. Dealing with some version of (0.1 a) (as [Fr, 21 illustrates using the theory of complex multiplication) is a necessity. There have been successful attempts to finesse around consideration of (0.1 a) for special cases in Hilbert's original paper [Hi] and in Shih's use of modular curves to realize PSL,(Z/p) as a Galois group wha7, 3 or 7 is a quadratic nonresidue modulo p [Sh]. Our example in direct approach to weakening (0.1. a). v t s of xl ,..., x are to be in Q, then (0.1. b) is necessary. Our r n , baternent in 3 5.1 (Prop. 5.2 and Prop. 5.4) relaxes the condition on the branch points being in Q, but the replacement condition will now be an absolute necessity. Finally, no one yet has shed m y serious light on relaxation of (0.1 c). What is surprising is how very often the conditions are satisfied in the case that r = 3 (e.g., Belyi [Be], Feit [F] among others, many papers of Malle and Matzat some of which are included in [Ma,l], and Thompson [TI). They are almost never satisfied when r exceeds 3 (e.g. no example has been found when G is a noncyclic simple group). Suppose that even every finite simple group is generated by three elements ri , T~ , T~ that give conjugaty classes that satisfy the necessary condition analogous to (0.1 b) and the mysterious condition (0.1 c). If all that were of concern were the inverse Galois theory problem, then it might make sense to concentrate all research efforts on relaxation of condition (0.1 a). The hypotheses, however, of these statements don't hold : generators of groups with such handy properties don't always exist; and few of the other applications allow the investigator to be so picky about the choice of generators (as in [DFr] and [FR,2 and 31, we are referring to applications to Hilbert's irreducibility theorem and Siegel's theorem, ranks of elliptic curves and values of rational functions over finite fields). Generalizations of rigidity for r > 3 : Fortunately there are generalizations of nffldity that hold quite frequently for r > 4 ([Fr,l : Theorem 5.1.1 and [Fr,3 : Theorem 1.51). Matzat has used versions of these [Ma,l] to realize several simple groups as Galois groups (among them the Matthew group of degree 24 [Ma,2]). Increasing r improves the possibility of satisfying all three of the conditions (0.1), as explained in [Fr,2; Remark 2.21. But there are two serious points. First : the generalization of (0.1 c) (condition 5.5 c)) works by asking for transitivity of a group that contains the Hum& monodromy group H of degree r (a quotient of the Artin braid group; (3 3.1). The calculations for this applied to one of the classical sequences of simple groups can be quite formidable (e.g., Ex. 2.3 of [Fr,3] to realize all of the A' s as Galois groups of 4 branch point covers of Pi). For any one group, Matzat, for example, has put together a computer program to test this transitivity, but experience with the calculations is still more of an art than a science. A later paper will consider the series of groups PSLy(U/p), p z  1 mod 24, and 7 a quadratic nonresidue modulo p. For the other primes this is Shih's result [Sh]. While the calculations aren't quite complete, it doesn't seem that it is possible to achieve the groups of this series with covers of fewer than 4 branch points. And for each of these primes there does exist ( C , q with r = 4 satisfying the analog of the 3 conditions of (0.1) (conditions 5.5. a*). Why this doesn't quite finish the job of realizing these groups as Galois groups comes from our second point. The analog of (0.1) includes a condition d) which we now explain. Parametrization of the covers associated to (C, T ) : The collection of equivalence classes of covers associated to (C, T) is naturally parametrized by the associated Hurwitz space &(C) (with T understood from the context). This arises as a cover of f ^ D coming from a representation of the Hurwitz monodromy group ( 3.2). Here D is the classical discriminant locus in the respective spaces. We note this existence of the Noether cover (PI) IPr, Galois with group 5, . When the analogs of the conditions 0.1) hold, (C) (with its maps to Pr) is defined over IQ. The extra condition d) for r > 3 demands that there be a Q point on a connected component of the pullback of &(C) to (P1)'. Below we refer to this space as 8 ( C ) ' . If all of the conditions (0.1) hold (with the Hurwitz monodromy action added to (0.1 c)), then condition d) is necessary (and sufficient) when the conjugacy classes in C are distinct. The problem with this is clear : the space &(C) is a production of such great abstraction that the diophantine reduction seems impossible to effect. The main result of this paper is an alternative description of (5.5 d) in the case that r = 4. THEOREM 1 (special case o f Conclusion 4.2). There is a curve cover $yj : Yc, Ñ Pi ramifiedovervist 0,1,os, suchthat &(C)' hasa is nonempty. Furthermore, Y& is identified with the projective normalization of a quotient of the upper half plane by a subgroup flc of PsL2(1) (of finite index), in such a way that it identifies the covered copy of IP1 with the classical Mine w i n a l l y , there is an explicit description of the branch cycles of the cover $6 given by an action of the Hurun.tz monodromy group H4 . There is an analogous curve cover ij)c Ñ IP' in which B" is identified with the classical J-line. Conclusion 4.2 is more general than Theorem 1 in that the former uses this cover as a replacement for that with Yb . This gives a necessary statement replacing condition (5.5 d) even when the 4 conjugacy classes of C are not distinct (when they are distinct, Yc = Yb). Congruence and noncongruence subgroups : A part of the proof of Conclusion 4.2 consists of showing that special values of C, Yc can be identified with the classical curve Yo(%) that arises from the quotient of the upper half plane by the subgroup called r0(n). Thus modular curves arise. But in general the curves Yc belong to noncongruence subgroups of PSLfl). Indeed, recently Diaz, Donagi and Harbater [DDH] have actually shown that every curve defined over the algebraic closure of () occurs as Yb for some choice of C . Their choice, however, of C has nothing to do with the classical modular curve arithmetic. An example where G = Ac appears in [FrT] to show how one might investigate (for the inverse Galois theory problem) the infinitely many totally nonsplit extensions of any given finite simple group. Here we use it for three straight forward reasons : to show in practice the distinction between the curves Yc and Yb ; to consider by example weakenings of condition (0.1 a); and to compare our results with the beginnings of this subject in [Hi]. 1.Basic data for covers. One way to give an (irreducible) algebraic curve is to give a polynomial (irreducible) in two variables f(x,y) 6 ([x,y] where C denotes the complex numbers. Then the curve is This curve, however, may have singular points : points (z0 ,yo) e X for which -r̂ , and &valuated at (xy ,yo) are both 0. Furthermore, we are missing the points at infinity obtained by taking the closure of X in the natural copy of 2 projective 2-space IP2 that contains the &ne space A with variables x and y (and these points, too, might be singular). The x-coordimte projection : After this we assume that our algebraic curves X don't have these defects; they will be projective nonsingular curves, so we may not be able to regard them as given by a single polynomial in 2-space. But the essential ingredient of this presentation, represented by the x-coordinate, will still be there. That is, we have a covering map given by projection of the point (x,y) onto its first coordinate. When the context 1 is clear we will identify I P with P1. We use this extra decoration by coordinate when it clarifies the context. The monodromy group of this cover is defined to be the Galois group G of the Galois closure of the field extension C(X)/t(x) where C(X) denotes the quotient field of the ring C[x, y]/(f (x, y)). In the sequel we will denote this Galois closure by C(X) or by the geometric version X, the smallest 1 Galois cover of I P; that factors through XiPx . Note that in this situation G automatically comes equipped with a transitive permutation representation T : G + S . Denote the stabilizer in G of an integer (say, 1) by G(T). Also, T is primitive (i.e., there are no proper groups between G and G(T)) if and only if there are no proper fields between 1 C(J) and C(x) (equivalently, no proper covers fitting between X Ix). 1.1.Branch points and the classical PS&(?j action : The first parameter for dealing with covers is the number r of branch points of a given cover : the number of distinct points x of P1 for which the fiber of X above x has fewer actual points than the degree of the map. We deal not with one polynomial at a time, but rather with a parametrized family of them. But clearly it is natural to assume that all members of the family have the same number of branch points. The Hurwitz monodromy groups H are the key for putting these covers into families. In 5 2 for r = 3 and in $ 3 for r = 4 we introduce these groups and their basic properties. Although $ 2.1 uses nothing more than the transitive action of /'SLj(C) on distinct triples of points of I P , the notation used here is the main tool for the rest of the paper. In classical algebraic geometry it has become a habit and a tradition to regard the parameter variety 3S for a family of covers with r branch points as 1 the source for a quotient 8/PSL2(C). Consider covers : Xi -+ f f x , i = 1,2, associated to two points m, ,re,? H. The action is the one that equivalences n+ and re, if and only if there exists a E PSL,(C) such that a o 4. = qi2 . In $ 2.3-2.4 we display the arithmetic and geometric subtleties that would make it a disaster to do this even in the case of families of 3 branch point covers. Here are some of the negatives for forming the quotient frivolously : (1.1 a) there are technical difficulties in giving o^/PSLo(C) the structure of an algebraic variety and in visualizing its properties; b) taking the quotient often destroys subtle finite group actions that are valuable for using the parameter space as a moduli space; c) there are few quotable sources on the enriched family of covers structure; and d) forming the quotient often wipes out the possibility of dealing with problems of considerable consequence. Our first 3 branch point example in $ 2.3 should go a long way to make our case for (1.1 d). It is the other points, of course, that cause the lengthy preambles to this subject with so many down to earth applications. 1.2.R.iemannts existence theorem and Nielsen classes : The classical discussion of maps of degree n from curves of genus g to projective l-space gives us data for a natural collection of covers. We call the data a Nielsen class (below), and it is this that we shall regard as being fixed in the consideration of any family of covers. Suppose that we are given a finite set z = { x ~ , ..., xT} of distinct points of ffi. For any element CG ST denote the group generated by its coordinate 1 entries by G(c). Consider 4 : X -+ ff , ramified only over z up to the relation 1 that regards # : X -+ 5' ; and # ' : X -+ IPx as equivalent if there exists a homeomorphism A : X -+ X 8 such that 4 ' o A = #. These equivalence classes are in one-one correspondence with (1.2){<7 = (u,, ...,i~) 6 s"] u, ... u = 1, G(F) is a transitive subgroup of S) modulo the relation that regards 5 and a' as equivalent if there is 76 5% with wyml = o". This correspondence goes under the heading of Riemann's existence theorem [Gro]. The collection of ramified points x will be called the 1 branch points of the cover # : X+ I P . (In most practical situations we shall mean that there truly is ramification over each of the points xi , i = 1, ..., r). E-'s exi8tence theorem for f 4 i a : Riemann's existence theorem generalizes through a combinatorial group situation to consider the covers above, not one at a time, but as topologized collections of families. That is, the branch points s run over the set ( P ' ) ~ \ A ~ with A r the +tupla with two or more coordinates equal. In 5 2 and 5 3, respectively, we will introduce the coordinates for these families in the cases r = 3 and 4. Suppose that T : G Ñ 3% is any faithful transitive permutation representation of a group G. Let C = (C,, ..., C) be an r-tuple of conjugacy classes from G. I t is understood in our next definition that we have fixed the group G before introducing conjugacy classes from it. DEFINITION 1.1. The Nielsen class of C is Ni\C} g f { r e ~ ^ l G { r ) = G a n d t h e r e i s C ~ S w i t h T 6 C i , i = l , ..., r}. ( 4 0 Relative to canonical generators a,,..., SF of the fundamental group 1 rl(iP x, x ), we say that a cover ramified only over x is in Ni(C) if the 0 classical representation of the fundamental group sends the respective canonical generators to an r-tuple u ? Ni(C). 2.Families for r = 3 and the Hurwitz monodromy group & . 2.1.Complete families for r = 3 from transport of structure : I t is clear that the fundamental group of ff3\D3 is of order 12 once it is shown that the fundamental group of (IP1)3\A3 is of order 2. But for any point (2, ,x2 ,x3) = x there is a unique element 0 = f S 6 PSL2(C) that maps (0,1, a) to x : Thus lP3\ A3 is a principal homogeneous space for PSZ2(C). They therefore have the same fundamental groups. As is well known, SL2(C) has trivial fundamental group. Thus the cover SL2(C) Ñ PSZ2(C) displays the representative permutation representation. Below we will use this in the manner of [Fr 1, p. 421. Let # : X -> V\ be any cover with three distinct branch points and order these as (4 ,g ,xS) = 8. Denote (resp., ff 3 \ ~ 3 ) by V (resp., V3). Also, denote the natural map PSZ2(C) -+ ~ut(ff1) by A. Form an irreducible family of covers from this data by transport of structure : where the down map on the far right takes x to f l -0 . The down maps indicate that the usual family notation (i.e. 3 denotes a total space) for the items in the bottom row is given in the top row. That is, with the identification of a' x 1'1 and PSLAC) x ff 1 based on a", 3 is the fiber product in the leftmost square of diagram (2 .1) . For each xl: V the points of 3 over x x P; 1 1 1 give a cover of Px equivalent to the cover hx o F8 o # : X -+ ffx . Let Ni(C) be the Nielsen class and G the monodromy group of 3 f : X + 6 Then V is the space X{C} s, (d. 5 3.2) much of the tine. Indeed, consider the straight absolute Nielsen classes of C The normalizer of G in S % , Ny(G) acts by conjugation on the r-tuples of elements in G. The subset that stabilizes Ni[C) is denoted by N d C ) . Form the quotient of SNi(Cl by the subgroup of Nrf(C) that leaves this set stable to get the absolute straight Nielsen classes, SNi{C) ab . Note that the quotient of ffy by the subgroup stabilizing each element of 5Ni(C) a0 is itself a quotient of S3 (and therefore is of order 1,2, 3 or 6). PROPOSITION 2.1. In the notation of section 2.1 assume that Thus H3 acts on N ~ ( c ) $ through a transitive permutation representation of S3 . Then, as covers of V3 , (%'(C) is isomorphic to ? d 3 (resp., Vi) i f and only if this is the regular represeatation (resp., the trivial representation). 2.2.Most 3 branch point families derive from transport of structure : A version of Proposition 2.1 appears in [BFr,l; 5 41. This analyzes when there exists a total representing family like that of (2.1) in the case when either (2.2) doesn't hold or when the action of H3 isn't through the regular representation of S3 . Below we will use a converse. That is, suppose that is any family of 3 branch point covers with Sf and 1% irreducible nonsingular complex manifolds. We assume that all morphisms are smooth. Also, for each m e 8, restriction of prl o $ to the fiber 3 gives a 3 branch point cover YmE' 1 x ' As above consider the following natural maps : V V3 ; and : 1% -+ U y by me 1% goes to the unordered collection of branch points of the corresponding cover. Any connected component 1%' of the fiber product c%'x V 3 has over it a connected component Y p that gives a family of 3 V. branch point covers. Suppose that m' e 1%', that x' is the image of projection of m' on V 3, and that 3 : = X -+ P: is the corresponding cover. Apply the transport of structure construction to canonically form a family of three branch point covers over V 3 having the fiber X-+V', over zz. Then take a connected component of its pulback to a'. PROPOSITION 2.2. Consider an zmducible family 9' of 3 branch point covers over 8' which has Ym; = X-+ 1'1 as a fiber. Then all covers X-+ that appear in such a family have Xf analytically isomorphic to X. Furthermore all such families are in one-one correspondence with the elements of the set