this paper we present a new and elementary approach for proving the main results of Katz (1996) using the Jordan-Pochhammer matrices of Takano and Bannai (1976) and Haraoka (1994). We find an explicit version of the middle convolution of Katz (1996) that connects certain tuples of matrices in linear groups. Our approach is valid for fields of any characteristic and it can be shown that this operation on tuples commutes with the braid group action. This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q: This approach is then applied to realize numerous series of orthogonal and symplectic groups regularly as Galois groups over Q: In the appendix we treat an additive version of the convolution. 1. Introduction

D. Fried Abstract. Each finite p-perfect group G (p a prime) has a universal central p-extension coming from the p part of its Schur multiplier. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups (p = 2) aimed at geometric covering space problems that included their regular realization for the Inverse Galois Problem. A special case of a general result is that any finite simple group with a nontrivial p part to its Schur multiplier has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime p. Sequences of moduli spaces of curves attached to G and p, called Modular Towers, capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of Harbater-Mumford cover representatives. These are modular curve tower generalizations. So, they inspire conjectures akin to Serre’s open image theorem, including that at suitably high levels we expect no rational points.

We show that the absolute Galois group of a countable Hilbertian P(seudo)-A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G ( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., ∏ ∞ n=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RG-Hilbertian property (Theorem B).

The paper [FGS] uses the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). We consider only separable polynomials; their derivative is nonzero. Then, f # Fq [x] is exceprional if it acts as a permutation map on infinitely many finite extensions of the finite field Fq , q = p a for some prime p. Carlitz's conjecture says f must be of odd degree (if p is odd). The main theorem of [FGS; Theorem 14.1] restricts the list of possible geometric monodromy groups of exceptional indecomposable polynomials (§1.1): either p = 2 or 3 or these must be affine groups. The proof of Carlitz's conjecture motivates considering general exceptional covers of nonsingular projective algebraic curves. For historical reasons we sometimes call these Schur covers [Fr2]. Suppose # : X # P 1 is an exceptional cover over Fq . Then, for some integer s, there is au/I--L x x x # X(F q t ) over each z # P 1 (F q t ) foreac h integer t with (t, s) = 1. In particular /5/ |X(F q t )| = q t +1 when (t, s) = 1. We include a complete proof that exceptionality is equivalent to a statement about the geometric/arithmetic monodromy pair of the cover. Theorem 2.5 shows all geometric/arithmetic monodromy pairs satisfying necessary conditions (§1.1-§1.2) derive from covers over Fp for all suitably large primes p. Other topics: (i) How modular curve points over finite fields explicitly produce rational function exceptional covers of prime degree (Corollary 3.5). (ii) How fiber products produce abundant general exceptional covers (Lemma 3.7). (iii) How Müller-Chen-Matthews produced exceptional polynomials with nonsolvable monodromy group ($1.7). (iv) How general exceptional covers realize curves of high genus over Fq with q small and |X(F q t # )| large for...

The context for this paper is the Inverse Galois Problem. First we give an if and only if condition that a finite group is the group of a Galois regular extension of R(X) with only real branch points. It is that the group is generated by elements of order 2 (Theorem 1.1 (a)). We use previous work on the action of the complex conjugation on covers of P 1 [FrD]. We also use Fried and Völklein [FrV] and Pop [P] to show each finite group is the Galois group of a Galois regular extension of Q tr (X). Here Q tr is the field of all totally real algebraic numbers (Theorem 5.7). §1, §2 and §3 discuss consequences, generalizations and related questions. The second part of the paper, §4 and §5, concerns descent of fields of definition from R to Q. Use of Hurwitz families reduces the problem to finding Q-rational point on a special algebraic curve. Our first application considers realizing the symmetric group Sm as the group of a Galois extension of Q(X), regular over Q, satisfying two further conditions. These are that the extension has four branch points, and it also has some totally real residue class field specializations. Such extensions exist for m = 4, 5, 6, 7, 10 (Theorem 4.11). Suppose that m is a prime larger than 7. Theorem 5.1 shows that the dihedral group

We prove that for almost all oeoe oe 2 G(Q) the field oe) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating . We then say that oe) is PAC over Z.

Let G be a finite (possibly simple) group, and let p be a prime dividing the order of G. The characteristic finite quotients k p ˜ G of the universal p-Frattini cover of G are strikingly similar groups. It takes an effort to distinguish them for finding Q regular realizations for the Inverse Galois problem. This paper starts a program to show one can’t realize all these groups as Galois groups of extensions L/Q(x) with at most r (fixed) branch points. Let C be an r-tuple of p-regular conjugacy classes of G. To compare realizations of these groups we use a sequence of varieties—a Modular Tower—attached to (G, p,C). The notation for this sequence is H ( k p ˜ G,C), k =0, 1,...: H ( k p ˜ G,C) isthekth level of the Modular Tower. Crucial properties of level k translate to properties of the characteristic modular representation of k p ˜ G. Properties of these representations support the following statement. Conjecture. For each r there exists kr so that for k>kr, Q regular realization of k p ˜ G requires more than r branch points.

Abstract. We study “pure-cycle ” Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the higher-genus case, requiring at least 3g simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of Conway-Fried-Parker-Völklein. 1.

this paper we shall use either quartic coverings of the projective line or quadratic coverings of elliptic curves with enough ramification points (instead of quadratic coverings of the projective line as in the class field tower constructions) to find for every genus g 3 curves with infinite geometric fundamental group defined over Q(i) or over F q (i) (where i is a forth root of unity) and we find even parametric families of such curves over every ground field containing i (cf. Theorem 5.22).

Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=0 2. We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the j-line. The latest Nielsen class techniques apply to these dessins d’enfant to see their subtle arithmetic and interesting cusps. John Thompson earlier considered another genus 0 problem: To find θfunctions uniformizing certain genus 0 (near) modular curves. We call this Problem g=0 1. We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=0