Results 1  10
of
51
An algorithm of Katz and its application to the inverse Galois problem
 J. Symb. Comput
, 1999
"... this paper we present a new and elementary approach for proving the main results of Katz (1996) using the JordanPochhammer 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 ..."
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Cited by 39 (11 self)
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this paper we present a new and elementary approach for proving the main results of Katz (1996) using the JordanPochhammer 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
Hurwitz monodromy, spin separation and higher levels of a modular tower
 Proc. Sympos. Pure Math
, 2002
"... D. Fried Abstract. Each finite pperfect group G (p a prime) has a universal central pextension coming from the p part of its Schur multiplier. Serre gave a StiefelWhitney class approach to analyzing spin covers of alternating groups (p = 2) aimed at geometric covering space problems that included ..."
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Cited by 39 (15 self)
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D. Fried Abstract. Each finite pperfect group G (p a prime) has a universal central pextension coming from the p part of its Schur multiplier. Serre gave a StiefelWhitney 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 HarbaterMumford 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.
The embedding problem over a Hilbertian PACfield
 Annals of Math
, 1992
"... 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 ..."
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Cited by 27 (15 self)
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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 RGHilbertian property (Theorem B).
Global Construction of General Exceptional Covers  With Motivation For Applications To Encoding
 in: Finite Fields: Theory, Applications, and Algorithms American Mathematical Socity
, 1994
"... 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 ..."
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Cited by 17 (10 self)
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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/IL 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 MüllerChenMatthews 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 locus of curves with prescribed automorphism group
 RIMS Series, Communications in Arithmetic Fundamental Groups and Galois
, 2002
"... Abstract: Let G be a finite group, and g ≥ 2. We study the locus of genus g curves that admit a Gaction of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g = 3 (including equa ..."
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Cited by 17 (5 self)
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Abstract: Let G be a finite group, and g ≥ 2. We study the locus of genus g curves that admit a Gaction of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g = 3 (including equations for the corresponding curves), and for g ≤ 10 we classify those loci corresponding to “large ” G. 1
Pseudo Algebraically Closed Fields Over Rings
 ISRAEL JOURNAL OF MATHEMATICS
, 1994
"... 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. ..."
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Cited by 16 (8 self)
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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.
The Irreducibility of Certain Purecycle Hurwitz Spaces
, 2006
"... Abstract. We study “purecycle ” Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irredu ..."
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Cited by 15 (4 self)
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Abstract. We study “purecycle ” Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus0 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 highergenus case, requiring at least 3g simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of ConwayFriedParkerVölklein. 1.
Applying modular towers to the inverse Galois problem
 Geometric Galois Actions II Dessins d’Enfants, Mapping Class Groups and Moduli
, 1997
"... 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 pFrattini cover of G are strikingly similar groups. It takes an effort to distinguish them for finding Q regular realizations for the Inverse Galois ..."
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Cited by 13 (6 self)
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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 pFrattini 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 rtuple of pregular 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.
Nonrigid constructions in Galois theory
 PAC. JOUR
, 1994
"... 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 o ..."
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Cited by 13 (8 self)
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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 Qrational 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
Relating two genus 0 problems of John Thompson
 IN PROGRESS IN GALOIS THEORY, H. VOELKLEIN AND T. SHASKA EDITORS 2005 SPRINGER SCIENCE
"... 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 0spo ..."
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Cited by 12 (5 self)
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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 0sporadic. Example: Finitely many Chevalley groups are 0sporadic. A proven result: Among polynomial 0sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the jline. 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 jline covers in two cases. First: From the three 0sporadic examples of Problem g=0