Kronecker conjugacy of polynomials

Müller, Peter

doi:10.1090/s0002-9947-98-02123-0

Cited by 9 publications

(2 citation statements)

References 29 publications

(29 reference statements)

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“…[Fr99, §8], [Mü98a] and [Mu95, §2.7] say much on the group theory of the indecomposable polynomial SDPs over number fields. Yet, we now say something new on the definition field of these families, a subtlety on dessins d'enfant, presented as genus 0 j-line covers.…”

Section: Exceptional Correspondences With Pmentioning

confidence: 99%

The place of exceptional covers among all diophantine relations

Fried

2005

Finite Fields and Their Applications

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Let F q be the order q finite field. An F q cover : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, is exceptional if it maps one-one on F q t points for ∞-ly many t over this locus. Lenstra suggested a curve Y may have an Exceptional (cover)Tower over F q Lenstra Jr. [Talk at Glasgow Conference, Finite Fields III, 1995]. We construct it, and its canonical limit group and permutation representation, in general. We know all onevariable tamely ramified rational function exceptional covers, and much on wildly ramified one variable polynomial exceptional covers, from Fried et al. [Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993) 157-225], Guralnick et al. [The rational function analogue of a question of Schur and exceptionality of permutations representations, Mem. Amer. Math. Soc. 162 (2003) 773, ISBN 0065-9266] and Lidl et al. [Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics , vol. 65, Longman Scientific, New York, 1993]. We use exceptional towers to form subtowers from any exceptional cover collections. This gives us a language for separating known results from unsolved problems.We generalize exceptionality to p(ossibly)r(educible)-exceptional covers by dropping irreducibility of X. Davenport pairs (DPs) are significantly different covers of Y with the same ranges (where maps are nonsingular) on F q t points for ∞-ly many t. If the range values have the same multiplicities, we have an iDP. We show how a pr-exceptional correspondence on F q covers characterizes a DP.You recognize exceptional covers and iDPs from their extension of constants series. Our topics include some of their dramatic effects • How they produce universal relations between Poincaré series. • How they relate to the Guralnick-Thompson genus 0 problem and to Serre's open image theorem. Historical sections capture Davenport's late 1960s desire to deepen ties between exceptional covers, their related cryptology, and the Weil conjectures.

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Section: Exceptional Correspondences With Pmentioning

confidence: 99%

The place of exceptional covers among all diophantine relations

Fried

2005

Finite Fields and Their Applications

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“…Besides the alternating, symmetric, cyclic and dihedral groups only finitely many cases arise. For applications of these results see [15,26,47,49].…”

Section: Introductionmentioning

confidence: 97%

Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials

Müller¹

2013

Annali Scuola Normale Superiore - Classe Di Scienze

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We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities for these monodromy groups, and we give applications to Hilbert's irreducibility theorem.

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Relating Two Genus 0 Problems of John Thompson

Fried¹

Developments in Mathematics

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Abstract. 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 θ-functions uniformizing certain genus 0 (near) modular curves. We call this Problem g=01 . We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=0 2 . Second: From finite collections of genus 0 curves with aspects of Problem g=01 .

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Kronecker conjugacy of polynomials

Cited by 9 publications

References 29 publications

The place of exceptional covers among all diophantine relations

The place of exceptional covers among all diophantine relations

Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials

Relating Two Genus 0 Problems of John Thompson

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