Exceptional polynomials and monodromy groups in positive characteristic [Elektronische Ressource] / vorgelegt von Florian Möller

Florian Möller

Sujets

• Mathematics

Informations

• Publié par: julius-maximilians-universitat_wurzburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 46
• Langue: Deutsch

Extrait

Exceptional polynomials and monodromy groups in positive characteristic

Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades derJulius-Maximilians-Universit¨atWu¨rzburg

vorgelegt von

FlorianM¨oller

Institutf¨urMathematikderUniversita¨tWu¨rzburg

Wu¨rzburg,Ma¨rz2009

ii

Danksagung Ichm¨ochtemichherzlichbeiProfessorPeterM¨ullerf¨urseineUnterstu¨tzunginallenPhasen meiner Promotion bedanken. AlsBetreuermeinerDissertationlasteteaufHerrnMu¨llerdieAufgabe,mireinerseitszu zeigenwiesch¨onfreieundeigensta¨ndigeForschungsta¨tigkeitseinkann,mirandererseitsaber auch Probleme vorzulegen, die mich in meiner Promotion voranbrachten. Dies ist Herrn M¨ullerausgezeichnetgelungen. Besonderserwa¨hnenm¨ochteichnoch,dassProfessorMu¨llermehrerekompliziertemath- ematische Situationen, mit denen ich mich konfrontiert sah, stark vereinfachte. Erst durch seine Hilfe sind mir einige Beweise gelungen oder in ihre jetzige lesbare(re) Form gebracht worden.

WeiterhindankeichProfessorTheoGrundho¨fer,derfu¨rmeineFragenimmereinoffenesOhr hatte und mir an mancher Stelle mit wertvollen Literaturhinweisen half. Weiter habe ich ihmeinezusa¨tzlicheAnstellungamLehrstuhlf¨urMathematikIIIzuverdanken.Ohnedie Unterst¨utzungvonHerrnGrundho¨ferw¨aremirmeinePromotionerheblichschwerergefallen.

MeinbesondererDankgiltmeinerFamilie,diemichinmeinemVorhabenstetsbest¨arkteund mireinenAusgleichzurmathematischenBescha¨ftigungbot.

iii

iv

Contents

1. Introduction

1

I. General Results 5 2. Monodromy groups and affine polynomials 7 2.1. Polynomials and monodromy groups . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Affine groups and affine polynomials . . . . . . . . . . . . . . . . . . . . . . . 8 2.3. Generic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. Calculation of differents and the genus-0 condition 13 3.1. The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2. Applications to the case of affine Galois groups . . . . . . . . . . . . . . . . . 18 3.2.1. Consequences of Hasse-Arf and Herbrand . . . . . . . . . . . . . . . . 18 3.2.2. Bounds for the number of fixed points for elements ofG 20. . . . . . .. 3.2.3. Bounds for ind(∞. . . . . . . . . . . . . . . . . . . . . . . 20) . . . . . . 3.2.4. Bounds for ind(p) withpa finite place. . . . . . . . . . . . . . . . . .  21

II. Application to specific problems 23 4. Affine polynomials withg≤225 4.1.g= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.1. Case (A):H ∼ = C m  .is cyclic 26. . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Case (B):His an elementary abelianp-group . . . . . . . . . . . . . . 28 4.1.3. Case (C):H ∼ = (C p × ∙ ∙ ∙ ×C p )oC m  28 . . . . .is a semidirect product 4.1.4. Case (D):H. . . . . . . . . 28is dihedral in even characteristic . . . . . 4.1.5. Case (E):H. . . . . . . . . . . 28 . . . is dihedral in odd characteristic 4.1.6. Case (F):H ∼ = A 5 in characteristic 3 . . . . . . . . . . . . . . . . . . . 28 4.1.7. Case (G):H = ∼ PSL(2 q) withp6. . . . . . . . . . . . . . . 29 . . . . = 2 4.1.8. Case (H):H ∼ = PGL(2 q) withp6 37= 2 . . . . . . . . . . . . . . . . . . . 4.1.9. Case (I):H = ∼ PGL(22 m  40. . . . . . . . . . . . . . . . . . . . . .) . . . 4.2.g= 1 . 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Ramification inE|K(t . . . . . ) . 41. . . . . . . . . . . . . . . . . . . . 4.2.2. Cases in odd characteristicp >2 . . . . . . . . . . . . . . . . . . . . . 43 4.2.3. Cases in even characteristicp 43. . . . . . . . . . . . . . . .= 2 . . . . . 4.3.g 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = 2 . 4.3.1. Cases in odd characteristicp >2 . 47 . . . . . . . . . . . . . . . . . . . . 4.3.2. Cases in even characteristicp . . . . = 2 . 51. . . . . . . . . . . . . . . . 5.AGLas a monodromy group 53

v

Contents

6.

7.

8.

vi

Affine polynomials of degreep 2 6.1. Cases in odd characteristicp6 . . . . . . . . . .= 2 6.2. Cases in even characteristicp .= 2. . . . . . . . .

Exceptional polynomials of degreep 3 7.1. The cases in characteristicp >3 . . . . . . . . . . 7.2. The cases in characteristicp∈ {23}. . . . . . . .

Exceptional polynomials of degreep r ,ran odd prime,

. . . . . .

. . . . . .

with

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

2-transitive

. . . . . . . . . .

. . . . . . . . . .

groupA

61 62 63

65 65 68

69

1. Introduction

In 1923 Schur [43] requested a description of all polynomialsf∈Z[X] that induce a bijection onZ/pZfor infinitely many primesp. He proved that iffis of prime degree, then it is – up to linear changes over the algebraic closure ofQ– either a cyclic polynomialX q or a Chebychev polynomialT q (X) (defined implicitly byT q (X+ 1/X) =X q + X 1 q ). He conjectured that in the general casef  Thiswas proved by Fried [13] inis a composition of such polynomials. 1970. Schur’s original question has been generalized in several ways: Turnwald [48] discusses theproblemoverintegraldomains,Guralnick,Mu¨ller,andSaxl[19]characterizerational functionsrover number fieldsKso thatrinduces a bijection on the residue fieldK p for infinitely many placesp∈ P(K). There is a different generalization of interest to us: we assume the base field to be of positive characteristic and search forexceptional polynomials, i.e. polynomials that fulfill the following Definition (Exceptionality)Assumek Letis a finite field.f∈k[X]be a polynomial with coefficients ink. IfKis an extension field ofk, thenfis called apermutation polynomial overKiffinduces a bijection onK. f∈k[X]is calledexceptional overkif it is a permutation polynomial over infinitely many finite extensions ofk. The classification of permutation polynomials has a long tradition and goes back to Her- mite [24] who noticed that any functionk→kwithka finite field can be represented by a polynomial. A broad survey of permutation polynomials can be found in Lidl and Niederreiter [35]. An important step forward concerning the theory of exceptional polynomials was the refor- mulation of the original definition in terms of Galois theory and covering theory in positive characteristic. A first consequence was the proof of the following Exceptionality Lemma (cf. [14])Letkbe a finite field of characteristicpandta tran- scendental overk. Letf∈k[X]be a polynomial. Denote the set of zeros off(X)−tbyZ. Fix a zerox∈Z. Then: (1) Supposefis not ap-th power ink[X]. Thenfis exceptional overkif and only if the xof the arithmetic monodromy group of-stabilizer ffixes no orbit of thex-stabilizer of the geometric monodromy group offonZ\ {x} definition of monodromy groups. (A is given on page 7.) (2) Supposefis a compositionf=f 1 ◦f 2 withf 1  f 2 ∈k[X]. Thenfis exceptional over kif and only if bothf 1 andf 2 are exceptional overk. In 1993 Fried, Guralnick, and Saxl [14] realized that this result reduces the classification of exceptional polynomials essentially to a question about primitive groups. They used the

1