# The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations

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*Robert M. Guralnick; Peter Müller; Jan Saxl*

In 1923 Schur considered the problem of which polynomials
\(f\in\mathbb{Z}[X]\) induce bijections on the residue fields
\(\mathbb{Z}/p\mathbb{Z}\) for infinitely many primes \(p\). His
conjecture, that such polynomials are compositions of linear and Dickson
polynomials, was proved by M. Fried in 1970. Here we investigate the analogous
question for rational functions, and also we allow the base field to be any
number field. As a result, there are many more rational functions for which the
analogous property holds. The new infinite series come from rational isogenies
or endomorphisms of elliptic curves. Besides them, there are finitely many
sporadic examples which do not fit in any of the series we obtain.

The Galois theoretic translation, based on Chebotarëv's density theorem,
leads to a certain property of permutation groups, called exceptionality. One
can reduce to primitive exceptional groups. While it is impossible to describe
explicitly all primitive exceptional permutation groups, we provide certain
reduction results, and obtain a classification in the almost simple case.

The fact that these permutation groups arise as monodromy groups of covers
of Riemann spheres \(f:\mathbb{P}^1\to\mathbb{P}^1\), where \(f\) is the rational
function we investigate, provides genus \(0\) systems. These are
generating systems of permutation groups with a certain combinatorial property.
This condition, combined with the classification and reduction results of
exceptional permutation groups, eventually gives a precise geometric
classification of possible candidates of rational functions which satisfy the
arithmetic property from above. Up to this point, we make frequent use of the
classification of the finite simple groups.

Except for finitely many cases, these remaining candidates are connected to
isogenies or endomorphisms of elliptic curves. Thus we use results about
elliptic curves, modular curves, complex multiplication, and the techniques
used in the inverse regular Galois problem to settle these finer arithmetic
questions.

#### Table of Contents

# Table of Contents

## The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Arithmetic-Geometric Preparation 615 free
- 2.1. Arithmetic and geometric monodromy groups 615
- 2.2. Distinguished conjugacy classes of inertia generators 615
- 2.3. Branch cycle descriptions 615
- 2.4. The branch cycle argument 716
- 2.5. Weak rigidity 817
- 2.6. Topological interpretation 918
- 2.7. Group theoretic translation of arithmetic exceptionality 1019
- 2.8. Remark about exceptional functions over finite fields 1019

- Chapter 3. Group Theoretic Exceptionality 1221
- Chapter 4. Genus 0 Condition 3645
- Chapter 5. Dickson Polynomials and Rédei Functions 5160
- Chapter 6. Rational Functions with Euclidean Signature 5362
- Chapter 7. Sporadic Cases of Arithmetic Exceptionality 7079
- 7.1. G = C[sub(2)] x C[sub(2)] (Theorem 4.13(a)(iii)) 7079
- 7.2. G = (C[sup(2)][sub(11)]) x GL[sub(2)(3) (Theorem 4.13(c)(1)) 7180
- 7.3. G = (C[sup(2)][sub(11)]) x S[sub(3)] (Theorem 4.13(c)(ii)) 7281
- 7.4. G = (C[sup(2)][sub(5)]) x ((C[sub(4)] x C[sub(2)]) x C[sub(2)]) (Theorem 4.13(c)(iii)) 7382
- 7.5. G = (C[sup(2)][sub(5)]) x D[sub(12)] (Theorem 4.13(c)(iv)) 7382
- 7.6. G = (C[sup(2)][sub(3)]) x D[sub(8)] (Theorem 4.13(c)(v)) 7382
- 7.7. G = (C[sup(4)][sub(2)]) x (C[sup(5)] x C[sub(2)]) (Theorem 4.13(c)(vi)) 7382
- 7.8. G = PSL[sub(2)](8) (Theorem 4.10(a)) 7382
- 7.9. G = PSL[sub(2)](9) (Theorem 4.10(b)) 7584
- 7.10. A remark about one of the sporadic cases 7584

- Bibliography 7786