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

Robert M. Guralnick University of Southern California, Los Angeles, CA
Peter Müller University of Heidelberg, Heidelberg, Germany
Jan Saxl , Cambridge, England
Available Formats:
Electronic ISBN: 978-1-4704-0371-3
Product Code: MEMO/162/773.E
List Price: $57.00 MAA Member Price:$51.30
AMS Member Price: $34.20 Click above image for expanded view The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations Robert M. Guralnick University of Southern California, Los Angeles, CA Peter Müller University of Heidelberg, Heidelberg, Germany Jan Saxl , Cambridge, England Available Formats:  Electronic ISBN: 978-1-4704-0371-3 Product Code: MEMO/162/773.E  List Price:$57.00 MAA Member Price: $51.30 AMS Member Price:$34.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1622003; 79 pp
MSC: Primary 12; 20; Secondary 11; 14;

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.

Graduate students and research mathematicians interested in algebra.

• Chapters
• 1. Introduction
• 2. Arithmetic-geometric preparation
• 3. Group theoretic exceptionality
• 4. Genus 0 condition
• 5. Dickson polynomials and Rédei functions
• 6. Rational functions with Euclidean signature
• 7. Sporadic cases of arithmetic exceptionality
• Requests

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Volume: 1622003; 79 pp
MSC: Primary 12; 20; Secondary 11; 14;

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.

Graduate students and research mathematicians interested in algebra.

• Chapters
• 1. Introduction
• 2. Arithmetic-geometric preparation
• 3. Group theoretic exceptionality
• 4. Genus 0 condition
• 5. Dickson polynomials and Rédei functions
• 6. Rational functions with Euclidean signature
• 7. Sporadic cases of arithmetic exceptionality
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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