How Long to Read The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations

How Long Does it Take to Read The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations?

It takes the average reader 1 hour and 21 minutes to read The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations by Robert M. Guralnick

Assuming a reading speed of 250 words per minute. Learn more

Description

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 Chebotarev'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.

How long is The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations?

The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations by Robert M. Guralnick is 79 pages long, and a total of 20,461 words.

This makes it 27% the length of the average book. It also has 25% more words than the average book.

How Long Does it Take to Read The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations Aloud?

The average oral reading speed is 183 words per minute. This means it takes 1 hour and 51 minutes to read The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations aloud.

What Reading Level is The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations?

The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations is suitable for students ages 10 and up.

Note that there may be other factors that effect this rating besides length that are not factored in on this page. This may include things like complex language or sensitive topics not suitable for students of certain ages.

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