Schur问题的有理函数模拟与置换表示的例外性
The rational function analogue of a question of Schur and exceptionality of permutation representations
摘要：
舒尔在1923年考虑了以下问题。设f（X）是一个整系数多项式，它对无穷多个素数p在剩余域Z/pZ上诱导一个双射。M. 1970年炸的。这里我们研究有理函数的类似问题，也允许基域是任意数域。结果，有更多的有理函数具有类似的性质。某些无穷级数来自于椭圆曲线的有理同构和形变。有几个零星的例子不适合我们得到的任何系列。首先将算术性质转化为有限置换群的问题，并对满足必要条件的群进行分类。证明依赖于有限单群的分类。然后，我们使用算术论证来排除许多情况，或者证明剩余的情况确实产生了例子。这一部分是基于Mazur关于模曲线X_0（p）和X_1（p）上有理点的经典结果、由Q的绝对Galois群作用于椭圆曲线p-扭点而得到的GL_2（p）中的Galois像的结果、复乘理论以及正则Galois反问题中的技巧。
英文摘要：
In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ 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, 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. Some infinite series come from rational isogenies of elliptic curves and deformations. There are several sporadic examples which do not fit in any of the series we obtain. First we translate the arithmetic property to a question about finite permutation groups, and classify those groups which fulfill the necessary conditions. The proofs depend on the classification of the finite simple groups. Then we use arithmetic arguments to either rule out many cases, or to prove that the remaining cases indeed give rise to examples. This part is based on Mazur's classical results about rational points on the modular curves X_0(p) and X_1(p), results about Galois images in GL_2(p) coming from action of the absolute Galois group of Q on p-torsion points of elliptic curves, the theory of complex multiplication, and the techniques used in the inverse regular Galois problem.
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