Galois theory of quadratic rational functions
Galois theory of quadratic rational functions
复制标题
二次有理函数的伽罗瓦理论
DOI：
10.4171/cmh/316
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发表时间：
2011
影响因子：
0.9
通讯作者：
M. Manes
中科院分区：
文献类型：
--
作者：
Rafe Jones;M. Manes
对于具有绝对伽罗瓦群G_K的数域K，在二阶有理函数下考虑G_K对K中点的无限原像树的作用，特别注意与非平凡莫比乌斯变换交换时的情况。从某种意义上说，这是一个动力系统类似于椭圆曲线上的l进伽罗瓦表示，特别注意CM的情况。利用关于迭代的分子的判别式的一个结果，给出了动作像尽可能大的判别式。这个准则是用两个临界点的正向轨道的算术来表示的。在与非平凡莫比乌斯变换交换的情况下，实际上只有一个临界轨道，我们给出了极大值准则的修改版本。在后一种情况下，我们证明了一个serre型有限指数结果。
For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of phi. In the case where phi commutes with a non-trivial Mobius transfomation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.