JournalscmhVol. 89, No. 1pp. 173–213

Galois theory of quadratic rational functions

  • Rafe Jones

    Carleton College, Northfield, USA
  • Michelle Manes

    University of Hawaii, Honolulu, USA
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Abstract

For a number field KK with absolute Galois group GKG_K, we consider the action of GKG_K on the infinite tree of preimages of αK\alpha \in K under a degree-two rational function ϕK(x)\phi \in K(x), with particular attention to the case when ϕ\phi commutes with a non-trivial Möbius transformation. In a sense this is a dynamical systems analogue to the \ell-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 ϕ\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 ϕ\phi. In the case where ϕ\phi commutes with a non-trivial Möbius transformation, 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.

Cite this article

Rafe Jones, Michelle Manes, Galois theory of quadratic rational functions. Comment. Math. Helv. 89 (2014), no. 1, pp. 173–213

DOI 10.4171/CMH/316