Galois theory of quadratic rational functions
By Rafe Jones, Michelle Manes. 2011
49 citations recorded by the source corpus.
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.
Subfields: Computational Mechanics, Computational Theory and Mathematics, Theoretical Computer Science
View Paper: https://www.semanticscholar.org/paper/e66b8c8cbcd7f3126649b22b6c7ede2bd0ad6db8
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In Computational Mechanics
In Computational Theory and Mathematics
- Joseph H. Silverman
- Robert L. Benedetto
- Rafe Jones
- T. Tucker
- Laura Demarco
- Matthew Baker
- Samir Siksek
- Sergei V. Kozyrev
In Theoretical Computer Science
Citations in the corpus
- Current trends and open problems in arithmetic dynamics (2018, background): “Over number fields K, it is known that the Odoni index ιK(f, P ) is finite for certain families of quadratic polynomials [123,227], as well as for various quadratic rational functions [128] and higher-degree polynomials [157], and there are examples over global function fields [111].”