Periods of rational maps modulo primes

By Robert L. Benedetto, Dragos Ghioca, Benjamin Hutz, Pär Kurlberg, Thomas Scanlon, T. Tucker. 2011

53 citations recorded by the source corpus.

Let K be a number field, let $${\varphi \in K(t)}$$ be a rational map of degree at least 2, and let $${\alpha, \beta \in K}$$ . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes $${\mathfrak{p}}$$ of K such that $${\alpha {\rm mod} \mathfrak{p}}$$ is not in the forward orbit of $${\beta {\rm mod} \mathfrak{p}}$$ . Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace α by a hypersurface, such as the ramification locus of a morphism $${\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}}$$ .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/43143b680bc5c00ef7bdd9589d7a0a8ea4860abc

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Citations in the corpus

The Dynamical Mordell-lang Conjecture (2007, methodology): “Heuristically it is even expected that this method might never work for endomorphisms of P if N ≥ 5, [4].”
Linear relations between polynomial orbits (2008, unknown): “A G ] 2 3 Ju l 2 00 8”