Periods of rational maps modulo primes

2011/07/14 by Benedetto, Robert L., Ghioca, Dragos, Hutz, Benjamin +3 · 1 citation
#37P55 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 14G25 #Secondary: 37F10

paper · doi:10.48550/arxiv.1107.2816

Abstract

Let K be a number field, let ϕ∈ K(t) be a rational map of degree at least 2, and let α, β∈ 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 α\mod \mathfrak p is not in the forward orbit of β\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 ϕ: \mathbb Pⁿ → \mathbb Pⁿ.

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• Preperiodic points, finiteness, and structures of semigroups of algebraic morphisms

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