Periods of rational maps modulo primes

Abstract

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 p of K such that α mod p is not in the forward orbit of β mod 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 : P n → P n .

A large arboreal Galois representation for a cubic postcritically finite polynomial

“…Page 11 of 21 Write f −1 (x) = {u 1 , u 2 , u 3 }, and for each i = 1, 2, 3, write 1 : K ] = 6. By Cauchy's Theorem, there is some τ 1 ∈ Gal(K 2 /K 1 ) of order 2.…”

“…Clearly, the set of primes P for which w i ≡ ∞ (mod P) is zero, since f is a polynomial, and so the proof is complete. 1 Amherst College, Amherst, MA, USA, 2 Center for Computing Sciences, Institute for Defense Analyses, Bowie, MD, USA, 3 Saint Louis University, Saint Louis, MO, USA, 4 College of Science and Technology, Nihon University, Tokyo, Japan.…”

p-adic interpolation of iterates

“…Inspired by the work of Skolem [12], Mahler [9], and Lech [8] on linear recursive sequences, Bell [2] proved that for a suitable p-adic analytic function f and starting point x, the iteratecomputing map n → f n (x) extends to a p-adic analytic function g(n) defined for n ∈ Z p . This result, along with its generalization by Bell, Ghioca, and Tucker [3, § 3] and earlier linearization results by Herman and Yoccoz [7, Theorem 1] and Rivera-Letelier [11, § 3.2], has significance beyond its intrinsic interest, because of its applications towards the dynamical Mordell-Lang conjecture [1,[3][4][5][6].…”

Evidence for the Dynamical Brauer–Manin Criterion

“…For instance, by a result of Pink (Pink, 2004, Corollary 4.3), such points cannot exist for the multiplication-by-d map on an abelian variety. Furthermore, by (Benedetto et al, 2013, Corollary 1.2), such points also cannot exist for a self-map of P 1 of degree at least two (though such a map is notétale, and thus 4.16 does not apply directly).…”