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$$ABC$$ implies primitive prime divisors in arithmetic dynamics. (English) Zbl 1291.37121
This article proves the existence of primitive prime divisors in orbits of rational functions. Over a number field $$K$$, the proof assumes the $$abc$$ conjecture, while the proof is unconditional when $$K$$ is a characteristic $$0$$ function field of transcendence degree $$1$$. The result does not assume that $$0$$ is a preperiodic point or a ramification point, unlike many of the previous results.
More specifically, let $$\varphi$$ be a rational function of degree $$d>1$$, and denote its $$n$$-th iterate by $$\varphi^n$$. We say that a prime $$\mathfrak p$$ of $$K$$ is a primitive prime of $$\varphi^n(\alpha)-\beta$$ if $$v_{\mathfrak p}(\varphi^n(\alpha) - \beta) > 0$$ and $$v_{\mathfrak p}(\varphi^m(\alpha) - \beta) \leq 0$$ for all $$m<n$$. We say that $$\mathfrak p$$ is a square-free primitive prime if further $$v_{\mathfrak p}(\varphi^n(\alpha) - \beta) = 1$$. Theorem 1.1 shows that there is a primitive prime of $$\varphi^n(\alpha) - \beta$$ for all sufficiently large $$n$$ if the following all hold: (1) $$\alpha$$ is not preperiodic, (2) $$\varphi$$ is non-isotrivial if $$K$$ is a function field, (3) $$\beta$$ is not in the orbit of $$\alpha$$, (4) $$(\varphi^2)^{-1}(\beta) \neq \{\beta\}$$. Furthermore, Theorem 1.2 shows that $$\varphi^n(\alpha) - \beta$$ has a square-free primitive prime for all sufficiently large $$n$$ if (4) above is replaced by the stronger (4’): there exists an infinite sequence $$\{\beta_n\}$$ of noncritical points such that $$\cdots \mapsto \beta_n \mapsto \cdots \mapsto \beta_1 \mapsto \beta$$.
The key of the proof is to show that given $$\epsilon>0$$ and a polynomial $$F$$ without multiple roots, there exists a constant $$C$$ such that $\sum_{\mathfrak p: v_{\mathfrak p}(F(z)) >0} \lambda_{\mathfrak p}^{(1)}(F(z)) \geq (\deg F - 2 - \epsilon) h(z)+C$ for all $$z\in K$$, where $$\lambda_{\mathfrak p}^{(1)}$$ is a local height (with respect to $$0$$) truncated at $$1$$. Over number fields (Proposition 3.4), this is based on [A. Granville, Int. Math. Res. Not. 1998, No. 19, 991–1009 (1998; Zbl 0924.11018)], and uses the $$abc$$ conjecture as well as the Belyi map. Over function fields (Proposition 4.2), because of the absence of Belyi maps, the authors instead use Vojta’s $$1+\epsilon$$ conjecture, proved by K. Yamanoi [Acta Math. 192, No. 2, 225–294 (2004; Zbl 1203.30035)].
The dynamical input comes in Proposition 5.1, which shows that the sum of $$\lambda_{\mathfrak p}^{(1)}(\varphi^n(\alpha))$$ over good-reduction primes which divide $$\varphi^m(\alpha)$$ for some $$m<n$$ grow only $$o(h(\varphi^n(\alpha)))$$. By choosing $$F$$ to be a suitable factor of some iterate of $$\varphi$$, the main results follow.
Building on [M. Stoll, Arch. Math. 59, No. 3, 239–244 (1992; Zbl 0758.11045)], the authors apply Theorem 1.2 to show that when $$a\in \mathbb Z$$ is such that $$-a$$ is not $$2$$ or a perfect square, $$f(x) = x^2 + a$$ has the property that the splitting field of $$f^{n+1}$$ is a degree-$$2^{2^n}$$ extension of the splitting field of $$f^n$$ for all sufficiently large $$n$$ (Proposition 6.1). This has also been proved in [W. Hindes, Acta Arith. 159, No. 2, 149–167 (2013; Zbl 1296.14017)] by a different method, also assuming the $$abc$$ conjecture.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 11G50 Heights 14G25 Global ground fields in algebraic geometry
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