\(ABC\) implies primitive prime divisors in arithmetic dynamics.

*(English)*Zbl 1291.37121This 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.

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.

Reviewer: Yu Yasufuku (Tokyo)