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\(ABC\) implies a Zsigmondy principle for ramification. (English) Zbl 1407.37128
The authors derive various results in arithmetic dynamics, conditional on the \(abc\) conjecture. Let \(k\) be a number field or function field of characteristic zero. Let \(\phi\) be a non-isotrivial polynomial over \(k\) of degree at least \(2\) that is not postcritically finite. Then for any \(\beta\in\mathbb{P}^1(k)\), and for all large enough \(n\), there is a prime of \(k\) that is ramified in \(k(\phi^{-n}(\beta))\) but not in \(k(\phi^{-m}(\beta))\) for any \(m<n\). In other words, eventually, the field \(k(\phi^{-n}(\beta))\) is always ramified at some new prime.
The authors prove the analogous result for rational functions, under some additional mild genericity assumptions on \(\beta\) and \(\phi\). They can also remove those assumptions for general rational functions, at the expense of getting new ramified primes every two steps instead of every one.
These results imply the exponential growth of the degree of the fields of definition of the preimages of points of \(k\).

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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