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Heights and totally $p$-adic numbers

Volume 171 / 2015

Lukas Pottmeyer Acta Arithmetica 171 (2015), 277-291 MSC: Primary 37P30, 11S82; Secondary 11R04. DOI: 10.4064/aa171-3-5

Abstract

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.

Authors

  • Lukas PottmeyerFachbereich Mathematik
    Universit\"at Basel
    Spiegelgasse 1
    4051 Basel, Switzerland
    e-mail

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