Heights and totally $p$-adic numbers
Tom 171 / 2015
Acta Arithmetica 171 (2015), 277-291
MSC: Primary 37P30, 11S82; Secondary 11R04.
DOI: 10.4064/aa171-3-5
Streszczenie
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.
